An Introduction to Theoretical Astrophysics
Master’s Revision Sheet
Everything to know by heart before the final examination
Module PH2080 — An Introduction to Theoretical Astrophysics (5 CP)
Scope Cosmic scales; gravitation & the virial theorem; timescales; stellar structure;
equation of state & degenerate matter; radiative transfer & energy transport;
nuclear energy; stellar evolution & scaling; compact objects; hydrodynamics &
shocks; accretion & the Eddington limit; dark matter; neutrinos & supernovae.
Format Self-contained, exam-focused. Per chapter: core concepts, full formula sheet,
derivations, scaling/OOM, diagrams, likely exam questions. Then a high-yield
compilation and a last-day sheet.
Legend red bold = memorize exactly; blue bold = safe to re-derive in the exam. Boxes:
Key result, Memorize exactly, Derivation, Order-of-magnitude, Conceptual trap,
Physical interpretation.
A condensed self-study sheet. Worked derivations are complete; every symbol is defined; numerical estimates are carried
out explicitly so they can be reproduced under exam conditions. Constants are collected at the end of the last-day sheet.
Contents
1 Constants and the art of estimation 2
2 Cosmic structures, scales, and dimensional astrophysics 2
3 Gravitation, the two-body problem, and the virial theorem 4
4 Timescales in astrophysics 6
5 Stellar structure: the equations of stellar equilibrium 8
6 The equation of state of stellar matter 9
7 Energy transport and radiative transfer 11
8 Nuclear energy generation 13
9 Stellar evolution and scaling relations 15
10 Compact objects: white dwarfs, neutron stars, black holes 17
11 Hydrodynamics: fluids, sound, and shocks 19
12 Accretion physics and the Eddington limit 21
,Theoretical Astrophysics — Master Revision Sheet PH2080
13 Mass determination and dark matter 23
14 Neutrinos and core-collapse supernovae 25
15 High-yield compilation 27
16 Last-day revision sheet 36
1. Constants and the art of estimation
A handful of constants and one habit — dimensional analysis — underlie the whole course. Keep
these at your fingertips; almost every order-of-magnitude question is solved by combining them.
Physical constants to know to one significant figure
Constant Symbol Value (SI) In cgs / handy form
Gravitational constant G 6.67 × 10−11 m3 kg−1 s−2 —
Speed of light c 3.00 × 108 m s−1 —
Planck (reduced) h̄ 1.05 × 10−34 J s —
Boltzmann kB 1.38 × 10−23 J K−1 8.6 × 10−5 eV K−1
Proton/atomic mass m p , mu 1.67 × 10−27 kg —
Electron mass me 9.11 × 10−31 kg 511 keV/c2
Thomson cross-section σT 6.65 × 10−29 m2 —
Radiation constant a 7.57 × 10−16 J m−3 K−4 a = 4σSB /c
Stefan–Boltzmann σSB 5.67 × 10−8 W m−2 K−4 —
Solar mass M⊙ 2.0 × 1030 kg —
Solar radius R⊙ 7.0 × 108 m —
Solar luminosity L⊙ 3.8 × 1026 W —
Solar eff. temperature Teff,⊙ 5800 K —
Planck mass MPl 2.2 × 10−8 kg (h̄c/G )1/2
Astronomical unit AU 1.5 × 1011 m —
Parsec pc 3.1 × 1016 m 3.26 ly
Year yr 3.15 × 107 s π × 107 s (mnemonic)
2. Cosmic structures, scales, and dimensional astrophysics
2.1 Core concepts
Fundamental idea. The universe is organized into a hierarchy of self-gravitating structures, each
existing on a characteristic mass, length and density scale set by the balance between gravity (always
attractive, always present) and some pressure or kinetic support. Astrophysics is, to first order, the
study of this competition.
Physical interpretation. Each object “chooses” its scale where inward gravity equals outward sup-
port. Different support mechanisms (thermal gas pressure, degeneracy pressure, radiation pressure,
random stellar motions) produce the different rungs of the cosmic ladder: planets, stars, white dwarfs,
neutron stars, star clusters, galaxies, clusters of galaxies.
Key assumptions. (i) Newtonian gravity suffices except for compact objects and cosmology; (ii)
most systems are in (quasi-)hydrostatic or virial equilibrium; (iii) spherical symmetry is an excellent
first approximation; (iv) matter can be treated as a fluid or as a collisionless ensemble of point masses.
Connections. This chapter is the scaffold: the virial theorem (Ch. 2) quantifies “gravity vs. sup-
port”; stellar structure (Ch. 4) and the equation of state (Ch. 5) specify the support inside stars; com-
pact objects (Ch. 9) are where degeneracy and relativity set the scale; dark matter (Ch. 12) is inferred
precisely because the dynamical scale disagrees with the luminous one.
2
,Theoretical Astrophysics — Master Revision Sheet PH2080
2.2 Essential formula sheet
Formula Symbols, units, meaning Validity / exam
use
3M
ρ̄ = M mass [kg], R radius [m], ρ̄ mean density Always. Use to
4πR3
[kg m−3 ]. The single most useful substitution in eliminate ρ in
the course. every estimate.
GM
g= 2
surface gravity [m s−2 ]. Spherical mass.
Rr
2GM
vesc = escape speed [m s−1 ]. Test mass from
R
surface to infinity.
r
GM
vcirc = circular orbital speed. Bound circular
R
orbit; √
vesc = 2 vcirc .
GM2
Egrav ∼ − gravitational (binding) energy [J]. Exact for Energy budget of
R
uniform sphere: − 35 GM2 /R. any
self-gravitating
body.
Rs 2GM
= compactness; Rs = 2GM/c2 Schwarzschild Dimensionless
R Rc2
radius. “how relativistic”.
∼ 10−6 Sun, ∼ 0.4
NS.
D
θ= angular size [rad], D physical size, d distance. Small-angle;
d
converting observ.
to physical.
What “characteristic scale” means
A characteristic value is the order of magnitude obtained by setting the one-zone version of the
governing balance, replacing gradients d/dr → 1/R, masses m(r ) → M, densities ρ → ρ̄ =
M/R3 . It is never exact (prefactors depend on the density profile) but is almost always right to a
factor of a few — which is all an oral or estimate question asks for.
2.3 Fundamental derivations
Escape and circular speeds (must be instant)
Steps.
√ (1) Circular orbit: gravity provides centripetal force, GMm/R2 = √mv2 /R ⇒ vcirc =
GM/R. (2) Escape:√set kinetic = binding, 12 mv2 = GMm/R ⇒ vesc = 2GM/R. Shortcut.
Memorize the factor 2 between them.
2.4 Scaling relations and order-of-magnitude estimates
Characteristic masses, radii, densities — know these cold
Star (Sun): M ∼ 1030 kg, R ∼ 109 m, ρ̄ ∼ 103 kg m−3 (water). White dwarf: M ∼ M⊙ , R ∼ 107 m
(Earth), ρ̄ ∼ 109 kg m−3 . Neutron star: M ∼ 1.4 M⊙ , R ∼ 104 m, ρ̄ ∼ 1017 kg m−3 (nuclear). Galaxy:
M ∼ 1011 M⊙ , R ∼ 10 kpc. Cluster: M ∼ 1014 M⊙ , R ∼ Mpc.
Estimate (Schwarzschild radius of the Sun): Rs = 2GM/c2 ∼ 2(6.7 × 10−11 )(2 × 1030 )/(9 × 1016 ) ∼ 3 km
— hence compactness ∼ 10−6 , Newtonian gravity is safe for the Sun.
3
, Theoretical Astrophysics — Master Revision Sheet PH2080
2.5 Diagrams and concept maps
THE COSMIC LADDER (support mechanism sets each rung)
density rho
high | neutron star ........ neutron degeneracy + nuclear forces
^ | white dwarf ......... electron degeneracy
| | main-seq star ....... thermal gas + radiation pressure
| | brown dwarf ......... electron degeneracy (failed star)
| | planet ............. Coulomb (solid/liquid) pressure
low | star cluster ....... random stellar motions (collisional)
| galaxy ............ random motions + rotation (collisionless)
| galaxy cluster .... random galaxy motions (virialised, + DM)
+------------------------------------------------> size R
small large
MASTER BALANCE: GRAVITY (inward) <==> PRESSURE / KINETIC (outward)
| |
GM^2/R (energy) which P? -> EOS (Ch.5)
2.6 Typical exam questions
Conceptual: Why do self-gravitating objects exist only on discrete scales? What support holds up each
rung of the ladder? Why is the Sun’s interior safely Newtonian but a neutron star’s is not?
Calculation: Estimate the mean density of the Sun; of a white dwarf; of a neutron star. Compute
the Schwarzschild radius of the Sun and of a 108 M⊙ black hole. Given a galaxy’s angular size and
distance, find its physical radius. √
Derivation: Derive vesc and vcirc and the factor 2 between them.
3. Gravitation, the two-body problem, and the virial theorem
3.1 Core concepts
Fundamental ideas. (i) The two-body Kepler problem: bound orbits are ellipses, with period set by
the semi-major axis and total mass. (ii) The virial theorem: for any gravitationally bound system in
steady state, the time-averaged kinetic and potential energies obey 2⟨K ⟩ + ⟨U ⟩ = 0. This is the single
most powerful tool in the course — it converts the impossible problem “solve the full structure” into
the trivial one “balance two energies”.
Physical interpretation. The virial theorem says a bound system is exactly half as kinetically “hot”
as it is gravitationally “deep”: ⟨K ⟩ = − 21 ⟨U ⟩, so total energy E = ⟨K ⟩ + ⟨U ⟩ = 12 ⟨U ⟩ = −⟨K ⟩ < 0.
Bound means negative total energy; the deeper the well, the faster the constituents move.
Key assumptions. Isolated system (no external torque/force), bound (E < 0), steady state so the
long-term average of Ï vanishes, and U ∝ r −1 gravity (the −1 power is what gives the clean factor 2).
Connections. The virial theorem powers: stellar central temperatures (Ch. 4), the negative heat ca-
pacity and Kelvin–Helmholtz contraction (Ch. 3,4), cluster and galaxy masses and hence dark matter
(Ch. 12), and the Jeans criterion for collapse. Whenever a question says “estimate the temperature/ve-
locity/mass of a bound system”, reach for it first.
3.2 Essential formula sheet
Formula Symbols, units, meaning Validity / exam
use
GMm
F=− r̂ Newton’s law; r separation [m]. Point/spherical
r2
masses.
4
Master’s Revision Sheet
Everything to know by heart before the final examination
Module PH2080 — An Introduction to Theoretical Astrophysics (5 CP)
Scope Cosmic scales; gravitation & the virial theorem; timescales; stellar structure;
equation of state & degenerate matter; radiative transfer & energy transport;
nuclear energy; stellar evolution & scaling; compact objects; hydrodynamics &
shocks; accretion & the Eddington limit; dark matter; neutrinos & supernovae.
Format Self-contained, exam-focused. Per chapter: core concepts, full formula sheet,
derivations, scaling/OOM, diagrams, likely exam questions. Then a high-yield
compilation and a last-day sheet.
Legend red bold = memorize exactly; blue bold = safe to re-derive in the exam. Boxes:
Key result, Memorize exactly, Derivation, Order-of-magnitude, Conceptual trap,
Physical interpretation.
A condensed self-study sheet. Worked derivations are complete; every symbol is defined; numerical estimates are carried
out explicitly so they can be reproduced under exam conditions. Constants are collected at the end of the last-day sheet.
Contents
1 Constants and the art of estimation 2
2 Cosmic structures, scales, and dimensional astrophysics 2
3 Gravitation, the two-body problem, and the virial theorem 4
4 Timescales in astrophysics 6
5 Stellar structure: the equations of stellar equilibrium 8
6 The equation of state of stellar matter 9
7 Energy transport and radiative transfer 11
8 Nuclear energy generation 13
9 Stellar evolution and scaling relations 15
10 Compact objects: white dwarfs, neutron stars, black holes 17
11 Hydrodynamics: fluids, sound, and shocks 19
12 Accretion physics and the Eddington limit 21
,Theoretical Astrophysics — Master Revision Sheet PH2080
13 Mass determination and dark matter 23
14 Neutrinos and core-collapse supernovae 25
15 High-yield compilation 27
16 Last-day revision sheet 36
1. Constants and the art of estimation
A handful of constants and one habit — dimensional analysis — underlie the whole course. Keep
these at your fingertips; almost every order-of-magnitude question is solved by combining them.
Physical constants to know to one significant figure
Constant Symbol Value (SI) In cgs / handy form
Gravitational constant G 6.67 × 10−11 m3 kg−1 s−2 —
Speed of light c 3.00 × 108 m s−1 —
Planck (reduced) h̄ 1.05 × 10−34 J s —
Boltzmann kB 1.38 × 10−23 J K−1 8.6 × 10−5 eV K−1
Proton/atomic mass m p , mu 1.67 × 10−27 kg —
Electron mass me 9.11 × 10−31 kg 511 keV/c2
Thomson cross-section σT 6.65 × 10−29 m2 —
Radiation constant a 7.57 × 10−16 J m−3 K−4 a = 4σSB /c
Stefan–Boltzmann σSB 5.67 × 10−8 W m−2 K−4 —
Solar mass M⊙ 2.0 × 1030 kg —
Solar radius R⊙ 7.0 × 108 m —
Solar luminosity L⊙ 3.8 × 1026 W —
Solar eff. temperature Teff,⊙ 5800 K —
Planck mass MPl 2.2 × 10−8 kg (h̄c/G )1/2
Astronomical unit AU 1.5 × 1011 m —
Parsec pc 3.1 × 1016 m 3.26 ly
Year yr 3.15 × 107 s π × 107 s (mnemonic)
2. Cosmic structures, scales, and dimensional astrophysics
2.1 Core concepts
Fundamental idea. The universe is organized into a hierarchy of self-gravitating structures, each
existing on a characteristic mass, length and density scale set by the balance between gravity (always
attractive, always present) and some pressure or kinetic support. Astrophysics is, to first order, the
study of this competition.
Physical interpretation. Each object “chooses” its scale where inward gravity equals outward sup-
port. Different support mechanisms (thermal gas pressure, degeneracy pressure, radiation pressure,
random stellar motions) produce the different rungs of the cosmic ladder: planets, stars, white dwarfs,
neutron stars, star clusters, galaxies, clusters of galaxies.
Key assumptions. (i) Newtonian gravity suffices except for compact objects and cosmology; (ii)
most systems are in (quasi-)hydrostatic or virial equilibrium; (iii) spherical symmetry is an excellent
first approximation; (iv) matter can be treated as a fluid or as a collisionless ensemble of point masses.
Connections. This chapter is the scaffold: the virial theorem (Ch. 2) quantifies “gravity vs. sup-
port”; stellar structure (Ch. 4) and the equation of state (Ch. 5) specify the support inside stars; com-
pact objects (Ch. 9) are where degeneracy and relativity set the scale; dark matter (Ch. 12) is inferred
precisely because the dynamical scale disagrees with the luminous one.
2
,Theoretical Astrophysics — Master Revision Sheet PH2080
2.2 Essential formula sheet
Formula Symbols, units, meaning Validity / exam
use
3M
ρ̄ = M mass [kg], R radius [m], ρ̄ mean density Always. Use to
4πR3
[kg m−3 ]. The single most useful substitution in eliminate ρ in
the course. every estimate.
GM
g= 2
surface gravity [m s−2 ]. Spherical mass.
Rr
2GM
vesc = escape speed [m s−1 ]. Test mass from
R
surface to infinity.
r
GM
vcirc = circular orbital speed. Bound circular
R
orbit; √
vesc = 2 vcirc .
GM2
Egrav ∼ − gravitational (binding) energy [J]. Exact for Energy budget of
R
uniform sphere: − 35 GM2 /R. any
self-gravitating
body.
Rs 2GM
= compactness; Rs = 2GM/c2 Schwarzschild Dimensionless
R Rc2
radius. “how relativistic”.
∼ 10−6 Sun, ∼ 0.4
NS.
D
θ= angular size [rad], D physical size, d distance. Small-angle;
d
converting observ.
to physical.
What “characteristic scale” means
A characteristic value is the order of magnitude obtained by setting the one-zone version of the
governing balance, replacing gradients d/dr → 1/R, masses m(r ) → M, densities ρ → ρ̄ =
M/R3 . It is never exact (prefactors depend on the density profile) but is almost always right to a
factor of a few — which is all an oral or estimate question asks for.
2.3 Fundamental derivations
Escape and circular speeds (must be instant)
Steps.
√ (1) Circular orbit: gravity provides centripetal force, GMm/R2 = √mv2 /R ⇒ vcirc =
GM/R. (2) Escape:√set kinetic = binding, 12 mv2 = GMm/R ⇒ vesc = 2GM/R. Shortcut.
Memorize the factor 2 between them.
2.4 Scaling relations and order-of-magnitude estimates
Characteristic masses, radii, densities — know these cold
Star (Sun): M ∼ 1030 kg, R ∼ 109 m, ρ̄ ∼ 103 kg m−3 (water). White dwarf: M ∼ M⊙ , R ∼ 107 m
(Earth), ρ̄ ∼ 109 kg m−3 . Neutron star: M ∼ 1.4 M⊙ , R ∼ 104 m, ρ̄ ∼ 1017 kg m−3 (nuclear). Galaxy:
M ∼ 1011 M⊙ , R ∼ 10 kpc. Cluster: M ∼ 1014 M⊙ , R ∼ Mpc.
Estimate (Schwarzschild radius of the Sun): Rs = 2GM/c2 ∼ 2(6.7 × 10−11 )(2 × 1030 )/(9 × 1016 ) ∼ 3 km
— hence compactness ∼ 10−6 , Newtonian gravity is safe for the Sun.
3
, Theoretical Astrophysics — Master Revision Sheet PH2080
2.5 Diagrams and concept maps
THE COSMIC LADDER (support mechanism sets each rung)
density rho
high | neutron star ........ neutron degeneracy + nuclear forces
^ | white dwarf ......... electron degeneracy
| | main-seq star ....... thermal gas + radiation pressure
| | brown dwarf ......... electron degeneracy (failed star)
| | planet ............. Coulomb (solid/liquid) pressure
low | star cluster ....... random stellar motions (collisional)
| galaxy ............ random motions + rotation (collisionless)
| galaxy cluster .... random galaxy motions (virialised, + DM)
+------------------------------------------------> size R
small large
MASTER BALANCE: GRAVITY (inward) <==> PRESSURE / KINETIC (outward)
| |
GM^2/R (energy) which P? -> EOS (Ch.5)
2.6 Typical exam questions
Conceptual: Why do self-gravitating objects exist only on discrete scales? What support holds up each
rung of the ladder? Why is the Sun’s interior safely Newtonian but a neutron star’s is not?
Calculation: Estimate the mean density of the Sun; of a white dwarf; of a neutron star. Compute
the Schwarzschild radius of the Sun and of a 108 M⊙ black hole. Given a galaxy’s angular size and
distance, find its physical radius. √
Derivation: Derive vesc and vcirc and the factor 2 between them.
3. Gravitation, the two-body problem, and the virial theorem
3.1 Core concepts
Fundamental ideas. (i) The two-body Kepler problem: bound orbits are ellipses, with period set by
the semi-major axis and total mass. (ii) The virial theorem: for any gravitationally bound system in
steady state, the time-averaged kinetic and potential energies obey 2⟨K ⟩ + ⟨U ⟩ = 0. This is the single
most powerful tool in the course — it converts the impossible problem “solve the full structure” into
the trivial one “balance two energies”.
Physical interpretation. The virial theorem says a bound system is exactly half as kinetically “hot”
as it is gravitationally “deep”: ⟨K ⟩ = − 21 ⟨U ⟩, so total energy E = ⟨K ⟩ + ⟨U ⟩ = 12 ⟨U ⟩ = −⟨K ⟩ < 0.
Bound means negative total energy; the deeper the well, the faster the constituents move.
Key assumptions. Isolated system (no external torque/force), bound (E < 0), steady state so the
long-term average of Ï vanishes, and U ∝ r −1 gravity (the −1 power is what gives the clean factor 2).
Connections. The virial theorem powers: stellar central temperatures (Ch. 4), the negative heat ca-
pacity and Kelvin–Helmholtz contraction (Ch. 3,4), cluster and galaxy masses and hence dark matter
(Ch. 12), and the Jeans criterion for collapse. Whenever a question says “estimate the temperature/ve-
locity/mass of a bound system”, reach for it first.
3.2 Essential formula sheet
Formula Symbols, units, meaning Validity / exam
use
GMm
F=− r̂ Newton’s law; r separation [m]. Point/spherical
r2
masses.
4
