Astrophysics
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Astrophysics Overview
- Table of contents for Stan Owocki’s Fundamentals of Astrophysics.
- Organized into five major parts: stellar properties, stellar structure and evolution, interstellar matter and planet formation, galaxies, and cosmology.
- Covers methods for inferring stellar distances, luminosities, temperatures, radii, masses, rotation, ages, and velocities.
- Introduces broader astrophysical topics including the Sun, star formation, exoplanets, the Milky Way, external galaxies, dark matter, and active galactic nuclei.
- Concludes with cosmology topics such as universal expansion, dark energy, the Big Bang, cosmic microwave background, nucleosynthesis, and inflation, plus appendices on atomic physics and radiative transfer.
Fundamentals of Astrophysics
Stan Owocki
Contents
Part I Stellar Properties page 1
1 Introduction 3
1.1 Observational vs. Physical Properties of Stars 3
1.2 Scales and Orders of Magnitude 5
1.3 Questions and Exercises 9
2 Inferring Astronomical Distances 10
2.1 Angular size 10
2.2 Trignonometric parallax 12
2.3 Determining the Astronomical Unit (au) 15
2.4 Solid angle 15
2.5 Questions and Exercises 16
3 Inferring Stellar Luminosity 18
3.1 \Standard Candle" methods for distance 18
3.2 Intensity or Surface Brightness 19
3.3 Apparent and absolute magnitude and the distance modulus 20
3.4 Questions and Exercises 21
4 Inferring Surface Temperature from a Star's Color and/or Spectrum 23
4.1 The wave nature of light 24
4.2 Light quanta and the Black-Body emission spectrum 24
4.3 Inverse-temperature dependence of wavelength for peak
ux 26
4.4 Inferring stellar temperatures from photometric colors 26
4.5 Questions and Exercises 27
5 Inferring Stellar Radius from Luminosity and Temperature 29
5.1 Stefan-Boltzmann law for surface
ux from a blackbody 29
5.2 Questions and Exercises 30
6 Absorption Lines in Stellar Spectra 31
6.1 Elemental composition of the Sun and stars 33
4 Contents
6.2 Stellar spectral type: ionization abundances as temperature
diagnostic 34
6.3 Hertzsprung-Russell (H-R) diagram 35
6.4 Questions and Exercises 36
7 Surface Gravity and Escape/Orbital Speed 37
7.1 Newton's law of gravitation and stellar surface gravity 37
7.2 Surface escape speed Vesc 38
7.3 Speed for circular orbit 39
7.4 Virial Theorum for bound orbits 39
7.5 Questions and Exercises 40
8 Stellar Ages and Lifetimes 42
8.1 Shortness of chemical burning timescale for Sun and stars 42
8.2 Kelvin-Helmholtz timescale for gravitational contraction 42
8.3 Nuclear burning timescale 43
8.4 Age of stellar clusters from main-sequence turno point 44
8.5 Questions and Exercises 45
9 Inferring Stellar Space Velocities 47
9.1 Transverse speed from proper motion observations 47
9.2 Radial velocity from Doppler shift 49
9.3 Questions and Exercises 50
10 Using Binary Systems to Determine Masses and Radii 51
10.1 Visual binaries 51
10.2 Spectroscopic binaries 53
10.3 Eclipsing binaries 55
10.4 Mass-Luminosity scaling from astrometric and eclipsing binaries 56
10.5 Questions and Exercises 57
11 Inferring Stellar Rotation 59
11.1 Rotational broadening of stellar spectral lines 59
11.2 Rotational period from starspot modulation of brightness 61
11.3 Questions and Exercises 62
12 Light Intensity and Absorption 63
12.1 Intensity vs. Flux 63
12.2 Absorption mean-free-path and optical depth 65
12.3 Inter-stellar extinction and reddening 67
12.4 Questions and Exercises 68
13 Observational Methods 69
13.1 Telescopes as light buckets 69
Contents 5
13.2 Angular resolution 70
13.3 Space-based missions 72
13.4 Questions and Exercises 73
14 Our Sun 74
14.1 Imaging the solar disk 74
14.2 Corona and solar wind 76
14.3 Convection as a driver of solar structure and activity 78
14.4 Questions and Exercises 80
Part II Stellar Structure & Evolution 81
15 Hydrostatic Balance between Pressure and Gravity 83
15.1 Hydrostatic equilibrium 83
15.2 Pressure scale height and thinness of surface layer 85
15.3 Hydrostatic balance in stellar interior and the virial temperature 86
15.4 Questions and Exercises 87
16 Transport of Radiation from Interior to Surface 88
16.1 Random walk of photon diusion from stellar core to surface 88
16.2 Diusion approximation at depth 90
16.3 Atmospheric variation of temperature with optical depth 91
16.4 Questions and Exercises 91
17 Structure of Radiative vs. Convective Stellar Envelopes 92
17.1LM3relation for hydrostatic, radiative stellar envelopes 92
17.2 Horizontal-track Kelvin-Helmholtz contraction to the main sequence 93
17.3 Convective instability and energy transport 94
17.4 Fully convective stars { the Hayashi track for proto-stellar
contraction 96
18 Hydrogen Fusion and the Mass Range of Stars 98
18.1 Core temperature for H-fusion 99
18.2 Main sequence scalings for radius-mass and luminosity-temperature 100
18.3 Lower mass limit for hydrogen fusion: Brown Dwarf stars 101
18.4 Upper mass limit for stars: the Eddington Limit 102
19 Post-Main-Sequence Evolution: Low-Mass Stars 104
19.1 Core-Hydrogen burning and evolution to the Red Giant branch 105
19.2 Helium
ash and core-Helium burning on the Horizontal Branch 106
19.3 Asymptotic Giant Branch to Planetary Nebula to White Dwarf 108
19.4 White Dwarf stars 108
19.5 Chandasekhar limit for white-dwarf mass: M < 1:4M 109
6 Contents
20 Post-Main-Sequence Evolution: High-Mass Stars 111
20.1 Multiple shell burning and horizontal loops in H-R diagram 111
20.2 Core-collapse supernovae 112
20.3 Neutron stars 114
20.4 Black Holes 114
20.5 Observations of stellar remnants 116
20.6 Gravitational Waves from Merging Black Holes or Neutron Stars 118
20.7 Questions and Exercises 121
Part III Interstellar Medium & Formation of Stars and Planets 123
21 The Interstellar Medium 125
21.1 Star-gas cycle 125
21.2 Cold-Warm-Hot phases of nearly isobaric ISM 126
21.3 Molecules and dust in cold ISM: Giant Molecular Clouds 129
21.4 HII regions 132
21.5 Galactic organization of ISM and star-gas interaction along spiral
arms 134
22 Star Formation 136
22.1 Jeans Criterion for gravitational contraction 136
22.2 Cooling by molecular emission 137
22.3 Free-fall timescale and the galactic star formation rate 138
22.4 Fragmentation into cold cores and the Initial Mass Function (IMF) 139
22.5 Angular momentum conservation of rotating cores and disk
formation 140
22.6 Questions and Exercises 142
23 Origin of Planetary Systems 144
23.1 The Nebular Model 144
23.2 Observations of Protoplanetary Disks 145
23.3 Our Solar System 146
23.4 The Ice Line: Gas Giants vs. Rocky Dwarfs 147
23.5 Equilibrium Temperature 148
23.6 Questions and Exercises 148
24 Water Planet Earth 149
24.1 Formation of Moon by Giant Impact 149
24.2 Water from Icy Asteroids 150
24.3 Our Magnetic Shield 151
24.4 Life from Oceans: Earth vs. Icy Moons 151
24.5 Questions and Exercises 152
25 Extra-Solar Planets 153
Contents 7
25.1 Direct Imaging Method 153
25.2 Radial Velocity Method 154
25.3 Transit Method 155
25.4 The Exoplanet Census: 4000+ and counting 157
25.5 Search for Earth-sized Planets in the Habitable Zone 158
25.6 Questions and Exercises 159
Part IV Our Milky Way & Other Galaxies 161
26 Our Milky Way Galaxy 163
26.1 Disk, halo, and bulge components of the Milky Way 163
26.2 Virial mass for cluster from stellar velocity dispersion inferred
from Doppler shifts 166
26.3 Galactic rotation curve & dark matter 168
26.4 Super-massive black hole at the galactic center 171
27 External Galaxies 174
27.1 Cepheid variables as standard candle for distances to external
galaxies 174
27.2 Galactic redshift and Hubble's law for expansion 175
27.3 Tully-Fisher Relation: Lgal/V4
rot 177
27.4 Spiral, Elliptical, & Irregular galaxies 179
27.5 Role of Galaxy Collisions 181
28 Active Galactic Nuclei (AGNs) and Quasars 182
28.1 Basic properties and model 182
28.2 Lyman alpha clouds 183
28.3 Gravitational lensing of quasar light by foreground Galaxy Clusters 185
28.4 Gravitational redshift 187
28.5 Apparent \super-luminal" motion of quasar jets 187
29 Large Scale Structure and Eras in the Evolution of the Universe 191
29.1 Galaxy clusters & super-clusters 191
29.2 Dark matter: Hot vs. Cold, WIMPs vs. MACHOs 192
Part V Cosmology 195
30 Newtonian Dynamical Model of Universe Expansion 197
30.1 Critical Density 197
30.2 Gravitational deceleration of increasing scale factor 198
30.2.1 Critical Universe,
m= 1 200
30.2.2 Closed Universe,
m>1 200
30.2.3 Open Universe,
m<1 201
30.3 Redshift vs. distance: Hubble law for various expansion models 201
8 Contents
30.4 Questions and Exercises 203
31 Accelerating Universe with a Cosmological Constant 205
31.1 White-dwarf supernova as distant standard candles 205
31.2 Cosmological Constant and Dark Energy 206
31.3 Flat Universe with Dark Energy 208
31.3.1 Exponential expansion of
at, matter-empty universe 208
31.3.2 General solutions for
at universe with dark energy 208
31.4 The \Flatness" problem 209
32 The Hot Big Bang 211
32.1 The temperature history of the universe 211
32.2 Discovery of the Cosmic Microwave Background (CMB) 212
32.3 Fluctuation Maps from COBE, WMAP, Planck 213
33 Eras in the Evolution of the Universe 216
33.1 Matter-dominated vs. Radiation-dominated eras 216
33.2 The recombination era 217
33.3 Era of nucleosynthesis 219
33.4 The particle era 220
33.5 Questions and Exercises 222
34 Cosmic in
ation 223
34.1 Problems for standard Hot Big Bang model 223
34.2 The era of cosmic in
ation 223
Appendix A Atomic Energy Levels and Transitions 226
Appendix B Equilibrium Excitation and Ionization Balance 231
Appendix C Atomic origins of opacity 234
Appendix D Radiative Transfer 238
Part I
Stellar Properties
1 Introduction
1.1 Observational vs. Physical Properties of Stars
Fundamental Properties of Stars
- Stars appear as points of light due to their immense distance, which prevents telescopes from resolving their physical surfaces like we do with the Sun.
- The Sun serves as a critical local benchmark, allowing astronomers to scale physical properties like mass and temperature to more distant stellar objects.
- Stellar positions are measured using celestial coordinates, with modern space telescopes achieving milli-arcsec precision by avoiding atmospheric distortion.
- Trigonometric parallax uses the Earth's orbital motion to calculate the distance to nearby stars based on shifts in their apparent position.
- Apparent brightness is quantified as energy flux, though the historical magnitude system remains in use due to the human eye's logarithmic response to light.
Of course, when we actually do so, the values we obtain dwarf anything we have direct experience with, thus stretching our imagination, and challenging the physical intuition and insights we instinctively draw upon to function in our own everyday world.
What are the key physical properties we can aspire to know about a star? When
we look up at the night sky, stars are just little \points of light", but if we look
carefully, we can tell that some appear brighter than others, and moreover that
some have distinctly di
erent hues or colors than others. Of course, in modern
times we now know that stars are really \Suns", with properties that are similar
{ within some spread { to our own Sun. They only appear much much dimmer
because they are much much further away. Indeed they appear as mere \points"
because they are so far away that ordinary telescopes almost never can actually
resolve a distinct visible surface, the way we can resolve, even with our naked
eye, that the Sun has a
nite angular size.
Because we can resolve the Sun's surface and see that it is nearly round, it is
perhaps not too hard to imagine that it is a real, physical object, albeit a very
special one, something we could, in principle \reach out and touch". (Indeed a
small amount of solar matter can even travel to the vicinity of the Earth through
the solar wind, coronal mass ejections, and energetic particles.) As such, we can
more readily imagine trying to assign values of common physical properties {
e.g. distance, size, temperature, mass, age, energy emission rate, etc. { that we
regularly use to characterize objects here on Earth. Of course, when we actually
do so, the values we obtain dwarf anything we have direct experience with, thus
stretching our imagination, and challenging the physical intuition and insights we
instinctively draw upon to function in our own everyday world. But once we learn
to grapple with these huge magnitudes for the Sun, we then have at our disposal
that example to provide context and a relative scale to characterize other stars.
And eventually as we move on to still larger scales involving stellar clusters or
even whole galaxies, which might contain thousands, millions, or indeed billions
of individual stars, we can try at each step to develop a relative characterization
of the scales involved in these same physical quantities of size, mass, distance,
etc.
So let's consider here the properties of stars, identifying
rst what we can
directly observe about a given star. Since, as we noted above, most stars are
e
ectively a \point" source without any (easily) detectable angular extent, we
might summarize what can be directly observed as three simple properties:
4 Introduction
1.Position on the Sky: Once corrected for the apparent movement due to
the Earth's own motion from rotation and orbiting the Sun, this can be char-
acterized by two coordinates { analogous to latitude and longitude { on a
\celestial sphere". Before modern times, measurements of absolute position
on the sky had accuracies on order an arcmin; nowadays, it is possible to get
down to a few hundreths of an arcsec from ground-based telescopes, and even
to about a milli-arcsec (or less in the future) from telescopes in space, where
the lack of a distorting atmosphere makes images much sharper. As discussed
below, the ability to measure an annual variation in the apparent position of
a star due to the Earth's motion around the Sun { a phenomena known as
\trignonometric parallax" { provides a key way to infer distance to at least
the nearby stars.
2.Apparent Brightness: The ancient Greeks introduced a system by which
the apparent brightness of stars is categorized in 6 bins called \magnitude",
ranging from m= 1 for the brightest to m= 6 for the dimmest visible to the
naked eye. Nowadays we have instruments that can measure a star's brightness
quantitatively in terms of the energy per unit area per unit time, a quantity
known as the \energy
ux" F, with units erg/cm2/s in CGS or W/m2in MKS.
Because the eye is adapted to distinguish a large dynamic range of brightness,
it turns out its response is logarithmic . And since the Greeks decided to give
dimmer stars a higher magnitude, we
Observational Properties of Stars
- The historical magnitude system for star brightness is logarithmic, where a five-magnitude difference corresponds to a factor of 100 in physical energy flux.
- Modern telescopes can detect stars as dim as magnitude +21, which is a million times fainter than what the naked eye can perceive.
- Astronomers use filters and diffraction gratings to analyze a star's spectrum, providing high-resolution data on how flux varies across narrow wavelength bins.
- High spectral resolution allows for the detection of spectral lines, which reveal the chemical composition and physical conditions of a star.
- The three primary observational inputs—position, brightness, and spectrum—are the foundation for inferring complex physical properties like mass, age, and luminosity.
And since the Greeks decided to give dimmer stars a higher magnitude, we find that magnitude scales with the log of the inverse flux.
the apparent brightness of stars is categorized in 6 bins called \magnitude",
ranging from m= 1 for the brightest to m= 6 for the dimmest visible to the
naked eye. Nowadays we have instruments that can measure a star's brightness
quantitatively in terms of the energy per unit area per unit time, a quantity
known as the \energy
ux" F, with units erg/cm2/s in CGS or W/m2in MKS.
Because the eye is adapted to distinguish a large dynamic range of brightness,
it turns out its response is logarithmic . And since the Greeks decided to give
dimmer stars a higher magnitude, we
nd that magnitude scales with the log
of the inverse
ux,mlog(1=F) log(F), with the m= 5 steps between
the brightest ( m= 1) to dimmest ( m= 6) naked-eye star representing a factor
100 decrease in physical
ux F. Using long exposures on large telescopes with
mirrors several meters in diameter, we can nowadays detect individual stars
with magnitudes m> +21, representing
uxes a million times dimmer than
the limiting magnitude m+6 visible to the naked eye.
3.Color or \Spectrum": Our perception of light in three primary colors comes
from the di
erent sensitivity of receptors in our eyes to light in distinct wave-
length ranges within the visible spectrum, corresponding to Red, Green, and
Blue (RGB). Similarly, in astronomy, the light from a star is often passed
through di
erent sets of
lters designed to transmit only light within some
characteristic band of wavelengths, for example the UBV (Ultraviolet, Blue,
Visible)
lters that make up the so-called \Johnson photometric system". But
much more information can be gained by using a prism or (more commonly) a
di
raction grating to split the light into its spectrum, de
ning the variation in
wavelength of the
ux,F, by measuring its value within narrow wavelength
bins of width
. The \spectral resolution" =available depends on
the instrument (spectrometer) as well as the apparent brightness of the light
source, but for bright stars with modern spectrometers, the resolution can be
10,000 or more, or indeed, for the Sun, many millions. As discussed below,
a key reason for seeking such high spectral resolution is to detect \spectral
lines" that arise from the absorption and emission of radiation via transitions
between discrete energy levels of the atoms within the star. Such spectral lines
1.2 Scales and Orders of Magnitude 5
can provide an enormous wealth of information about the composition and
physical conditions in the source star.
Indeed, a key theme here is that these 3 apparently rather limited observational
properties of point-stars { position, apparent brightness, and color spectrum {
can, when combined with a clear understanding of some basic physical principles,
allow us to infer many of the key physical properties of stars, for example:
1.Distance
2.Luminosity
3.Temperature
4.Size (i.e. Radius)
5.Elemental Composition (denoted as X,Y,Z for mass fraction of H, He, and
of heavy \metals")
6.Velocity (Both radial (toward/away) and transverse (\proper motion" across
the sky)
7.Mass (and surface gravity )
8.Age
9.Rotation (PeriodPand/or equatorial rotation speed Vrot)
10.Mass loss properties (e.g., rate _Mand out
ow speed V)
11.Magnetic
eld
These are ranked roughly in order of diculty for inferring the physical prop-
erty from one or more of the three types of observational data. It also roughly
describes the order in which we will examine them below. In fact, except for
perhaps the last two, which we will likely discuss only brie
y if at all (though
they happen to be two specialities of my own research), a key goal is to provide a
basic understanding of the combination of physical theories, observational data,
and computational methods that make it possible to infer each of the
rst 9
physical properties, at least for some stars.
1.2 Scales and Orders of Magnitude
Scales and Orders of Magnitude
- The text outlines a pedagogical approach to understanding stellar properties through physical theories, observational data, and computational methods.
- A geometric progression using powers of ten is employed to bridge the gap between human scales and the vastness of the universe.
- The scale of the Earth is seven orders of magnitude larger than a human, while the Sun is approximately one hundred times the diameter of Earth.
- Distances in the solar system are often measured in light-travel time, with the Sun being eight light minutes away from Earth.
- Interstellar distances represent a massive jump in scale, with the nearest star being five orders of magnitude further than the Earth-Sun distance.
- The Milky Way galaxy, containing 100 billion stars, extends the cosmic scale another five orders of magnitude beyond the distance between individual stars.
As a mneumonic, this is cast as a 10-digit "telephone number", with the 3-digit "area code" representing the 3 steps of 10-5 from us down to the nucleus, and 7-digit main-number representing 7 key steps to the scale of the universe.
These are ranked roughly in order of diculty for inferring the physical prop-
erty from one or more of the three types of observational data. It also roughly
describes the order in which we will examine them below. In fact, except for
perhaps the last two, which we will likely discuss only brie
y if at all (though
they happen to be two specialities of my own research), a key goal is to provide a
basic understanding of the combination of physical theories, observational data,
and computational methods that make it possible to infer each of the
rst 9
physical properties, at least for some stars.
1.2 Scales and Orders of Magnitude
Before proceeding, let us make a brief aside to discuss ways to get our heads
around the enormous scales we encounter in astrophysics.
As illustrated in
gure 1.1, one approach is to use a geometric progression
through powers of ten1, from the scale from our own bodies, which in standard
metric (MKS) units is of order 1 meter (m), to the progressively larger scales in
our universe.
For example, the meter itself was originally de
ned (in 1793!) as one ten mil-
lionth, or 10 7, of the distance from Earth's equator to poles; this thus means
1There are many online versions, including a rather dated (1977) but still informative movie
titled \Powers of Ten", which you can readily
nd by google; for a modern version, see
http://www.htwins.net/scale2/.
6 Introduction
Figure 1.1 Graphic to illustrate key powers-of-ten steps between our own human scale
of 1 meter, both upward to the scale of the universe (1026m), and also downward to
the scale of an atomic nucleus (10 15m). As a mneumonic, this is cast as a 10-digit
\telephone number", with the 3-digit \area code" representing the 3 steps of 10 5
from us down to the nucleus, and 7-digit main-number representing 7 key steps to the
scale of the universe.
Figure 1.2 Graphics to illustrate the range of scales for time (left) and speed (right).
a total of seven steps in powers of ten from the scale of us to that of our Earth.
1.2 Scales and Orders of Magnitude 7
This is the largest scale for which most of us have direct experience, e.g., from
overseas plane travel, or a cross country drive.
The other, rocky inner planets are somewhat smaller but same order as Earth;
among the outer, gas giant planets Jupiter is the largest, about a factor ten
larger than Earth, while the Sun is about another factor ten larger still, with a
diameterD
1:4106km, about a factor hundred bigger than Earth, or of
order 109m.
The Earth-Sun distance, dubbed an \astronomical unit" (AU), is about about
a hundred solar diameters, at 150 million km. This is of order 108km = 1011m,
or four further powers of ten beyond the scale of our Earth, and so a total of
eleven orders of magnitude bigger in scale than our own bodies.
An alternative way to characterize this is in terms of the time it takes light,
which propagates at a speed c= 300;000 km/s, to reach us from the Sun; a simple
calculation gives t=d=c= 1:5e8=3e5 = 500 s, which is about eight minutes; so
we can say the Sun is 8 light minutes from Earth.
By contrast, it takes light from the next nearest star, Proxima Centauri, about
four years to reach us, meaning it is at a distance of 4 light years (ly). A simple
calculation shows that one year is 1 yr = 365 2460603107s; so
multiplying by the speed of light c= 3105km/s gives that 1 ly 91012km,
or of order 1016m. Thus the scale between the stars is another
ve order of
magnitude greater than that the Earth-Sun distance, or sixteen orders greater
than that of ourselves.
The Sun is only one of about 100 billion (1011) stars in our Milky Way galaxy,
a disk that is about 1000 ly thick, and about 100,000 ly across. Thus our galaxy is
another
ve orders of magnitude bigger than the scale between individual stars,
Scales of the Universe
- The distance between stars is roughly 10 to the 16th power meters, which is five orders of magnitude greater than the Earth-Sun distance.
- The Milky Way galaxy is a disk 100,000 light-years across, representing another five-order jump in scale to 10 to the 21st power meters.
- The observable universe spans 26 orders of magnitude from the human scale, reaching approximately 10 to the 26th power meters.
- A '10-digit phone number' mnemonic (555-711-2555) maps the powers of ten from atomic nuclei up to the entire universe.
- Time and speed scales are equally vast, ranging from a single human heartbeat to the 14-billion-year age of the universe and the speed of light.
The full sequence of steps over this span thus looks something like a 10-digit phone number with area code: 555-711-2555.
calculation shows that one year is 1 yr = 365 2460603107s; so
multiplying by the speed of light c= 3105km/s gives that 1 ly 91012km,
or of order 1016m. Thus the scale between the stars is another
ve order of
magnitude greater than that the Earth-Sun distance, or sixteen orders greater
than that of ourselves.
The Sun is only one of about 100 billion (1011) stars in our Milky Way galaxy,
a disk that is about 1000 ly thick, and about 100,000 ly across. Thus our galaxy is
another
ve orders of magnitude bigger than the scale between individual stars,
or about 1021m, thus twenty-one orders bigger than us.
The universe itself is about 14 billion years old (14 Gyr), meaning that the
most distant galaxies we can see are of order 1010ly1026m away. We thus
see that twenty-six powers of ten takes us from our own scale to the scale of the
entire observable universe!
To recap, powers of ten steps of 7 takes us from us to the Earth; then powers
of ten steps 1, 1 and 2 takes us from Earth to the size of Jupiter, Sun, and Earth-
Sun distance. Then 3 successive power-ten steps of 5 take us to the distance of
the nearest other star; to the size of our galaxy; and
nally to the size of the
universe. It can be helpful to remember this 711-2555 rule as a mnemonic { like
a 7-digit telephone number { to capture the progression between key scales that
characterize our place in the universe.
Indeed, we can extend this even to small scales, by noting that 5 powers of ten
smaller takes us successively to the characteristic size of cell, 10 5m=10 micron;
then to the size of atoms, 10 10m = 0.1 nanometer; and
nally to the scale of an
atomic nucleus,10 femtometer (a.k.a. \fermi") or 1 fm = 10 15m.
The full sequence of steps over this span thus looks something like a 10-digit
phone number with area code: 555-711-2555, representing the power of ten steps
from scales of nuclei to atoms to cells to us to Earth to Jupiter to Sun to au
8 Introduction
(distance to Sun) to light-year ( distance between stars) to our Galaxy to the
Universe.
Finally, the enormous timescales at play in the universe can likewise be dicult
to grasp.
As illustrated in the left panel of
gure 1.2, humans experience time in our
everyday world on the scale of a second, which is roughly the order of a single
heartbeat. We live a maximum of about 100 years, or about 3 billion seconds . In
comparison, it is estimated that the Earth is about 4.4 billion years old, almost
as old as the Sun and the rest of the solar system. The Sun is expected to sustain
its current energy output for about another 5 billion years, and so have a full
lifetime of about 10 billion years. And as discussed below (see x8), the lifetimes
of other stars can depend strongly on their mass; the most massive stars (about
a hundred solar masses) live only about ten million years, while those with mass
less than the Sun are expected to last for up to hundred billion years, much much
longer than the current age of the universe!
The right panel of
gure 1.2 gives a similar graphic for the range of speeds,
from our own slow walk, through others (bicycles, cars, airplanes) we experience,
then ranging to speeds of the moon, earth and Sun in their orbits, to stellar winds
and supernovae, and
nally ending with the maximum possible speed, the speed
of light,c= 3108m/s. The right axis relates the fraction of the light speed for
each of the progression of nine powers from walking to light itself.
The remaining sections below explain how we are able to discover these fun-
damental properties of stars, beginning with their distance.
1.3 Questions and Exercises 9
1.3 Questions and Exercises
Do the following computations by hand (without a calculator), to obtain results
good to just one or two signi
cant
gures, but clearly showing the correct order
of magnitude.
Quick Question 1:
Inferring Distances and Angular Size
- The text introduces fundamental astronomical calculations involving the speeds of Earth's rotation, its orbit around the Sun, and the Sun's orbit within the Milky Way.
- It provides exercises for calculating distances and travel times using light-seconds, solar radii, and astronomical units (AU).
- The concept of angular size is introduced as a primary method for intuitively and mathematically estimating the distance of an object.
- The small-angle approximation is explained, showing how the tangent and sine functions simplify to a linear relationship for distant objects.
- Geometric formulas are established to relate an object's physical size and its distance to the angle it subtends in a viewer's field of vision.
The apparent angular size that object subtends in our overall field of view is then used intuitively by our brains to infer the object's distance, based on our extensive experience that a greater distance makes the object subtend a smaller angle.
of light,c= 3108m/s. The right axis relates the fraction of the light speed for
each of the progression of nine powers from walking to light itself.
The remaining sections below explain how we are able to discover these fun-
damental properties of stars, beginning with their distance.
1.3 Questions and Exercises 9
1.3 Questions and Exercises
Do the following computations by hand (without a calculator), to obtain results
good to just one or two signi
cant
gures, but clearly showing the correct order
of magnitude.
Quick Question 1:
a. What speed does a person at the equator move due to Earth's rotation? Give your
answer in mi/hr, km/hr, and m/s.
b. What is the speed of the Earth in its orbit around the Sun? Give your answer in
AU/yr, km/s, mi/hr, and in terms of the fraction of the speed of light vorb=c?
c. The Sun is about 25,000 ly from the center of the Milky Way, and takes about 200
million years to complete one \Galactic year". What is the speed of Sun in its orbit
around the Milky Way, in km/s. In ly/yr? In terms of the fraction of the speed of light
vorb=c?
Quick Question 2: The Sun has a radius of about 700,000 km.
a. How many solar radii in 1 AU? In 1 ly?
b. How many Earth radii REin one solar radius R
?
c. Solar neutrinos created in the Sun's core travel at very nearly speed of light but
hardly interact with solar matter. How long does it take such core neutrinos to reach
the solar surface? How long to reach us on Earth?
d. What then is the solar radius in light-seconds?
Quick Question 3: The Moon is about 240,000 miles from Earth.
a. What is the Earth-Moon distance in km? In light-seconds? In Earth radii RE? In
solar radiiR
?
b. How many Earth-Moon distances in 1 AU?
2 Inferring Astronomical Distances
2.1 Angular size
To understand ways we might infer stellar distances, let's
rst consider how
we intuitively estimate distance in our everyday world. Two common ways are
through apparent angular size , and/or using our stereoscopic vision .
For the
rst, let us suppose we have some independent knowledge of the phys-
ical size of a viewed object. The apparent angular size that object subtends in
our overall
eld of view is then used intuitively by our brains to infer the object's
distance, based on our extensive experience that a greater distance makes the
object subtend a smaller angle.
s α AB
Cd
Figure 2.1 Angular size and parallax: The triangle illustrates how an object of
physical size s(BC) subtends an angular size
when viewed from a point A that is
at a distance d. Note that the same triangle can also illustrate the parallax angle
toward the point A at distance dwhen viewed from two points B and C separated by
a lengths.
As illustrated in
gure 2.1, we can, with the help of some elementary geometry,
formalize this intuition to write the speci
c formula. The triangle illustrates
the angle
subtended by an object of size sfrom a distance d. From simple
trigonometry, we
nd
tan(
=2) =s=2
d: (2.1)
For distances much larger than the size, d
s, the angle is small,
1,
for which the tangent function can be approximated (e.g. by
rst-order Tay-
lor expansion) to give tan(
=2)
=2, where
is measured here in radians.
2.1 Angular size 11
(1 rad = (180 =)57). The relation between distance, size, and angle thus
becomes simply
s
d: (2.2)
Of course, if we know the physical size and then measure the angular size, we
can solve the above relation to determine the distance d=s=
.
0.20.611.40.511.52Tan x
Sin xx
xx+x3/3
x-x3/6
Figure 2.2 Taylor expansion of trig functions sin xand tanx, aboutx= 0 to order x
and orderx3.
dR
RαAB
Csin(α/2)=R/d
for R<<D , α≈2R/d
Figure 2.3 Diagram to illustrate the relation between angular size
and diameter 2 R
for a sphere at distance d.
As illustrated in
gure 2.3, for a spherical object the angular size
is related
to the distance dand radius Rthrough the sine function,
sin(
=2) =R=d: (2.3)
From
Measuring the Cosmos
- The distance to a celestial object can be calculated by comparing its known physical size to its observed angular size.
- Small-angle approximations simplify complex trigonometric relations into linear equations for distant astronomical bodies.
- Atmospheric seeing blurs images to about 1 arcsecond, preventing ground-based telescopes from resolving the tiny angular diameters of most stars.
- Trigonometric parallax mimics human stereoscopic vision by using two different viewpoints to perceive depth and distance.
- The parallax effect is inversely proportional to distance, meaning closer objects exhibit a larger shift in apparent position.
Neurosensors in the eye muscles that effect this inward pointing relay this inward angle to our brain, where it is processed to provide our sense of 'depth' perception.
Of course, if we know the physical size and then measure the angular size, we
can solve the above relation to determine the distance d=s=
.
0.20.611.40.511.52Tan x
Sin xx
xx+x3/3
x-x3/6
Figure 2.2 Taylor expansion of trig functions sin xand tanx, aboutx= 0 to order x
and orderx3.
dR
RαAB
Csin(α/2)=R/d
for R<<D , α≈2R/d
Figure 2.3 Diagram to illustrate the relation between angular size
and diameter 2 R
for a sphere at distance d.
As illustrated in
gure 2.3, for a spherical object the angular size
is related
to the distance dand radius Rthrough the sine function,
sin(
=2) =R=d: (2.3)
From
gure 2.2 we see that, for the small angles that apply at large distances
d
R, this again reduces to a simple linear form,
2R=d, that relates size
to distance.
12 Inferring Astronomical Distances
For example, the distance from the Earth to the Sun, known as an \astronom-
ical unit" (abbreviated \au"), is d= 1 au150106km, much larger than the
Sun's physical size (i.e. diameter), which is about s= 2R
1:4106km. This
means that the Sun has an apparent angular diameter of
2R
1au0:009 rad0:5= 30 arcmin = 1800 arcsec : (2.4)
However, as noted in x1.2 (and illustrated in
gure 1.1), even the nearest stars
are more than 200,000 times further away than the Sun. If we assume a similar
physical radius (which actually is true for one of the components of the nearest
star system,
Centauri A), then
=2R
200;000au0:009 arcsec: (2.5)
For ground-based telescopes, the distorting e
ect of the Earth's atmosphere,
known as \atmospheric seeing" (see x13.2), blurs images over an angle size of
about 1 arcsec, making it very dicult to infer the actual angular size. There are
some specialized techniques, e.g. \speckle interferometry", that can just barely
resolve the angular diameter of a few nearby giant stars (e.g. Betelgeuse, a.k.a.
Ori). But generally the diculty of measuring a star's angular size means that,
even if we knew its physical size, we can not use this angular-size method to infer
its distance.
2.2 Trignonometric parallax
Fortunately, there is a practical, quite direct way to infer distances to at least
relatively nearby stars, namely through the method of trigonometric parallax .
This is physically quite analogous to the stereoscopic vision by which we use
our two eyes to infer distances to objects in our everyday world. To understand
this parallax e
ect, we can again refer to
gure 2.1. If we now identify sas the
separation between the eyes, then when we view objects at some nearby distance
d, the two eyes, in order to combine the separate images as one, have to point
inward an angle
= 2 arctan( s=2d). Neurosensors in the eye muscles that e
ect
this inward pointing relay this inward angle to our brain, where it is processed
to provide our sense of \depth" (i.e. distance) perception.
You can easily experiment with this e
ect by placing your
nger a few inches
from your face, then blinking between your left and right eye, which thus causes
the image of your
nger to jump back a forth by the angle
= 2 arctan( s=2d).
The eye separation sis
xed, but as you move the
nger closer and further away,
the angle shift will become respectively larger and smaller.
Home Experiment: To illustrate this close link between parallax and angular size,
try the following experiment. In front of a wall mirror, close one eye and then extend
a
nger from either arm to the mirror, covering the image of your closed eye. Without
moving your
nger, now switch the closure to the other eye. Note that the
nger has
2.2 Trignonometric parallax 13
also switched to cover the other (now closed) eye, even though you didn't physically
move it! Note further that this even still works as you decrease the distance from your
face to the mirror. The key point here is that the \parallax" angle shift of your
nger,
which results from switching perspective from one eye to the other, exactly
ts the
Trigonometric Parallax and Stellar Distance
- Trigonometric parallax uses a change in perspective to calculate the distance of an object based on an angular shift.
- While human binocular vision is limited to about 10 meters, larger baselines allow for the measurement of astronomical distances.
- Early 19th-century astronomers used the Earth's diameter as a baseline to calculate the distance to Mars during opposition.
- Stellar parallax utilizes the Earth's orbital radius (1 AU) as a baseline, observing stars at six-month intervals to maximize the shift.
- The 'parsec' is defined as the distance at which a star exhibits a parallax angle of one arcsecond, equivalent to roughly 3.26 light-years.
- Atmospheric blurring and the tiny scale of parallax angles (all less than one arcsecond) make these measurements extremely challenging.
The key point here is that the parallax angle shift of your finger, which results from switching perspective from one eye to the other, exactly fits the apparent angular separation between your own mirror-image eyes.
nger from either arm to the mirror, covering the image of your closed eye. Without
moving your
nger, now switch the closure to the other eye. Note that the
nger has
2.2 Trignonometric parallax 13
also switched to cover the other (now closed) eye, even though you didn't physically
move it! Note further that this even still works as you decrease the distance from your
face to the mirror. The key point here is that the \parallax" angle shift of your
nger,
which results from switching perspective from one eye to the other, exactly
ts the
apparent angular separation between your own mirror-image eyes.
Of course, for distances much more than the separation between our eyes,
d
s, the angle becomes too small to perceive, and so we can only use this
approach to infer distances of about, say, 10 m. But if we extend the baseline
to much larger sizes s, then when coupled with accurate measures of the angle
shift
, this method can be used to infer much larger distances.
For example, in the 19th century, there were e
orts to use this approach to
infer the distance to Mars at time when it was relatively close to Earth, namely
at opposition (i.e. when Mars is on the opposite side of the Earth from the Sun).
Two expeditions tried to measure the position of Mars at the same time from
widely separated sites on Earth. If the distance between the sites is known, the
angle di
erence in the measured directions to Mars, which turns out to be about
an arcmin, yields a distance to Mars.
The largest separation possible from two points on the surface of the Earth
is limited by the Earth's diameter. But to apply this method of trigonometric
parallax to infer distances to stars, we need to use a much bigger baseline than
the Earth's diameter. Fortunately though, we don't need then to go into space.
au
Jan.dJuly
*p*
**
nearby
stard/pc = arcsec/pbackground stars *
***
****
Figure 2.4 Illustration of stellar parallax, in which a relatively nearby star appears to
shift against background stars by a parallax angle pas the earth moves through the 1
au radius of earth's orbit. The distance din parsec (pc) is given by the inverse of p
measured in arcsec.
As illustrated in
gure 2.4, just waiting a half year from one place on the Earth
allows us, as a result of the Earth's orbit around the Sun, to view the stars from
two points separated by twice the Earth's orbital radius, i.e. 2 au. By convention,
however, the associated \parallax angle"
of a star is traditionally quoted in
terms of the shift from a baseline sof just oneau. If we scale the parallax angle
in units of an arcsec, the distance is
d=s
=206;265 arcsec=radian
=radianauarcsec
parsec; (2.6)
14 Inferring Astronomical Distances
where we note that the conversion between arcsec and radian is given by (180/ )
degree/radian60 arcmin/degree 60 arcsec/arcmin = 206,265 arcsec/radian.
In the last equality, we have also introduced the distance unit parsec (short for
\parallax second", and often further abbreviated as \pc"), which is de
ned as
the distance at which the parallax angle is 1 arcsec. It is thus apparent that
1 pc = 206;265 au, which works out to give 1 pc 31016m.
The \parsec" is one of the two most common units used to characterize the
huge distances we encounter in astronomy. The other is the light-year , which is
the distance light travels in a year, at the speed of light c= 3108m/s. The
number of seconds in a year is given by 1 yr = 365 246060 = 3:15107s,
which, coincidentally, can be remembered as 1 yr 107s (or sincep
103:16,
1 yr107:5s). Thus a light-year is roughly 1 ly 3108+79:51015
1016m. In terms of parsecs, we can see that 1 pc 3:26 ly.
The parallax for even the nearest star is less than an arcsec, implying stars are
all at distances more (generally much more) than a parsec. By repeated observa-
tion, the roughly 1 arcsec overall blurring of single stellar images by atmospheric
Measuring Cosmic Distances
- The parsec and light-year are the primary units for astronomical distance, with one parsec equaling approximately 3.26 light-years.
- Ground-based parallax measurements are limited to about 100 parsecs due to atmospheric blurring, though averaging techniques can improve accuracy.
- Space-based missions like Hipparchus and Gaia significantly extend our reach by measuring parallax angles down to a milliarcsecond.
- The Astronomical Unit (au) serves as the fundamental baseline for all stellar distance calculations.
- Modern radar ranging of planets like Venus allows for the precise trigonometric derivation of the au's physical length.
- Solid angle, measured in steradians, provides a two-dimensional generalization for describing the size of extended objects in the sky.
The parallax for even the nearest star is less than an arcsec, implying stars are all at distances more (generally much more) than a parsec.
the distance at which the parallax angle is 1 arcsec. It is thus apparent that
1 pc = 206;265 au, which works out to give 1 pc 31016m.
The \parsec" is one of the two most common units used to characterize the
huge distances we encounter in astronomy. The other is the light-year , which is
the distance light travels in a year, at the speed of light c= 3108m/s. The
number of seconds in a year is given by 1 yr = 365 246060 = 3:15107s,
which, coincidentally, can be remembered as 1 yr 107s (or sincep
103:16,
1 yr107:5s). Thus a light-year is roughly 1 ly 3108+79:51015
1016m. In terms of parsecs, we can see that 1 pc 3:26 ly.
The parallax for even the nearest star is less than an arcsec, implying stars are
all at distances more (generally much more) than a parsec. By repeated observa-
tion, the roughly 1 arcsec overall blurring of single stellar images by atmospheric
seeing can be averaged to give a position accuracy of about
0:01 arcsec,
implying that one can estimate distances to stars out to about d100 pc. The
Hipparchus satellite orbiting above Earth's atmosphere can measure parallax an-
gles approaching a milliarcsec (1 mas = 10 3arcsec), thus potentially extending
distance measurements for stars out to about a kiloparsec, d1 kpc. However,
parallax measurements out to such distances typically require a relatively bright
source. In practice, only a fraction of all the stars (those with the highest intrin-
sic brightness, or \luminosity") with distances near d1 kpc have thus far had
accurate measurements of their parallax1.
Again, from the above discussion it should be apparent that parallax is really
the \
ip slide" of the angular size vs. distance relation. That is, the triangle in
gure 2.1 was initially used to illustrate how, from the perspective of a given
point A, the angle
subtended by an object is set by the ratio of its size sto its
distanced. But if we consider a simple change of observer's perspective to the
two endpoints (B and C) of the size seqment s, then the same triangle can be
used equally well to illustrate the observed parallax angle
for the point A at a
distanced.
For the large ( >parsec) distances in astronomy, it is convenient to rewrite
our simple equation (2.2) to scale angular size in arcsec, with the size in au and
distance in pc:
arcsec=s=au
d=pc: (2.7)
1Since 2013 a follow-up satellite mission call Gaia has been in the process of measuring the
absolute position and parallax to roughly one billion stars; see http://sci.esa.int/gaia/.
2.3 Determining the Astronomical Unit (au) 15
2.3 Determining the Astronomical Unit (au)
We thus see that determining the distance of the Earth to the Sun, i.e. measuring
the physical length of an au, provides a fundamental basis for determining the
distances to stars and other objects in the universe. In modern times, one way this
is computed involves
rst measuring the distance from the Earth to the planet
Venus through \radar ranging", i.e. measuring the time tit takes a radar signal
to bounce o
Venus and return to Earth. The associated Earth-Venus distance
is then given by
dEV=ct
2: (2.8)
If this distance is measured at the time when Venus has its \maximum elonga-
tion", or maximum angular separation, from the Sun, which is found to be about
47o, then one can use simple trigonometry to derive a physical value of the au.
The details are left as an exercise for the reader. (See Exercise 2-1 at the end of
this section.)
2.4 Solid angle
In general objects that have a measurable angular size on the sky are extended
intwoindependent directions. As the 2D generalization of an angle along just
one direction, it is useful then to de
ne for such objects a 2D solid angle
,
measured now in square radians , but more commonly referred by the shorthand
\steradians ".
Just as projected area Ais related to the square of physical size s(or radius
R), so is solid angle
related to the square of the angular size
Solid Angles and Celestial Geometry
- The distance to Venus at maximum elongation can be used with simple trigonometry to calculate the physical value of an astronomical unit (au).
- Solid angle is introduced as the two-dimensional generalization of an angle, measured in steradians or square radians.
- The solid angle of an object is mathematically defined as the ratio of its projected area to the square of its distance.
- A full sphere contains 4π steradians, which is equivalent to approximately 41,253 square degrees.
- The Sun and Moon each cover a solid angle of about 0.2 square degrees, representing only 1/200,000th of the total sky.
- Calculations of solid angles help explain physical phenomena, such as why the full moon is significantly dimmer than the sun.
Integration over a full sphere shows that there are 4π steradians in the full sky.
If this distance is measured at the time when Venus has its \maximum elonga-
tion", or maximum angular separation, from the Sun, which is found to be about
47o, then one can use simple trigonometry to derive a physical value of the au.
The details are left as an exercise for the reader. (See Exercise 2-1 at the end of
this section.)
2.4 Solid angle
In general objects that have a measurable angular size on the sky are extended
intwoindependent directions. As the 2D generalization of an angle along just
one direction, it is useful then to de
ne for such objects a 2D solid angle
,
measured now in square radians , but more commonly referred by the shorthand
\steradians ".
Just as projected area Ais related to the square of physical size s(or radius
R), so is solid angle
related to the square of the angular size
. For an object
at a distance dwith projected area A, the solid angle is just
=A
d2R2
d2=
2; (2.9)
where the latter equalities assume a sphere (or disk) with projected radius R
and associated angular radius
=R=d.
For more general shapes,
gure 2.5 illustrates how a small solid-angle patch
is de
ned in terms of ranges in the standard spherical angles representing
co-latitude and azimuth
on a sphere. An extended object would then have a
solid angle given by the integral
=Z
d
sind: (2.10)
Integration over a full sphere shows that there are 4steradians in
the full sky. This represents the 2D analog to the 2 radians around the full
circumference of a circle.
For our example of a circular patch of angular radius
, let us assume the
16 Inferring Astronomical Distances
θ
θ+δθ
δφ δΩ=sinθ δθ δφ
Figure 2.5 Diagram to illustrate a small patch of solid angle
seen by an observer at
the center of a sphere, with size de
ned by ranges in the co-latitude and azimuth
.
object is centered around the coordinate pole { representing perhaps the image
of a distant spherical object like the Sun or moon. The azimuthal symmetry
means the
integral evaluates to 2 , while carrying out the remaining integral
over co-latitude range 0 to
then gives
= 2[1 cos
]: (2.11)
In particular, applying the angular radius of the Sun
R
=au and expanding
the cosine to
rst order (i.e., cos x1 x2=2), we
nd
= 2[1 cos(R
=au)](R
=au)2
2
: (2.12)
One can alternatively measure solid angle in terms of square degrees. Since
there are 180 =57:3 degrees in a radian, there are (180 =)2= 57:323283
square degrees in a steradian; the number of square degrees in the 4 steradians
of the full sky is thus
4180
2
= 41;253 deg2: (2.13)
The Sun and moon both have angular radii of about 0 :25o, meaning they each
have a solid angle of about (0:25)2==16 = 0:2 deg2= 610 5ster, which
is about 1=200;000 of the full sky2.
2.5 Questions and Exercises
Quick Question 1: A helium party balloon of diameter 20 cm
oats 1 meter above
your head.
2If you think about it, you'll see that this helps explain why a full moon is about a million
times dimmer than full sunlight! See Exercise 2-3.
2.5 Questions and Exercises 17
a. What is its angular diameter, in degrees and radians?
b. What is its solid angle, in square degrees and steradians?
c. What fraction of the full sky does it cover?
d. At what height hwould its angular diameter equal that of the Moon and Sun?
Quick Question 2:
a. What angle
would the Earth-Sun separation subtend if viewed from a distance
ofd= 1 pc? Give your answer in both radian and arcsec.
b. How about from a distance of d= 1 kpc?
Quick Question 3: Over a period of several years, two stars appear to go around each
other with a
xed angular separation of 1 arcsec.
a. What is the physical separation, in au, between the stars if they have a distance
d= 10 pc from Earth?
b. If they have a distance d= 100 pc?
Exercise 1: At the time when Venus exhibits its maximum elongation angle of about
Luminosity and Distance Measurement
- The text provides practical exercises for calculating astronomical distances using angular separation, radar timing, and parallax errors.
- Apparent brightness is defined as the flux of light, which is the energy per unit time per unit area captured by a detector.
- The inverse-square law dictates that light intensity decreases in proportion to the square of the distance as energy spreads over a spherical area.
- The 'Standard Candle' method allows astronomers to calculate distance by comparing an object's known intrinsic luminosity to its observed flux.
- When distance is known via trigonometric parallax, the same physical relationship is used to calculate a star's total power output or luminosity.
This is a profoundly important equation in astronomy, and so you should not just memorize it, but embed it completely and deeply into your psyche.
would the Earth-Sun separation subtend if viewed from a distance
ofd= 1 pc? Give your answer in both radian and arcsec.
b. How about from a distance of d= 1 kpc?
Quick Question 3: Over a period of several years, two stars appear to go around each
other with a
xed angular separation of 1 arcsec.
a. What is the physical separation, in au, between the stars if they have a distance
d= 10 pc from Earth?
b. If they have a distance d= 100 pc?
Exercise 1: At the time when Venus exhibits its maximum elongation angle of about
47ofrom the Sun, a radar signal is found to take a round trip time t= 667 sec to
return to Earth. Assuming both Earth and Venus have circular orbits, and using the
speed of light c= 3105km/s, compute (in km) the Earth-Sun distance, 1 AU.
Exercise 2: With a suciently large telescope in space, with angle error
1 mas,
for how many more stars can we expect to obtain a measured parallax than we can
from ground-based surveys with
20 mas? (Hint: What assumption do you need
to make about the space density of stars in the region of the galaxy within 1 kpc from
the Sun/Earth?)
Exercise 3: a. Assuming the Moon re
ects a fraction a(dubbed the \albedo") of
sunlight hitting it, derive an expression for the ratio of apparent brightness ( Fmoon=F
)
between the full Moon and Sun, in terms of the Moon's radius Rmoon and its distance
from earth, dem
au. b. Derive the value of the albedo afor which this ratio equals
the fraction of sky subtended by the Moon's solid angle, i.e. for which Fmoon=F
=
moon=4.
3 Inferring Stellar Luminosity
3.1 \Standard Candle" methods for distance
In our everyday experience, there is another way we sometimes infer distance,
namely by the change in apparent brightness for objects that emit their own light,
with some known power or \luminosity". For example, a hundred watt light bulb
at a distance of d= 1 m certainly appears a lot brighter than that same bulb at
d= 100 m. Just as for a star, what we observe as apparent brightness is really
a measure of the
uxof light, i.e. energy per unit time per unit area (erg/s/cm2
in CGS units, or watt/m2in MKS).
When viewing a light bulb with our eyes, it's just the rate at which the light's
energy is captured by the area of our pupils. If we assume the light bulb's emission
isisotropic (i.e., the same in all directions), then as the light travels outward to
a distanced, its power or luminosity is spread over a sphere of area 4 d2. This
means that the light detected over a
xed detector area (like the pupil of our
eye, or, for telescopes observing stars, the area of the telescope mirror) decreases
in proportion to the inverse-square of the distance, 1 =d2. We can thus de
ne the
apparent brightness in terms of the
ux,
F=L
4d2: (3.1)
This is a profoundly important equation in astronomy, and so you should not
just memorize it, but embed it completely and deeply into your psyche.
In particular, it should become obvious that this equation can be readily used
to infer the distance to an object of known luminosity , an approach called the
standard candle method. (Taken from the idea that a candle, or at least a \stan-
dard" candle, has a known luminosity or intrinisic brightness.) As discussed
further in sections below, there are circumstances in which we can get clues to
a star's (or other object's) intrinsic luminosity L, for example through careful
study of a star's spectrum. If we then measure the apparent brightness (i.e.
ux
F), we can infer the distance through:
d=r
L
4F: (3.2)
Indeed, when the study of a stellar spectrum is the way we infer the luminos-
3.2 Intensity or Surface Brightness 19
ity, this method of distance determination is sometimes called \spectroscopic
parallax".
Of course, if we can independently determine the distance through the actual
trigonometric parallax, then such a simple measurement of the
ux can instead
be used to determine the luminosity,
L= 4d2F: (3.3)
Stellar Luminosity and Surface Brightness
- Luminosity can be determined through spectroscopic parallax by measuring a star's spectrum and apparent brightness.
- The Sun's luminosity is approximately 4 x 10^26 Watts, a scale equivalent to four million billion billion 100-watt light bulbs.
- Solar luminosity serves as a benchmark for other stars, which range from 1/1000th to one million times the Sun's power.
- Surface brightness, or specific intensity, is a unique quantity that remains constant regardless of the observer's distance from the object.
- While the flux of light decreases with distance, the solid angle it occupies shrinks proportionally, keeping the ratio of flux per solid angle stable.
- The surface brightness of the Sun viewed from Earth is identical to the brightness one would experience standing on the Sun's surface.
Thus we see that the Sun emits the power of about 4 x 10^24 100-watt light bulbs! In common language this corresponds to four million billion billion, a number so huge that it loses any meaning.
study of a star's spectrum. If we then measure the apparent brightness (i.e.
ux
F), we can infer the distance through:
d=r
L
4F: (3.2)
Indeed, when the study of a stellar spectrum is the way we infer the luminos-
3.2 Intensity or Surface Brightness 19
ity, this method of distance determination is sometimes called \spectroscopic
parallax".
Of course, if we can independently determine the distance through the actual
trigonometric parallax, then such a simple measurement of the
ux can instead
be used to determine the luminosity,
L= 4d2F: (3.3)
In the case of the Sun, the
ux measured at Earth is referred to as the \solar
constant", with a measured mean value of about
F
1:4kW
m2= 1:4106erg
cm2s: (3.4)
If we then apply the known mean distance of the Earth to the Sun, d= 1 au, we
obtain for the solar luminosity
L
41026W = 41033erg
s: (3.5)
Thus we see that the Sun emits the power of about 4 1024100-watt light bulbs!
In common language this corresponds to four million billion billion, a number so
huge that it loses any meaning. It illustrates again how in astronomy we have to
think on a entirely di
erent scale than we are used to in our everyday world.
But once we get used to the idea that the luminosity and other properties of
the Sun are huge but still
nite and measurable, we can use these as benchmarks
for characterizing analogous properties of other stars and astronomical objects.
In the case of stellar luminosities, for example, these typically range from about
L
=1000 for very cool, low-mass \dwarf" stars, to as high as 106L
for very hot,
high-mass \supergiants".
As discussed further below, the luminosity of a star depends directly on both
its size (i.e. radius) and surface temperature. But more fundamentally these in
turn are largely set by the star's mass, age, and chemical composition.
3.2 Intensity or Surface Brightness
For any object with a resolved solid angle
, an important
ux-related quantity
is the surface brightness { also known as the speci
c intensity I; this can be
roughly (though not quite exactly; see x12.1) thought of as the
ux per solid
angle , i.e.
IF
L
4d2(R=d)2L
42R2=F
; (3.6)
whereFF(R) =L=4R2is the surface
ux evaluated at the stellar radius R.
As illustrated in
gure 3.1, the surface brightness of any resolved radiating object
turns out, somewhat surprisingly, to be independent of distance . This is because,
even though the
ux declines with distance, the surface brightness `crowds' this
20 Inferring Stellar Luminosity
d1Ad2α2
Ω2=πα 22α1R
Ω1=πα 12Ω=πα 2α≈ R/d F=L/4πd2I=F/Ω=L/4πd2/πα 2=L/4π2R2=F */πSurface brightness I is
independent of distance d
angular
radiusSolid
angle Flux
Figure 3.1 Distance independence of surface brightness of a radiating sphere,
representing the
ux per solid angle, B=F=
. At greater distance d, the
ux
declines in proportion to 1 =d2; but because this
ux is squeezed into a smaller solid
angle
, which also declines as 1 =d2, the surface brightness Bremains constant,
independent of the distance.
ux into a proportionally smaller solid angle as the distance is increased. The
ratio of
ux per solid angle, or surface brightness, is thus constant.
In particular, if we ignore any absorption from earth's atmosphere, the surface
brightness of the Sun that we see here on earth is actually the same as if we were
standing on the surface of the Sun itself!
Of course, on the surface of the Sun, its radiation will
ll up half the sky {
i.e. 2steradians, instead of the mere 0 :2 deg2= 610 5steradians seen from
earth. The huge
ux from this large, bright solid angle would cause a lot more
than a mere sunburn!1
3.3 Apparent and absolute magnitude and the distance modulus
To summarize, we have now identi
ed 3 distinct kinds of \brightness" { abso-
lute, apparent, and surface { associated respectively with the luminosity (en-
ergy/time),
ux (energy/time/area), and speci
c intensity (
Stellar Magnitudes and Distance Modulus
- Astronomers distinguish between absolute, apparent, and surface brightness, which relate to luminosity, flux, and specific intensity respectively.
- The magnitude system is a logarithmic scale where a difference of 5 magnitudes corresponds to a factor of 100 in relative brightness.
- Apparent magnitude (m) measures how bright a star looks from Earth, while absolute magnitude (M) measures its brightness at a standard distance of 10 parsecs.
- The distance modulus (m - M) is a mathematical relationship used to determine the distance to a star based on its apparent and absolute magnitudes.
- The Sun has an absolute magnitude of approximately +4.8, serving as a baseline for calculating the absolute magnitudes of other stars.
The huge flux from this large, bright solid angle would cause a lot more than a mere sunburn!
Of course, on the surface of the Sun, its radiation will
ll up half the sky {
i.e. 2steradians, instead of the mere 0 :2 deg2= 610 5steradians seen from
earth. The huge
ux from this large, bright solid angle would cause a lot more
than a mere sunburn!1
3.3 Apparent and absolute magnitude and the distance modulus
To summarize, we have now identi
ed 3 distinct kinds of \brightness" { abso-
lute, apparent, and surface { associated respectively with the luminosity (en-
ergy/time),
ux (energy/time/area), and speci
c intensity (
ux emitted into a
given solid angle). Before moving on to examine additional properties of stel-
lar radiation, let us
rst discuss some speci
cs of how astronomers characterize
apparent vs. absolute brightness, namely through the so-called \magnitude" sys-
tem.
This system has some rather awkward conventions, developed through its long
history, dating back to the ancient Greeks. As noted in x1, they ranked the
apparent brightness of stars in 6 bins of magnitude, ranging from m= 1 for
the brightest to m= 6 for the dimmest. Because the human eye is adapted to
1NASA 's recently launched \Parker Solar Probe" will eventually
y within about 9 R
of
the solar surface, or about 1/20 au. So a key challenge has been to provide the shielding
to keep the factor >400 higher solar radiation
ux from frying the spacecraft's instruments.
3.4 Questions and Exercises 21
detect a large dynamic range in brightness, it turns out that our perception of
brightness depends roughly on the logarithm of the
ux.
In our modern calibration this can be related to the Greek magnitude system
by stating that a di
erence of 5 in magnitude represents a factor 100 in the
relative brightness of the compared stars, with the dimmer star having the larger
magnitude . This can be expressed in mathematical form as
m2 m1= 2:5 log(F1=F2): (3.7)
We can further extend this logarithmic magnitude system to characterize the
absolute brightness, a.k.a. luminosity, of a star in terms of an absolute magnitude.
To remove the inherent dependence on distance in the
ux F, and thus in the
apparent magnitude m, the absolute magnitude Mis de
ned as the apparent
magnitude that a star would have if it were placed at a standard distance, chosen
by convention to be d= 10 pc. Since the
ux scales with the inverse-square of
distance,F1=d2, the di
erence between apparent magnitude mand absolute
magnitude Mis given by
m M= 5 log(d=10 pc); (3.8)
which is known as the distance modulus .
The absolute magnitude of the Sun is M+4.8 (though for simplicity in
calculations, this is often rounded up to 5), and so the scaling for other stars can
be written as
M= 4:8 2:5 log(L=L
): (3.9)
Combining these relations, we see that the apparent magnitude of any star is
given in terms of the luminosity and distance by
m= 4:8 2:5 log(L=L
) + 5 log(d=10 pc): (3.10)
For bright stars, magnitudes can even become negative. For example, the (ap-
parently) brightest star in the night sky, Sirius, has an apparent magnitude
m= 1:42. But with a luminosity of just L23L
, its absolute magnitude is
still positive, M= +1:40. Its distance modulus, m M= 1:42 1:40 = 2:82,
is negative. Through eqn. (3.8), this implies that its distance, d= 101 2:82=5=
2:7 pc, is lessthan the standard distance of 10 pc used to de
ne absolute mag-
nitude and distance modulus [eqn. (3.8)].
3.4 Questions and Exercises
Quick Question 1: Recalling the relationship between an AU and a parsec from
eqn. (2.6), use eqns. (3.8) and (3.9) to compute the apparent magnitude of the Sun.
What then is the Sun's distance modulus?
Quick Question 2: Suppose two stars have a luminosity ratio L2=L1= 100.
22 Inferring Stellar Luminosity
a. At what distance ratio d2=d1would the stars have the same apparent brightness,
F2=F1?
b. For this distance ratio, what is the di
erence in their apparent magnitude, m2
m1?
c. What is the di
erence in their absolute magnitude, M2 M1?
Stellar Luminosity and Thermal Radiation
- Mathematical exercises explore the relationships between apparent magnitude, absolute magnitude, and distance modulus for stars and supernovae.
- Stellar brightness is a direct consequence of high surface temperatures, which cause atoms and electrons to collide violently and emit thermal radiation.
- Temperature in astronomy is measured in Kelvins, where absolute zero represents the theoretical limit where all thermal motion ceases.
- Light is defined as electromagnetic radiation consisting of oscillating electric and magnetic fields as described by Maxwell's equations.
- The electromagnetic spectrum spans from short-wavelength gamma rays to long-wavelength radio waves, with visible light occupying a narrow band between 400 and 750 nm.
- All electromagnetic waves travel at the constant speed of light (c) in a vacuum, maintaining a strict inverse relationship between wavelength and frequency.
The light they emit is called "thermal radiation", and arises from the jostling of the atoms (and particularly the electrons in and around those atoms) by the violent collisions associated with the star's high temperature.
Quick Question 1: Recalling the relationship between an AU and a parsec from
eqn. (2.6), use eqns. (3.8) and (3.9) to compute the apparent magnitude of the Sun.
What then is the Sun's distance modulus?
Quick Question 2: Suppose two stars have a luminosity ratio L2=L1= 100.
22 Inferring Stellar Luminosity
a. At what distance ratio d2=d1would the stars have the same apparent brightness,
F2=F1?
b. For this distance ratio, what is the di
erence in their apparent magnitude, m2
m1?
c. What is the di
erence in their absolute magnitude, M2 M1?
d. What is the di
erence in their distance modulus?
Quick Question 3: A white-dwarf supernova with peak luminosity L1010L
is
observed to have an apparent magnitude of m= +20 at this peak.
a. What is its Absolute Magnitude M?
b. What is its distance d(in pc and ly). s c. How long ago did this supernova explode
(in Myr)?
(For simplicity of computation, you may take the absolute magnitude of the Sun to
beM
+5.)
4 Inferring Surface Temperature from
a Star's Color and/or Spectrum
Let us next consider why stars shine with such extreme brightness. Over the
long-term (i.e., millions of years), the enormous energy emitted comes from the
energy generated (by nuclear fusion) in the stellar core, as discussed further in x18
below. But the more immediate reason stars shine is more direct, namely because
their surfaces are so very hot. The light they emit is called \thermal radiation",
and arises from the jostling of the atoms (and particularly the electrons in and
around those atoms) by the violent collisions associated with the star's high
temperature1.
Figure 4.1 The Electromagnetic Spectrum.
1In astronomy, temperature is measured in a degree unit called a Kelvin , abbreviated K,
and de
ned relative to the centigrade or \Celsius" scale C such that K=C+ 273. A
temperature of T= 0Kis called \absolute zero", and represents the ideal limit that all
thermal motion is completely stopped. To convert from our US use of the Fahrenheit scale
F, we
rst just convert to centigrade using C= (5=9)(F 32), and then add 273 to get the
temperature in K.
24 Inferring Surface Temperature from a Star's Color and/or Spectrum
4.1 The wave nature of light
To lay the groundwork for a general understanding of the key physical laws
governing such thermal radiation and how it depends on temperature, we have
to review what is understood about the basic nature of light, and the processes
by which it is emitted and absorbed.
The 19th century physicist James Clerk Maxwell developed a set of 4 equations
(Maxwell's equations) that showed how variations in Electric and Magnetic
elds
could lead to oscillating wave solutions, which he indeed indentifed with light,
or more generally Electro-Magnetic (EM) radiation . The wavelengths of these
EM waves are key to their properties. As illustrated in
gure 4.1, visible light
corresponds to wavelengths ranging from 400 nm (violet) to 750 nm
(red), but the full spectrum extends much further, including Ultra-Violet (UV),
X-rays, and gamma rays at shorter wavelengths, and InfraRed (IR), microwaves,
and radio waves at longer wavelengths. White light is made up of a broad mix
of visible light ranging from Red through Green to Blue (RGB).
In a vacuum, all these EM waves travel at the same speed , namely the speed
of light, customarily denoted as c, with a value c3105km/s = 3108m/s
= 31010cm/s. The wave period is the time it takes for a complete wavelength
to pass a
xed point at this speed, and so is given by P==c. We can thus
see that the sequence of wave crests passes by at a frequency of once per period,
= 1=P, implying a simple relationship between light's wavelength , frequency
, and speed c,
P==c: (4.1)
4.2 Light quanta and the Black-Body emission spectrum
The wave nature of light has been con
rmed by a wide range experiments. How-
Light Quanta and Black-Body Radiation
- Light exhibits a dual nature, behaving as both a wave with a specific frequency and wavelength and as discrete bundles of energy called photons.
- The energy of a photon is directly proportional to its frequency, a discovery by Planck and Einstein that established the quantization of energy.
- A Black Body is an idealized material in thermodynamic equilibrium that emits a Spectral Energy Distribution (SED) dependent solely on its temperature.
- The Planck function describes how the intensity or surface brightness of this radiation is distributed across different wavelengths or frequencies.
- Wien's Displacement Law shows that the peak wavelength of emission shifts to shorter, bluer wavelengths as the temperature of the object increases.
- By measuring the peak wavelength of a star's light, astronomers can calculate its surface temperature, such as the Sun's 5800K temperature based on its 500 nm peak.
Each photon carries a discrete, indivisible 'quantum' of energy that depends on the wave frequency.
of light, customarily denoted as c, with a value c3105km/s = 3108m/s
= 31010cm/s. The wave period is the time it takes for a complete wavelength
to pass a
xed point at this speed, and so is given by P==c. We can thus
see that the sequence of wave crests passes by at a frequency of once per period,
= 1=P, implying a simple relationship between light's wavelength , frequency
, and speed c,
P==c: (4.1)
4.2 Light quanta and the Black-Body emission spectrum
The wave nature of light has been con
rmed by a wide range experiments. How-
ever, at the beginning of the 20th century, work by Einstein, Planck, and others
led to the realization that light waves are also quantized into discrete wave \bun-
dles" called photons . Each photon carries a discrete, indivisible \quantum" of
energy that depends on the wave frequency as
E=h; (4.2)
wherehisPlanck's constant , with value h6:610 27erg s = 6:610 34
Joule s.
This quantization of light (and indeed of all energy) has profound and wide-
ranging consequences, most notably in the current context for how thermally
emitted radiation is distributed in wavelength or frequency. This is known as the
\Spectral Energy Distribution" (SED). For a so-called Black Body { meaning
4.2 Light quanta and the Black-Body emission spectrum 25
5500K
5000K
4000K
3500K4500K
0 500 1000 1500 200005.0×1061.0×1071.5×1072.0×107
λ(nm)Bλ(erg/s/cm2/nm)
Figure 4.2 The Planck Black-Body Spectral Energy Distribution (SED) vs.
wavelength , plotted for various temperatures T.
idealized material that is readily able to absorb and emit radiation of all wave-
lengths {, Planck showed that as thermal motions of the material approach a
Thermodynamic Equilibrium (TE) in the exchange of energy between radiation
and matter, the SED can be described by a function that depends onlyon the gas
temperature T(and not, e.g., on the density, pressure, or chemical composition).
In terms of the wave frequency , this Planck Black-Body function takes the
form
B(T) =2h3=c2
eh=kT 1; (4.3)
wherekis Boltzmann's constant, with value k= 1:3810 16erg/K = 1:38
10 23Joule/K. For an interval of frequency between and+d, the quantity
Bdgives the emitted energy per unit time per unit area per unit solid an-
gle. This means the Planck Black-Body function is fundamentally a measure of
intensity orsurface brightness , withBrepresenting the distribution of surface
brightness over frequency , having CGS units erg/cm2/s/ster/Hz (and MKS
units W/m2/ster/Hz).
Sometimes it is convenient to instead de
ne this Planck distribution in terms of
the brightness distribution in a wavelength interval between and+d,Bd.
Requiring that this equals Bd, and noting that =c=impliesjd=dj=c=2,
we can use eqn. (4.3) to obtain
B(T) =2hc2=5
ehc=kT 1: (4.4)
26 Inferring Surface Temperature from a Star's Color and/or Spectrum
4.3 Inverse-temperature dependence of wavelength for peak
ux
Figure 4.2 plots the variation of Bvs. wavelength for various temperatures
T. Note that for higher temperature, the level of Bis higher at allwavelengths,
with greatest increases near the peak level.
Moreover, the location of this peak shifts to shorter wavelength with higher
temperature. We can determine this peak wavelength maxby solving the equa-
tiondB
d
=max0: (4.5)
Leaving the details as an exercise, the result is
max=2:9106nm K
T=290 nm
T=10;000K500 nm
T=T
; (4.6)
which is known as Wien's displacement law .
For example, the last equality uses the fact that the observed wavelength peak
in the Sun's spectrum is max;
500 nm, very near the the middle of the visible
spectrum.2We can solve for a Black-Body-peak estimate for the Sun's surface
temperature
T
=2:9106nm K
500 nm= 5800K: (4.7)
By similarly measuring the peak wavelength maxin other stars, we can likewise
derive an estimate of their surface temperature by
T=T
max;
max5800K500 nm
max: (4.8)
Measuring Stellar Temperatures
- Wien's displacement law provides a direct mathematical relationship between a star's peak wavelength and its surface temperature.
- The Sun's peak wavelength of 500 nm corresponds to a surface temperature of approximately 5800 K, placing its brightest emission in the visible spectrum.
- Human vision likely evolved to be most sensitive to the specific wavelengths where solar illumination is at its peak intensity.
- In practice, astronomers use photometric color systems like the Johnson UBV filters because measuring a full spectral energy distribution is difficult and time-consuming.
- The 'color index' (such as B-V) serves as a distance-independent diagnostic for temperature, where negative values indicate hotter, bluer stars.
This is not entirely coincidental, since our eyes evolved to use the wavelengths of light for which the solar illumination is brightest.
T=10;000K500 nm
T=T
; (4.6)
which is known as Wien's displacement law .
For example, the last equality uses the fact that the observed wavelength peak
in the Sun's spectrum is max;
500 nm, very near the the middle of the visible
spectrum.2We can solve for a Black-Body-peak estimate for the Sun's surface
temperature
T
=2:9106nm K
500 nm= 5800K: (4.7)
By similarly measuring the peak wavelength maxin other stars, we can likewise
derive an estimate of their surface temperature by
T=T
max;
max5800K500 nm
max: (4.8)
4.4 Inferring stellar temperatures from photometric colors
In practice, this is not quite the approach to estimating a star's temperature
that is most commonly used in astronomy, in part because with real SEDs, it
is relatively dicult to identify accurately the peak wavelength. Moreover in
surveying a large number of stars, it requires a lot more e
ort (and telescope
time) to measure the full SED, especially for relatively faint stars. A simpler,
more common method is just to measure the stellar color .
But rather than using the Red, Green, and Blue (RGB) colors we perceive
with our eyes, astronomers typically de
ne a set of standard colors that extend
to wavebands beyond just the visible spectrum. The most common example is
the Johnson 3-color UBV (Ulraviolet, Blue, Visible) system. The left panel of
2This is not entirely coincidental, since our eyes evolved to use the wavelengths of light for
which the solar illumination is brightest.
4.5 Questions and Exercises 27
U
B
V
Eye
200 300 400 500 600 700 800 9000.00.20.40.60.81.0
Wavelength(nm)Sensitivity
⊙ ⊙
-0.5 0.0 0.5 1.0 1.5 2.0500010 00015 00020 000
Color Index B-VSurface Temperture(K)
Figure 4.3 Left: Comparison of the spectral sensitivity of the human eye with those
the UBV
lters in the Johnson photometric color system. Right: Temperature
dependence of the B-V color for a Black-Body emitted spectrum. The circle dot
marks the solar values T
5800 K and ( B V)
0:656.
gure 4.3 compares the wavelength sensitivity of such UBV
lters to that of the
human eye. By passing the star's light through a standard set of
lters designed
to only let through light for the de
ned color waveband, the observed apparent
brightness in each
lter can be used to de
ne a set of color magnitudes, e.g.
mU;mB, andmV.
The standard shorthand is simply to denote these color magnitudes just by
the capital letter alone, viz. U, B, and V. The di
erence between two color
magnitudes, e.g. B VmB mV, is independent of the stellar distance,
but provides a direct diagnostic of the stellar temperature, sometimes called the
\color temperature".
Because a larger magnitude corresponds to a lower brightness, stars with a
positive B-V actually are less bright in the blue than in the visible, implying
a relatively lowtemperature. On the other hand, a negative B-V means blue
is brighter, implying a high temperature. The right panel of
gure 4.3 shows
how the temperature of a Black-Body varies with the B-V color of the emitted
Black-Body spectrum.
4.5 Questions and Exercises
Quick Question 1: Two photons have wavelength ratio 2=1= 2.
a. What is the ratio of their period P2=P1?
b. What is the ratio of their frequency 2=1?
c. What is the ratio of their energy E2=E1?
Quick Question 2:
28 Inferring Surface Temperature from a Star's Color and/or Spectrum
a. Estimate the temperature of stars with max=100, 300, 1000, and 3000 nm. (To
simplify the numerics, you may take T
6000 K.)
b. Conversely, estimate the peak wavelengths maxof stars with T=2000, 10,000,
and 60, 000 K.
c. What parts of the EM spectrum (i.e. UV, visible, IR) do each of these lie in?
Quick Question 3:
a. Assuming the Earth has an average temperature equal to that of typical spring
day, i.e. 50F, compute the peak wavelength of Earth's Black-Body radiation.
b. What part of the EM spectrum does this lie in?
Stellar Radius and Stefan-Boltzmann Law
- The Stefan-Boltzmann law establishes that the total energy emitted by a blackbody increases sharply with the fourth power of its temperature.
- Stellar luminosity is determined by the combination of a star's surface temperature and its total surface area.
- By measuring a star's flux and temperature, astronomers can mathematically derive its physical radius.
- The relationship between luminosity, temperature, and radius is often simplified by scaling values against those of the Sun.
- While blackbody radiation provides a foundational model, real stellar spectra are complicated by discrete absorption lines.
The Stefan-Boltzmann law is one of the linchpins of stellar astronomy.
a. Estimate the temperature of stars with max=100, 300, 1000, and 3000 nm. (To
simplify the numerics, you may take T
6000 K.)
b. Conversely, estimate the peak wavelengths maxof stars with T=2000, 10,000,
and 60, 000 K.
c. What parts of the EM spectrum (i.e. UV, visible, IR) do each of these lie in?
Quick Question 3:
a. Assuming the Earth has an average temperature equal to that of typical spring
day, i.e. 50F, compute the peak wavelength of Earth's Black-Body radiation.
b. What part of the EM spectrum does this lie in?
Exercise 1: UsingBd=Bdand the relationship between frequency and wave-
length, derive eqn. (4.4) from eqn. (4.3).
Exercise 2: Derive eqn. (4.6) from eqn. (4.4) using the de
nition (4.5).
5 Inferring Stellar Radius from
Luminosity and Temperature
We see from
gure 4.2 that, in addition to a shift toward shorter peak wavelength
max, a higher temperature also increases the overall brightness of blackbody
emission at allwavelengths. This suggests that the total energy emitted over all
wavelengths should increase quite sharply with temperature. Leaving the details
as an exercise for the reader, let us quantify this expectation by carrying out
the necessary spectral integrals to obtain the temperature dependence of the
Bolometric intensity of a blackbody
B(T)Z1
0B(T)d=Z1
0B(T)d=sbT4
; (5.1)
withsb= 25k4=(15h3c2) known as the Stefan-Boltzmann constant, with nu-
merical value sb= 5:6710 5erg/cm2/s/K4= 5:6710 8J/m2/s/K4.
If we spatially resolve a pure blackbody with surface temperature T, then
B(T) represents the Bolometric surface brightness we would observe from each
part of the visible surface.
5.1 Stefan-Boltzmann law for surface
ux from a blackbody
Combining eqns. (3.6) and (5.1), we see that the radiative
uxat the surface
radiusRof a blackbody is given by
FF(R) =B(T) =sbT4; (5.2)
which is known as the Stefan-Boltzman law .
The Stefan-Boltzmann law is one of the linchpins of stellar astronomy. If we
now relate the surface
ux to the stellar luminosity Lover the surface area 4 R2,
then applying this to the Stefan-Boltzmann law gives
L=sbT44R2; (5.3)
which is often more convenient to scale by associated solar values,
L
L
=T
T
4R
R
2
: (5.4)
30 Inferring Stellar Radius from Luminosity and Temperature
We can also use eqn. (5.3) to solve for the stellar radius,
R=r
L
4sbT4=s
F(d)
sbT4d; (5.5)
where the latter equation uses the inverse-square-law to relate the stellar radius
to the
ux F(d) and distance d, along with the surface temperature T.
For a star with a known distance d, e.g. by a measured parallax, measurement
of apparent magnitude gives the
ux F(d), while measurement of the peak wave-
lengthmaxor color (e.g. B-V) provides an estimate of the temperature T(see
gure 4.3). Applying these in eqn. (5.5), we can thus obtain an estimate of the
stellar radius R.
5.2 Questions and Exercises
Quick Question 1: Compute the luminosity L(in units of the solar luminosity L
),
absolute magnitude M, and peak wavelength max(in nm) for stars with (a) T=T
;
R= 10R
, (b)T= 10T
;R=R
, and (c)T= 10T
;R= 10R
. If these stars all
have a parallax of p= 0:001 arcsec, compute their associated apparent magnitudes m.
Quick Question 2: Suppose a star has a parallax p= 0:01 arcsec, peak wavelength
max= 250 nm, and apparent magnitude m= +5 . About what is its:
a. Distance d(in pc)?
b. Distance modulus m M?
c. Absolute magntidue M?
d. Luminosity L(inL
)?
e. Surface temperature T(inT
)?
f. RadiusR(inR
)?
g. Angular radius
(in radian and arcsec)?
h. Solid angle
(in steradian and arcsec2)?
i. Surface brightness relative to that of the Sun, B=B
?
6 Absorption Lines in Stellar Spectra
Figure 6.1 The Sun's spectrum, showing the complex pattern of absorption lines at
discrete wavelength or colors. [NOAO/AURA/NSF]
In reality stars are not perfect blackbodies, and so their emitted spectra don't
Stellar Absorption Line Spectra
- Stars are not perfect blackbodies because their spectra contain detailed signatures of elemental composition rather than just temperature.
- The interior heat of a star diffuses outward through a temperature gradient, interacting with atoms and ions in the cooler surface layers.
- Atomic energy levels are quantized like steps in a staircase, meaning atoms only efficiently absorb photons that match specific energy differences.
- The resulting missing light appears as a complex series of dark absorption lines when a star's spectrum is spread out by a prism.
- These absorption lines act as a unique fingerprint or barcode that identifies the specific elements and ionization stages present in a star's atmosphere.
- Laboratory measurements of known elements allow scientists to decode these stellar fingerprints to determine chemical composition.
As such the associated wavelengths of the absorption lines in a star's spectrum provide a direct 'fingerprint' — perhaps even more akin to a supermarket bar code — for the presence of that element in the star's atmosphere.
max= 250 nm, and apparent magnitude m= +5 . About what is its:
a. Distance d(in pc)?
b. Distance modulus m M?
c. Absolute magntidue M?
d. Luminosity L(inL
)?
e. Surface temperature T(inT
)?
f. RadiusR(inR
)?
g. Angular radius
(in radian and arcsec)?
h. Solid angle
(in steradian and arcsec2)?
i. Surface brightness relative to that of the Sun, B=B
?
6 Absorption Lines in Stellar Spectra
Figure 6.1 The Sun's spectrum, showing the complex pattern of absorption lines at
discrete wavelength or colors. [NOAO/AURA/NSF]
In reality stars are not perfect blackbodies, and so their emitted spectra don't
just depend on temperature, but contain detailed signatures of key physical
properties like elemental composition. The energy we see emitted from a stellar
surface is generated in the very hot interior and then di
uses outward, following
the strong temperature decline to the surface. The atoms and ions that absorb
and emit the light don't do so with perfect eciency at all wavelengths, which
is what is meant by the \black" in \blackbody". We experience this all the
time in our everyday world, which shows that di
erent objects have distinct
\color", meaning they absorb certain wavebands of light, and re
ect others. For
example, a green leaf re
ects some of the \green" parts of the visible spectrum
{ with wavelengths near 5100 A{ and absorbs most of the rest.
For atoms in a gas, the ability to absorb, scatter and emit light can likewise
32 Absorption Lines in Stellar Spectra
Figure 6.2 Illustration of principals for producing an emission vs. an absorption line
spectrum. The left panel shows that an incandescent light passed through a prism
generally produces a featureless continuum spectrum, but a cold gas placed in front of
this yields an absorption line spectrum. That same gas when heated and seen on its
own against a dark background produces the same pattern of lines, but now in
emission instead of absorption. The right panel shows heuristically how the relatively
cool gas in the surface layers of a star leads to an absorption line spectrum from the
star.
depend on the wavelength, sometimes quite sharply. Just as the energy of light is
quantized into discrete bundles called photons, the energy of electrons orbiting an
atomic nucleus have discrete levels, much like the steps in a staircase. Absorption
or scattering by the atom is thus much more ecient for those select few photons
with an energy that closely matches the energy di
erence between two of these
atomic energy levels.
The evidence for this is quite apparent if we examine carefully the actual spec-
trum emitted by any star. Although the overall \Spectral Energy Distribution"
(SED) discussed above often roughly
ts a Planck Black-Body function, careful
inspection shows that light is missing or reduced at a number of discrete wave-
lengths or colors. As illustrated in
gure 6.1 for the Sun, when the color spectrum
of light is spread out, for example by a prism or di
raction grating, this missing
light appears as a complex series of relatively dark \absorption lines".
Figure 6.2 illustrates how the absorption by relatively cool, low-density atoms
in the upper layers of the Sun or a star's atmosphere can impart this pattern of
absorption lines on the continuum, nearly Black-Body spectrum emitted by the
denser, hotter layers.
A key point here is that the discrete energies levels associated with atoms
of di
erent elements (or, as discussed below, di
erent \ionization stages" of a
given element) are quite distinct. As such the associated wavelengths of the
absorption lines in a star's spectrum provide a direct \
ngerprint" { perhaps
even more akin to a supermarket bar code { for the presence of that element in
the star's atmosphere. The code \key" can come from laboratory measurement
of the line-spectrum from known samples of atoms and ions, or, as discussed in
Stellar Fingerprints and Composition
- Absorption lines in a star's spectrum act as a unique 'barcode' or fingerprint for identifying specific chemical elements.
- The Sun and most stars are composed almost entirely of hydrogen (90.9%) and helium (8.9%), with all other elements making up only 0.2% of the total atoms.
- In astronomy, all elements heavier than hydrogen and helium are collectively referred to as 'metals,' accounting for a mass fraction of about 2%.
- Earth's composition mirrors the Sun's heavier elements, but our planet lost its lighter gases like hydrogen and helium due to its weaker gravity.
- Stellar spectral types (OBAFGKM) are determined by surface temperature, which dictates the ionization stages of the elements present.
The associated wavelengths of the absorption lines in a star's spectrum provide a direct 'fingerprint'—perhaps even more akin to a supermarket bar code—for the presence of that element in the star's atmosphere.
denser, hotter layers.
A key point here is that the discrete energies levels associated with atoms
of di
erent elements (or, as discussed below, di
erent \ionization stages" of a
given element) are quite distinct. As such the associated wavelengths of the
absorption lines in a star's spectrum provide a direct \
ngerprint" { perhaps
even more akin to a supermarket bar code { for the presence of that element in
the star's atmosphere. The code \key" can come from laboratory measurement
of the line-spectrum from known samples of atoms and ions, or, as discussed in
xA.1, from theoretical models of the atomic energy levels using modern principles
of quantum physics.
6.1 Elemental composition of the Sun and stars 33
H H
He He
Li Li
Be BeB BC C
N NO O
F FNe Ne
Na NaMg Mg
Al AlSi Si
P PS S
Cl ClAr Ar
K KCa Ca
Sc ScTi Ti
V VCr CrMn MnFe Fe
Co CoNi Ni
Cu CuZn Zn
Ga GaGe Ge
As AsSe Se
Br BrKr Kr
Rb RbSr Sr
Y YZr Zr
Nb NbMo MoRu Ru
Rh RhPd Pd
Ag AgCd Cd
In InSn Sn
Sb SbTe Te
I IXe Xe
Cs CsBa Ba
La LaCe Ce
Pr PrNd NdSm Sm
Eu EuGd Gd
Tb TbDy Dy
Ho HoEr Er
Tm TmYb Yb
Lu LuHf Hf
Ta TaW WRe ReOs OsIr IrPt Pt
Au AuHg HgTl TlPb Pb
Bi BiTh ThU U
0 20 40 60 8010-1110-810-50.01
Atomic NumberRelative Abundance by Number
Figure 6.3 Number fractions of elements, plotted on a log scale vs. atomic number,
with data points labeled by the symbols for each element.
6.1 Elemental composition of the Sun and stars
With proper physical modeling, the relative strengths of the absorption lines
can even provide a quantitative measure of the relative abundance of the various
elements. A key result is that the composition of the Sun, which is typical of
most all stars, is dominated by just the two simplest elements, namely Hydrogen
(H) and Helium (He) { which make up respectively 90.9% and 8.9% of the atoms,
with all the other only about 0.2%. Figure 6.3 gives a log plot of these number
fractions vs. atomic number.
The corresponding mass fractions are X0:72 andY0:26 for Hydrogen
and Helium. All the remaining elements of the periodic table { commonly referred
to in astronomy as \metals" { make up just the
nal two percent of the mass.,
denoted as a \metalicity" Z0:02. Of these, the most abundant are Oxygen,
Carbon, and Iron, with respective mass fractions of 0.009, 0.003, and 0.001.
Like all the planets in our solar system, the Earth formed out of the same
material that makes up the Sun ( x23). But its relatively weak gravity has allowed
a lot of the light elements like Hydrogen and Helium to escape into space, leaving
behind the heavier elements that make up our world, and us ( x24). Indeed, once
the H and He are removed, the relative abundances of all these s elements are
roughly the same on the Earth as in the Sun!
34 Absorption Lines in Stellar Spectra
6.2 Stellar spectral type: ionization abundances as temperature
diagnostic
Figure 6.4 Stellar spectra for the full range of spectral types OBAFGKM,
corresponding to a range in stellar surface temperature from hot to cool.
[NOAO/AURA/NSF]
Another key factor in the observed stellar spectra is that the atomic elements
present are generally not electrically neutral, but typically have had one or more
electrons stripped { ionized| by thermal collisions with characteristic energies
set by the temperature. As such, the observed degree of ionization depends on
the temperature near the visible stellar surface. Figure 6.4 compares the spec-
tra of stars of di
erent surface temperature, showing that this leads to gradual
changes and shifts in the detailed pattern of absorption lines from the various
ionizations stages of the various elements. The letters \OBAFGKM" represent
various categories, known as spectral class or \spectral type", assigned to stars
with di
erent spectral patterns. It turns out that type O is the hottest, with tem-
peratures about 50,000 K, while M is the coolest1with temperatures of about
Spectral Classification and H-R Diagrams
- Stars are categorized into spectral types (OBAFGKM) based on surface temperature, ranging from 50,000 K to 3,500 K.
- Luminosity classes, denoted by Roman numerals I through V, distinguish between massive supergiants and smaller dwarf stars.
- The Sun is classified as a G2V star, representing an average temperature and a dwarf-scale luminosity.
- Absorption lines in stellar spectra serve as markers for measuring the Doppler effect and identifying chemical composition.
- The Hertzsprung-Russell (H-R) diagram is a fundamental tool that relates a star's luminosity to its surface temperature.
- Brown dwarfs (classes LTY) represent a bridge between the coolest stars and gas giant planets like Jupiter.
In summary, the appearance of absorption lines in stellar spectra provides a real treasure trove of clues to the physical properties of stars.
the temperature near the visible stellar surface. Figure 6.4 compares the spec-
tra of stars of di
erent surface temperature, showing that this leads to gradual
changes and shifts in the detailed pattern of absorption lines from the various
ionizations stages of the various elements. The letters \OBAFGKM" represent
various categories, known as spectral class or \spectral type", assigned to stars
with di
erent spectral patterns. It turns out that type O is the hottest, with tem-
peratures about 50,000 K, while M is the coolest1with temperatures of about
3500 K. The sequence is often remembered through the mnemonic2\Oh, Be A
Fine Gal/Guy Kiss Me". In keeping with its status as a kind of average star,
the Sun has spectral type G, just a bit cooler than type F in the middle of the
sequence.
In addition to the spectral classes OBAFGKM that depend on surface temper-
atureT, spectra can also be organized in terms of luminosity classes, convention-
1In recent years, it has become possible to detect even cooler \Brown dwarf" stars, with
spectral classes LTY, extending down to temperatures as low as 1000 K. Brown dwarf
stars have too low a mass ( <0:08M
) to force hydrogen fusion in their interior (see
x16.3). They represent a link to gas giant planets like Jupiter (for which MJ0:001M
).
2A student in one of my exams once o
ered an alternative mnemonic: \Oh Boy, Another
F's Gonna Kill Me".
6.3 Hertzsprung-Russell (H-R) diagram 35
ally denoted though Roman numerals I for the biggest, brightest \supergiant"
stars, to V for smaller, dimmer \dwarf" stars; in between, there are luminosity
classes II (bright giants), III (giants), and IV (sub-giants).
In this two-parameter scheme, the Sun is classi
ed as a G2V star.
Finally, in addition to giving information on the temperature, chemical compo-
sition, and other conditions of a star's atmosphere, these absorption lines provide
convenient \markers" in the star's spectrum. As discussed in x9.2, this makes
it possible to track small changes in the wavelength of lines that arise from the
so-called Doppler e
ect as a star moves toward or away from us.
In summary, the appearance of absorption lines in stellar spectra provides a
real treasure trove of clues to the physical properties of stars.
Figure 6.5 Left: Hertzsprung-Russel (H-R) diagram relating star's absolute magnitude
(or log luminosity ) vs. surface temperature, as characterized by the spectral type or
color, with hotter bluer stars on the left, and cooler redder stars on the right. The
main sequence (MS) represents stars burning Hydrogen into Helium in their core,
whereas the giants are supergiants are stars that have evolved away from the MS
after exhausting Hydrogen in their cores. The White Dwarf stars are dying remnants
of solar-type stars. Right: Observed H-R diagram for stars in the solar neighborhood.
The points include 22,000 stars from the Hipparcos Catalogue together with 1000
low-luminosity stars (red and white dwarfs) from the Gliese Catalogue of Nearby
Stars.
6.3 Hertzsprung-Russell (H-R) diagram
A key diagnostic of stellar populations comes from the Hertzsprung-Russel (H-
R) diagram, illustrated by the left panel of
gure 6.5. Observationally, it relates
(absolute) magnitude (or luminosity class) on the y-axis, to color or spectral
36 Absorption Lines in Stellar Spectra
type on the x-axis; physically, it relates luminosity to temperature. For stars in
the solar neighborhood with parallaxes measured by the Hipparchus astrometry
satellite, one can readily use the associated distance to convert observed apparent
magnitudes to absolute magnitudes and luminosities. The right panel of
gure 6.5
The Hertzsprung-Russell Diagram
- The H-R diagram relates a star's absolute magnitude or luminosity to its color, spectral type, or temperature.
- The main sequence represents the longest phase of a star's life, characterized by stable hydrogen burning in the core.
- Giant and supergiant stars represent later evolutionary stages where stars burn heavier elements or hydrogen in shells.
- White dwarfs are the final, cooling remnants of low-mass stars like the Sun, positioned below the main sequence.
- The H-R diagram serves as a vital link between observable surface light and the physical evolution of a star's interior.
- Mass and age are identified as the two primary parameters that differentiate stars across the diagram's various regions.
The reason there are so many stars in this main-sequence band is that it represents the long-lived phase when stars are stably burning Hydrogen into Helium in their cores.
R) diagram, illustrated by the left panel of
gure 6.5. Observationally, it relates
(absolute) magnitude (or luminosity class) on the y-axis, to color or spectral
36 Absorption Lines in Stellar Spectra
type on the x-axis; physically, it relates luminosity to temperature. For stars in
the solar neighborhood with parallaxes measured by the Hipparchus astrometry
satellite, one can readily use the associated distance to convert observed apparent
magnitudes to absolute magnitudes and luminosities. The right panel of
gure 6.5
shows the H-R diagram for these stars, plotting their known luminosities vs. their
colors or spectral types, with the horizontal lines showing the luminosity classes3.
The extended band of stars running from the upper left to lower right is known
as the main sequence , representing \dwarf" stars of luminosity class V. The
reason there are so many stars in this main-sequence band is that it represents
the long-lived phase when stars are stably burning Hydrogen into Helium in their
cores (x18).
The medium horizontal band above the main sequence represents \giant stars"
of luminosity class III. They are typically stars that have exhausted hydrogen in
their core, and are now getting energy from a combination of hydrogen burning in
a shell around the core, and burning Helium into Carbon in the cores themselves
(x19).
The relative lack here of still more luminous supergiant stars of luminosity class
I stems from both the relative rarity of stars with suciently high mass to become
this luminous, coupled with the fact that such luminous stars only live for a very
short time (x8.4). As such, there are only a few such massive, luminous stars
in the solar neighborhood. Studying them requires broader surveys extending to
larger distances that encompass a greater fraction of our galaxy.
The stars in the band below the main sequence are called white dwarfs ; they
represent the slowly cooling remnant cores of low-mass stars like the Sun ( x19.4).
This association between position on the H-R diagram, and stellar parameters
and evolutionary status, represents a key link between the observable properties
of light emitted from the stellar surface and the physical properties associated
with the stellar interior. Understanding this link through examination of stellar
structure and evolution will constitute the major thrust of our studies of stellar
interiors in part II of these notes.
But before we can do that, we need to consider ways that we can empirically
determine the two key parameters di
erentiating the various kinds of stars on
this H-R diagram, namely mass and age.
6.4 Questions and Exercises
Quick Question 1: On the H-R diagram, where do we
nd stars that are: a.) Hot and
luminous? b.) Cool and luminous? c.) Cool and Dim? d.) Hot and Dim?
Which of these are known as: 1.) White Dwarfs? 2.) Red Giants? 3.) Blue supergiants?
4.) Red dwarfs?
3The more recent GAIA satellite has provided an even more extensive H-R diagram
representing more than 4 million stars within 5000 pc. See
https://sci.esa.int/web/gaia/-/60198-gaia-hertzsprung-russell-diagram.
7 Surface Gravity and Escape/Orbital
Speed
So far we've been able to
nds ways to estimate the
rst
ve stellar parameters on
our list { distance, luminosity, temperature, radius, and elemental composition.
Moreover, we've done this with just a few, relatively simple measurements {
parallax, apparent magnitude, color, and spectral line patterns. But along the
way we've had to learn to exploit some key geometric principles and physical
laws { angular-size/parallax, inverse-square law, and Planck's, Wien's and the
Stellar Mass and Surface Gravity
- The text transitions from measuring stellar distance and luminosity to the fundamental physical parameter of stellar mass.
- Newton's law of gravitation provides the mathematical framework for calculating surface gravity based on a star's mass and radius.
- The Sun's surface gravity is approximately 27 times that of Earth, significantly increasing the theoretical weight of any object on its surface.
- Red Giant stars possess extremely low surface gravity due to their massive expansion, often leading to the loss of their outer envelopes into space.
- Compact remnants like white dwarfs and neutron stars exhibit surface gravities thousands to millions of times stronger than the Sun due to their extreme density.
- Surface gravity serves as a precursor to understanding more complex concepts like escape velocity and orbital speeds in binary systems.
Imagine what you'd weigh then on the surface of a neutron star!
7 Surface Gravity and Escape/Orbital
Speed
So far we've been able to
nds ways to estimate the
rst
ve stellar parameters on
our list { distance, luminosity, temperature, radius, and elemental composition.
Moreover, we've done this with just a few, relatively simple measurements {
parallax, apparent magnitude, color, and spectral line patterns. But along the
way we've had to learn to exploit some key geometric principles and physical
laws { angular-size/parallax, inverse-square law, and Planck's, Wien's and the
Stefan-Boltzman laws of blackbody radiation.
So what of the next item on the list, namely stellar mass? Mass is clearly a
physically important parameter for a star, since for example it will help determine
the strength of the gravity that tries to pull the star's matter together. To lay
the groundwork for discussing one basic way we can determine mass (from orbits
of stars in stellar binaries), let's
rst review the Newton's law of gravitation and
show how this sets such key quantities like the surface gravity, and the speeds
required for material to escape or orbit the star.
7.1 Newton's law of gravitation and stellar surface gravity
On Earth, an object of mass mhas a weight given by
Fgrav=mge; (7.1)
where the acceleration of gravity on Earth is ge= 980cm=s2= 9:8m=s2. But
this comes from Newtons's law of gravity, which states that for two point masses
mandMseparated by a distance r, the attractive gravitational force between
them is given by
Fgrav=GMm
r2; (7.2)
where Newton's constant of gravity is G= 6:710 8cm3=g=s2. Remarkably,
when applied to spherical bodies of mass Mand
nite radius R, the same formula
works for all distances rRat or outside the surface!1Thus, we see that the
1Even more remarkably, even if we are inside the radius, r<R , then we can still use
Newton's law if we just count that part of the total mass that is insider, i.e.Mr, and
completely ignore all the mass that is above r.
38 Surface Gravity and Escape/Orbital Speed
acceleration of gravity at the surface of the Earth is just given by the mass and
radius of the Earth through
ge=GMe
R2e: (7.3)
Similarly for stars, the surface gravity is given by the stellar mass Mand radius
R. In the case of the Sun, this gives g
= 2:6104cm/s227ge. Thus, if you
could stand on the surface of the Sun, your \weight" would be about 27 times
what it is on Earth.
For other stars, gravities can vary over a quite wide range, largely because of
the wide range in size. For example, when the Sun get's near the end of its life
about 5 billion years from now, it will swell up to more than 100 times its current
radius, becoming what's known as a \Red Giant" ( x19). Stars we see now that
happen to be in this Red Giant phase thus tend to have quite low gravity, about
a fraction 1/10,000 that of the Sun.
Largely because of this very low gravity, much of the outer envelope of such
Red Giant stars will actually be lost to space (forming, as we shall see, quite
beautiful nebulae; see x19 and
gure 20.5.) When this happens to the Sun, what's
left behind will be just the hot stellar core, a so-called \white dwarf", with about
2/3 the mass of the current Sun, but with a radius only about that of the Earth,
i.e.RRe7103km0:01R
. The surface gravities of white dwarfs are
thus typically 10 ;000 times higher than the current Sun ( x19.4).
For \neutron stars", which are the remnants of stars a bit more massive than
the Sun, the radius is just about 10 km, more than another factor 500 smaller
than white dwarfs ( x20.3). This implies surface gravities another 5-6 orders of
magnitude higher than even white dwarfs. (Imagine what you'd weigh then on
the surface of a neutron star!)
Gravity and Orbital Mechanics
- Stellar remnants like white dwarfs and neutron stars exhibit extreme surface gravities, with neutron stars reaching ten billion times the gravity of Earth.
- The log g scale provides a standardized way to compare gravitational strength across diverse celestial bodies, from Red Giants to dense stellar cores.
- Escape speed represents the velocity required for an object to overcome a body's gravitational pull and reach an infinite distance.
- Circular orbital speed is mathematically related to escape speed, specifically being the escape speed divided by the square root of two.
- The Virial Theorem establishes a fundamental relationship in bound orbits where kinetic energy is equal to half of the absolute gravitational binding energy.
Imagine what you'd weigh then on the surface of a neutron star!
left behind will be just the hot stellar core, a so-called \white dwarf", with about
2/3 the mass of the current Sun, but with a radius only about that of the Earth,
i.e.RRe7103km0:01R
. The surface gravities of white dwarfs are
thus typically 10 ;000 times higher than the current Sun ( x19.4).
For \neutron stars", which are the remnants of stars a bit more massive than
the Sun, the radius is just about 10 km, more than another factor 500 smaller
than white dwarfs ( x20.3). This implies surface gravities another 5-6 orders of
magnitude higher than even white dwarfs. (Imagine what you'd weigh then on
the surface of a neutron star!)
Since stellar gravities vary over such a large range, it is customary to quote
them in terms of the log of the gravity, log g, using CGS units. We thus have
gravities ranging from log g0 for Red Giants, to log g4 for normal stars
like the Sun, to log g8 for white dwarfs, to log g13 for neutron stars. Since
the Earth's gravity has log ge3, the di
erence of log gfrom 3 is the number of
order of magnitudes more/less that you'd weigh on that surface. For example,
for neutron stars the di
erence from Earth is 10, implying you'd weigh 1010, or
ten billion times more on a neutron star! On the other hand, on a Red Giant,
your weight would be about 1000 times lessthan on Earth.
7.2 Surface escape speed Vesc
Another measure of the strength of a gravitational
eld is through the surface
escape speed,
Vesc=r
2GM
R: (7.4)
7.3 Speed for circular orbit 39
A object of mass mlaunched with this speed has a kinetic energy mV2
esc=2 =
GMm=R . This just equals the work needed to lift that object from the surface
radiusRto escape at a large radius r!1 ,
W=Z1
RGMm
r2dr=GMm
R: (7.5)
Thus if one could throw a ball (or launch a rocket!) with this speed outward
from a body's surface radius R, then2by conservation of total energy, that ob-
ject would reach an arbitrarily large distance from the star, with however a
vanishingly small
nal speed.
For the earth, the escape speed is about 25,000 mph, or 11.2 km/s. By compar-
ison, for the moon, it is just 2.4 km/s, which is one reason the Apollo astronauts
could use a much smaller rocket to get back from the moon, than they used to
get there in the
rst place. However, escaping from the surface of the Sun (and
most any star), is much harder, requiring an escape speed of 618 km/s.
7.3 Speed for circular orbit
Let us next compare this escape speed with the speed needed for an object to
maintain a circular orbit at some radius rfrom the center a gravitating body of
massM. For an orbiting body of mass m, we require that the gravitational force
be balanced by the centrifugal force from moving along the circle of radius r,
GMm
r2=mV2
orb
r; (7.6)
which solves to
Vorb(r) =r
GM
r: (7.7)
Note in particular that the orbital speed very near the stellar surface, rR,
is given by Vorb(R) =Vesc=p
2. Thus the speed of satellites in low-earth-orbit
(LEO) is about 17,700 mph, or 7.9 km/s.
Of course, orbits can also be maintained at any radius above the surface radius,
r > R , and eqn. (7.7) shows that in this case, the speed needed declines as
1=pr. Thus, for example, the orbital speed of the earth around the Sun is about
30 km/s, a factor ofp
R
=au=p
1=215 = 0:0046 smaller than the orbital speed
near the Sun's surface, Vorb;
= 434 km/s.
7.4 Virial Theorum for bound orbits
If we de
ne the gravitational energy to be zero far from a star, then for an object
of massmat a radius rfrom a star of mass M, we can write the gravitational
2neglecting forces other than gravity, like the drag from an atmosphere
40 Surface Gravity and Escape/Orbital Speed
binding energy Uas the negative of the escape energy,
U(r) = GMm
r: (7.8)
If this same object is in orbit at this radius r, then the kinetic energy of the orbit
is
T(r) =mV2
orb
2= +GMm
2r= U(r)
2; (7.9)
where the second equation uses eqn. (7.7) for the orbital speed Vorb(r). We can
Gravitational Energy and the Virial Theorem
- The text defines gravitational binding energy as the negative of the energy required for an object to escape a star's pull.
- For a stable orbit, kinetic energy is shown to be exactly half of the absolute value of the gravitational binding energy.
- The Virial Theorem states that total energy in a stably bound system equals half of the gravitational binding energy, a principle applicable to both orbits and internal stellar structures.
- The theorem extends to stars by treating internal thermal energy as a form of kinetic energy that balances self-gravity.
- Mathematical exercises are provided to calculate surface gravity, escape speeds, and orbital velocities for various stellar masses and radii.
- The text transitions into the historical search for stellar energy sources, questioning if chemical burning could sustain a star's lifespan.
This fact that the total energy E just equals half the gravitational binding energy U is an example of what is known as the Virial Theorem.
ne the gravitational energy to be zero far from a star, then for an object
of massmat a radius rfrom a star of mass M, we can write the gravitational
2neglecting forces other than gravity, like the drag from an atmosphere
40 Surface Gravity and Escape/Orbital Speed
binding energy Uas the negative of the escape energy,
U(r) = GMm
r: (7.8)
If this same object is in orbit at this radius r, then the kinetic energy of the orbit
is
T(r) =mV2
orb
2= +GMm
2r= U(r)
2; (7.9)
where the second equation uses eqn. (7.7) for the orbital speed Vorb(r). We can
then write the total energy as
E(r)T(r) +U(r) = T(r) =U(r)
2: (7.10)
This fact that the total energy Ejust equals halfthe gravitational binding energy
Uis an example of what is known as the Virial Theorum . It is applicable broadly
to most any stably bound gravitational system. For example, if we recognize that
the thermal energy inside a star as a kind of kinetic energy, it even applies to
stars, in which the internal gas pressure balances the star's own self gravity. This
is discussed further in x8.2 and the part II notes on stellar structure.
7.5 Questions and Exercises
Quick Question 1: In CGS units, the Sun has log g
4:44. Compute the log gfor
stars with:
a.M= 10M
andR= 10R
b.M= 1M
andR= 100R
c.M= 1M
andR= 0:01R
Quick Question 2:
The Sun has an escape speed of Ve
= 618 km/s. Compute the escape speed Veof
the stars in parts a-c of QQ1.
Quick Question 3:
The earth has an orbital speed of Ve= 2au/yr = 30 km/s. Compute the orbital
speedVorb(in km/s) of a body at the following distances from the stars with the quoted
masses:
a.M= 10M
andd= 10 au.
b.M= 1M
andd= 100 au.
c.M= 1M
andd= 0:01 au.
Exercise 1:
a. During a solar eclipse, the moon just barely covers the visible disk of the Sun.
What does this tell you about the relative angular size of the Sun and moon?
b. Given that the moon is at a distance of 0.0024 au, what then is the ratio of the
physical size of the moon vs. Sun?
7.5 Questions and Exercises 41
c. Compared to earth, the Sun and moon have gravities of respectively 27 geand
ge=6. Using this and your answer above, what is the ratio of the mass of the moon to
that of the Sun?
d. Using the above, plus known values for Newton's constant G, earth's gravity
ge= 9:8 m/s2, and the solar radius R
= 700;000 km, compute the masses of the Sun
and moon in kg.
Exercise 2:
a. What is the ratio of the energy needed to escape the moon vs. the earth? What's
the ratio for the Sun vs. the earth?
b. What is the escape speed (in km/s) from a star with: (1) M= 10M
andR=
10R
; (2)M= 1M
andR= 100R
; (3)M= 1M
andR= 0:01R
?
c. To what radius (in km) would you have to shrink the Sun to make its escape speed
equal to the speed of light c?
Exercise 3:
a. What is the ratio of the energy needed to escape the the earth vs. that needed to
reach LEO?
b. What is the orbital speed (in km/s) of a planet that orbits at a distance afrom a
star with mass M, given: (1) M= 10M
anda= 10 au; (2) M= 1M
anda= 100;
au; (3)M= 1M
anda= 0:01 au?
8 Stellar Ages and Lifetimes
In our list of basic stellar properties, let us next consider stellar age. Just how
old are stars like the Sun? What provides the energy that keeps them shining?
And what will happen to them as they exhaust various available energy sources?
8.1 Shortness of chemical burning timescale for Sun and stars
When 19th century scientists pondered the possible energy sources for the Sun,
some
rst considered whether this could come from the kind of chemical reactions
(e.g., from fossil fuels like coal, oil, natural gas, etc.) that power human activities
on Earth. But such chemical reactions involve transitions of electrons among
Stellar Energy and Timescales
- Nineteenth-century scientists initially explored chemical reactions as a solar energy source but found they could only power the Sun for about 15,000 years.
- The Kelvin-Helmholtz timescale suggests that gravitational contraction could power the Sun for 30 million years by converting potential energy into radiation.
- Geological evidence of the Earth's age eventually proved that both chemical and gravitational energy sources are insufficient to explain solar longevity.
- Nuclear fusion provides the actual energy source, operating with a mass-energy efficiency roughly seven million times greater than chemical reactions.
- By fusing hydrogen into helium in its core, a star like the Sun can maintain its luminosity for approximately 10 billion years.
Even in the 19th century, it was clear, e.g. from geological processes like erosion, that the Earth — and so presumably also the Sun — had to be much older than this.
In our list of basic stellar properties, let us next consider stellar age. Just how
old are stars like the Sun? What provides the energy that keeps them shining?
And what will happen to them as they exhaust various available energy sources?
8.1 Shortness of chemical burning timescale for Sun and stars
When 19th century scientists pondered the possible energy sources for the Sun,
some
rst considered whether this could come from the kind of chemical reactions
(e.g., from fossil fuels like coal, oil, natural gas, etc.) that power human activities
on Earth. But such chemical reactions involve transitions of electrons among
various bound states of atoms, and, as discussed below ( xA.1) for the Bohr
model of the Hydrogen, the scale of energy release in such transitions is limited
to something on the order of an electron volt (eV). In contrast, the rest-mass
energy of the protons and neutrons that make up the mass is about 1 GeV, or
109times higher. With the associated mass-energy eciency of 10 9, we can
readily estimate a timescale for maintaining the solar luminosity from chemical
reactions,
tchem =M
c2
L
=4:51020s =1:51013yr15;000 yr: (8.1)
Even in the 19th century, it was clear, e.g. from geological processes like erosion,
that the Earth { and so presumably also the Sun { had to be much older than
this.
8.2 Kelvin-Helmholtz timescale for gravitational contraction
So let us consider whether, instead of chemical reactions, gravitational contrac-
tion might provide the energy source to power the Sun and other stars. As a star
undergoes a contraction in radius, its gravitational binding becomes stronger,
with a deeper gravitational potential energy, yielding an energy release set by
the negative of the change in gravitational potential ( dU > 0). If the contrac-
tion is gradual enough that the star roughly maintains dynamical equilibrium,
8.3 Nuclear burning timescale 43
then just half of the gravitational energy released goes into heating up the star1,
leaving the other half available to power the radiative luminosity, L= 1
2dU=dt .
For a star of observed luminosity Land present-day gravitational binding energy
U, we can thus de
ne a characteristic gravitational contraction lifetime,
tgrav= 1
2U
LtKH (8.2)
where the subscript \KH" refers to Kelvin and Helmholtz, the names of the
two scientists credited with
rst identifying this as an important timescale. To
estimate a value for the gravitational binding energy, let us consider the example
for the Sun under the somewhat arti
cial assumption that it has a uniform,
constant density, given by its mass over volume, =M
=(4R3
=3). Since the
gravity at any radius rdepends only on the mass m=4r3=3inside that
radius, the total gravitational binding energy of the Sun is given by integrating
the associated local gravitational potential Gm=r over all di
erential mass
shellsdm,
U=ZM
0Gm
rdm=162
3G2ZR
0r4dr=3
5GM2
R
; (8.3)
Applying this in eqn. (8.2), we
nd for the \Kelvin-Helmholtz" time of the
Sun,
tKH3
10GM2
R
L
30 Myr: (8.4)
Although substantially longer than the chemical burning timescale (8.1), this
is still much shorter than the geologically inferred minimum age of the Earth,
which is several Billion years.
8.3 Nuclear burning timescale
We now realize, of course, that the ages and lifetimes of stars like the Sun are set
by a much longer nuclear burning timescale. When four hydrogen nuclei are fused
into a helium nucleus, the helium mass is about 0 :7%lower than the original four
hydrogen. For nuclear fusion the above-de
ned mass-energy burning eciency
is thus now nuc0:007. But in a typical main sequence star, only some core
fractionf1=10 of the stellar mass is hot enough to allow Hydrogen fusion.
Applying this we thus
nd for the nuclear burning timescale
tnuc=nucfMc2
L10 GyrM=M
L=L
; (8.5)
Stellar Lifetimes and Cluster Ages
- The Sun's lifetime is determined by nuclear fusion efficiency, where 0.7% of mass is converted to energy, giving it a total lifespan of approximately 10 billion years.
- Currently 4.6 billion years old, the Sun is roughly halfway through its hydrogen-burning phase before it will exhaust its core fuel.
- Stellar luminosity scales steeply with mass (L ~ M³), meaning high-mass stars consume their fuel much faster and have significantly shorter lifespans than low-mass stars.
- The most massive stars may live for only 1 million years, a stark contrast to the multi-billion-year timescales of solar-mass stars.
- The age of a stellar cluster can be determined by identifying the 'turn-off point' on an H-R diagram, where stars begin to exhaust their hydrogen and exit the main sequence.
The most massive stars, of order 100 M, and thus with luminosities of order 106L, have main-sequence lifetimes of only about about 1 Myr, much shorter the multi-Gyr timescale for solar-mass stars.
We now realize, of course, that the ages and lifetimes of stars like the Sun are set
by a much longer nuclear burning timescale. When four hydrogen nuclei are fused
into a helium nucleus, the helium mass is about 0 :7%lower than the original four
hydrogen. For nuclear fusion the above-de
ned mass-energy burning eciency
is thus now nuc0:007. But in a typical main sequence star, only some core
fractionf1=10 of the stellar mass is hot enough to allow Hydrogen fusion.
Applying this we thus
nd for the nuclear burning timescale
tnuc=nucfMc2
L10 GyrM=M
L=L
; (8.5)
where Gyr109yr, i.e., a billion years, or a \Giga-year".
1This is another example of the Virial theorem for gravitationally bound systems, as
discussed in 7.4.
44 Stellar Ages and Lifetimes
We thus see that the Sun can live for about 10 Gyr by burning Hydrogen into
Helium in its core. It's present age of 4.6 Gyr2thus puts it roughly half way
through this Hydrogen-burning phase, with about 5.4 Gyr to go before it runs
out of H in its core.
8.4 Age of stellar clusters from main-sequence turno
point
As discussed below (see x10.4 and eqn. 10.11), observations of stellar binary
systems indicate that the luminosities of main-sequence stars scale with a high
power of the stellar mass { roughly LM3. In the present context, this implies
that high-mass stars should have much shorter lifetimes than low-mass stars.
If we make the reasonable assumption that the same
xed fraction ( f0:1) of
the total hydrogen mass of any star is available for nuclear burning into helium in
its stellar core, then the fuel available scales with the mass, while the burning rate
depends on the luminosity. Normalized to the Sun, the main-sequence lifetime
thus scales as
tms=tms;
M=M
L=L
10 GyrM
M2
: (8.6)
The most massive stars, of order 100 M
, and thus with luminosities of order
106L
, have main-sequence lifetimes of only about about 1 Myr, much shorter
the multi-Gyr timescale for solar-mass stars.
Figure 8.1 Left: H-R diagram for globular cluster M55, showing how stars on the
upper main sequence have evolved to lower temperature giant stars. Right: Schematic
H-R diagram for clusters, showing the systematic peeling o
of the main sequence
with increasing cluster age.
2As inferred, e.g., from radioactive dating of the oldest meteorites.
8.5 Questions and Exercises 45
This strong scaling of lifetime with mass can be vividly illustratied by plotting
the H-R diagram of stellar clusters. The H-R diagram plotted in
gure 6.5 is for
volume-limited sample near the Sun, consisting of stars of a wide range of ages,
distances, and perhaps even chemical composition. But stars often appear in
clusters, all roughly at the same distance, and, since they likely formed over a
relatively short time span out of the same interstellar cloud, they all have roughly
the same age and chemical composition. Using eqn. (8.6) together with the the
LM3relation, the age of a stellar cluster can be inferred from its H-R diagram
simply by measuring the luminosity Ltoof stars at the \ turn-o
" point from the
main sequence,
tcluster10 GyrL
Lto2=3
: (8.7)
The left panel of
gure 8.1 plots an actual H-R diagram for the globular cluster
M55. Note that stars to the upper left of the main sequence have evolved to a
vertical branch of cooler stars extending up to the Red Giants3. This re
ects the
fact that more luminous stars exhaust their hydrogen fuel sooner that dimmer
stars, as shown by the inverse luminosity scaling of the nuclear burning timescale
in eqn. (8.5). The right panel illustrates schematically the H-R diagrams for
various types of stellar clusters, showing how the turno
point from the main
sequence is an indicator of the cluster age. Observed cluster H-R diagrams like
this thus provide a direct diagnostic of the formation and evolution of stars with
various masses and luminosities.
8.5 Questions and Exercises
Stellar Lifetimes and Space Velocities
- The main-sequence turn-off point in H-R diagrams serves as a critical diagnostic for determining the age of stellar clusters.
- Luminous stars exhaust their hydrogen fuel significantly faster than dimmer stars due to inverse luminosity scaling.
- Blue stragglers are stars rejuvenated by mass transfer from a binary companion, making them appear younger and hotter than their peers.
- Stellar motion is measured through proper motion (transverse drift) and spectrometric techniques for radial velocity.
- Barnard's star exhibits the highest proper motion of any star, showing a detectable drift even with modest equipment.
This rejuvenated the mass gainer, making it again a hot, luminous blue star.
fact that more luminous stars exhaust their hydrogen fuel sooner that dimmer
stars, as shown by the inverse luminosity scaling of the nuclear burning timescale
in eqn. (8.5). The right panel illustrates schematically the H-R diagrams for
various types of stellar clusters, showing how the turno
point from the main
sequence is an indicator of the cluster age. Observed cluster H-R diagrams like
this thus provide a direct diagnostic of the formation and evolution of stars with
various masses and luminosities.
8.5 Questions and Exercises
Quick Question 1: What are the luminosities (in L
) and the expected main sequence
lifetimes (in Myr) of stars with masses: a. 10 M
? b. 0.1M
? c. 100M
?
Quick Question 2: Suppose you observe a cluster with a main-sequence turno
point
at a luminosity of 100 L
. What is the cluster's age, in Myr. What about for a cluster
with a turno
at a luminosity of 10 ;000L
?
Exercise 1: A cluster has a main-sequence turno
at a spectral type G2, corresponding
to stars of apparent magnitude m= +10.
(a) About what is the luminosity, in L
, of the stars at the turno
point?
(b) About what is the age (in Gyr) of the cluster?
(c) About what is the distance (in pc) of the cluster?
3Stars just above this main sequence turn-o
are dubbed \blue stragglers". They are stars
whose close binary companion became a Red Giant with a such big radius that mass from
its envelope spilled over onto it. This rejuvenated the mass gainer, making it again a hot,
luminous blue star.
46 Stellar Ages and Lifetimes
Exercise 2: Con
rm the integration result in eqn. (8.3).
9 Inferring Stellar Space Velocities
The next section ( x10) will use the inferred orbits of stars in binary star systems
to directly determine stellar masses. But
rst, as a basis for interpreting obser-
vations of such systems in terms of the orbital velocity of the component stars,
let us review the astrometric and spectrometric techniques used to measure the
motion of stars through space.
9.1 Transverse speed from proper motion observations
In addition to such periodic motion from binary orbits, stars generally also ex-
hibit some systematic motion relative to the Sun, generally with components
both transverse (i.e. perpendicular) to and along (parallel to) the observed line
of sight. For nearby stars, the perpendicular movement, called \proper motion",
can be observed as a drift in the apparent position in the star relative to the
more
xed pattern of more distant, background stars. Even though the associ-
ated physical velocities can be quite large, e.g. Vt10 100 km/s, the distances
to stars is so large that proper motions of stars { measured as an angular drift
per unit time, and generally denoted with the symbol { are generally no bigger
than about 1 arcsec/year. But because this is a systematic drift, the longer
the star is monitored, the smaller the proper motion that can be detected, down
to about1 arcsec/century or less for the most well-observed stars.
Figure 9.1 illustrates the proper motion for Barnard's star, which has the
highestvalue of any star in the sky. It is so high in fact, that its proper motion
can even be followed with a backyard telescope, as was done for this
gure. This
star is actually tracking along the nearly South-to-North path labeled as the
\Hipparcos1mean" in the
gure. The apparent, nearly East-West (EW) wobble
is due to the Earth's own motion around the Sun, and indeed provides a measure
of the star's parallax, and thus its distance. Referring to the arcsec marker in
the lower right, we can estimate the full amplitude of the wobble at a bit more
than an arcsec, meaning the parallax2isp0:55 arcsec, implying a distance
1Hipparcos is an orbiting satellite that, because of the absence of the atmospheric blurring,
can make very precise \astrometric" measurements of stellar positions, at precisions
Measuring Stellar Space Velocities
- The apparent wobble in a star's path, such as Barnard's star, is caused by Earth's orbital motion and provides a direct measure of stellar parallax.
- Stellar distance can be calculated from parallax, which then allows astronomers to convert observed proper motion into a physical transverse velocity.
- Barnard's star exhibits one of the fastest transverse speeds among nearby stars, moving at approximately 90 km/s.
- Radial velocity, the component of motion along the line of sight, is measured using the Doppler effect rather than direct positional shifts.
- The Doppler effect causes a shift in observed wavelength proportional to the object's speed relative to the speed of the signal, such as light or sound.
- In extreme cases where an object moves faster than the signal speed, such as supersonic travel, shock waves like sonic booms are created.
Consider the noise from a car on a highway, for which the 'vvvvrrrrrooomm' sound stems from just this shift in pitch from the car engine noise.
gure. The apparent, nearly East-West (EW) wobble
is due to the Earth's own motion around the Sun, and indeed provides a measure
of the star's parallax, and thus its distance. Referring to the arcsec marker in
the lower right, we can estimate the full amplitude of the wobble at a bit more
than an arcsec, meaning the parallax2isp0:55 arcsec, implying a distance
1Hipparcos is an orbiting satellite that, because of the absence of the atmospheric blurring,
can make very precise \astrometric" measurements of stellar positions, at precisions
approaching a milli-arcsec.
2given by half the full amplitude, since parallax assumes a 1 au baseline that is half the full
diameter of earth's orbit
48 Inferring Stellar Space Velocities
Figure 9.1 Proper motion of Barnard's star. The star is actually tracking along the
path labeled as the mean from the Hipparcos astrometric satellite. The apparent
wobble is due to the parallax from the Earth's own motion around the Sun. Referring
to lower right label showing one arcsec, we can estimate the full amplitude of the
parallax wobble as about 1.1 arcsec; but since this re
ects a baseline of 2 AU from
the earth's orbital diameter, the (one-AU) parallax angle is half this, or
p= 0:55 arcsec, implying a distance of d= 1=p1:8 pc.
ofd1:8 pc. By comparison, the roughly South-to-North proper motion has a
value10 arcsec/yr.
In general, with a known parallax pin arcsec, and known proper motion in
arcsec/yr, we can derive the associated transverse velocity Vtacross our line of
sight,
Vt=
pau=yr = 4:7
pkm=s; (9.1)
where the last equality uses the fact that the Earth's orbital speed VE= 2au=yr =
30 km/s. For Barnard's star this works out to give Vt90 km/s, or about 3
times the earth's orbital speed around the Sun. This among the fastest trans-
verse speeds inferred among the nearby stars.
9.2 Radial velocity from Doppler shift 49
9.2 Radial velocity from Doppler shift
We've seen how we can directly measure the transverse motion of relatively
nearby, fast-moving stars in terms of their proper motion. But how might we
measure the radial velocity component along our line of sight? The answer is:
via the \Doppler e
ect", wherein such radial motion leads to an observed shift
in the wavelength of the light.
To see how this e
ect comes about, we need only consider some regular signal
with period Pobeing emitted from an object moving at a speed Vrtoward
(Vr<0) or away ( Vr>0) from us. Let the signal travel at a speed Vs, where
Vs=cfor a light wave, but might equally as well be speed of sound if we were
to use that as an example. For clarity of language, let us assume the object is
moving away, with Vr>0. Then after any given pulse of the signal is emitted,
the object moves a distance VrPobefore emitting the next pulse. Since the pulse
still travels at the same speed, this implies it takes the second pulse an extra
time
P=VrPo
Vs(9.2)
to reach us. Thus the period we observe is longer, P0=Po+ P.
For a wave, the wavelength is given by =PVs, implying then an associated
stretch in the observed wavelength
0=P0Vs= (Po+ P)Vs= (Vs+Vr)Po=o+VrPo: (9.3)
whereo=PoVsis the rest wavelength. The associated relative stretch in wave-
length is thus just
o0 o
o=Vr
Vs: (9.4)
For sound waves, this formula works in principle as long as Vr>