Download –1– Order of Magnitude Astrophysics

–1–
Order of Magnitude Astrophysics - a.k.a. Astronomy 111
Stars
A star is a massive, luminous ball of plasma held together by gravity.
The Existence of Stars
To take into account both thermal and quantum degenerate contributions, we take the matter
pressure to be P ≈ nkB T + nεF , which is a simple interpolation between the two limits. This
pressure can balance the gravitational pressure if (kB T + εF ) ≈ Gm2p N 2/3 n1/3 . Using expression for
εF for non-relativistic electrons, we get
kB T ≈ Gm2p N 2/3 n1/3 −
(3π2 )2/3 ~2 2/3
n .
2 me
(1)
For a classical system, the first term on the right hand side dominates, and we see that the
gravitational potential energy and kinetic energy corresponding to the temperature T are
comparable; this is merely a restatement of the virial theorem. As the radius of the system R is
reduced, the second term on the right-hand side (∝ n2/3 ) grows faster than the first (∝ n1/3 ) and
the temperature of the system will increase, reach maximum and decrease again; equilibrium is
possible for any of these values with gravity balanced by thermal and degeneracy pressure. The
maximum temperature T max is reached when n = nc , with
!
αG
N 2/3
;
nc (3π2 )2/3 oe
αG2
kB T max ≈
N 4/3 me c2
2(3π2 )2/3
where oe ≡ ~/(me c) and αG ≡ Gm2p /(~c).
(2)
(3)
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An interesting phenomenon arises if the maximum temperature T max is sufficiently high
to trigger nuclear fusion in the system; then we obtain a gravitationally bound, self-sustained
nuclear reactor. The condition for triggering nuclear reaction occurs at energy scales higher than
εnucl ≈ ηα2 m p c2 , with η ≈ 0.1. The energy corresponding to kB T max will be larger than εnucl when
!3/4
!3/2
α
3/4
2 1/2 m p
N > (2η) (3π )
≈ 4 × 1056
(4)
me
αG
for η = 0.1. The corresponding condition on mass is M > M∗ , where
!3/4
!3/2
α
3/4
2 1/2 m p
m p ≈ 4 × 1032 g,
M∗ > (2η) (3π )
me
αG
(5)
which is comparable to the mass of the smallest stars observed in our Universe.
The H-R Diagram
Once reactions occur at the hot central region of the star, its structure changes significantly.
If the transport of this energy to the outer regions is through photon diffusion, then the opacity
matter will play a vital role in determining the stellar structure. In particular, opacities determine
the relation between the luminosity and mass of the star. A photon mean free path l = (σn)−1 ,
random walking through the plasma , will have Ncoll ≈ (R/l)2 collisions in traversing the radius
R. This will take the time tesc ≈ (l/c)Ncoll ≈ (R/c)(R/l) for the photon to escape. The luminosity
of a star L is the ration between the radiant energy content of the star Eγ and tesc . Because
Eγ ≈ (aT 4 )R3 , where a = π2 k2B /(15~3 c3 ) = (π2 /15) ≈ 1 (in units with κB = ~ = c = 1), we find
that L = (aT 4 R3 l/R2 ). For a wide class of stars, we may assume that the central temperature
T ≈ GMm p /R is reasonably constant because of nuclear reactions – which depend strongly on T
– act as a thermostat. If Thomson scattering dominates, then σ = σT , and we get
!3
aT 4 R G4 5 2 3
34 M
L≈
≈ 2 m p me M ≈ 10
erg s−1 .
nσT
α
M∗
(6)
It is convenient to define the surface temperature T s of the star by the relation L ∝ R2 T s4 , so that
T s ∝ L1/4 R−1/2 ∝ L1/4 M −1/2 . Combining this result with the relation M ∝ L1/3 , we get T s ∝ L1/12 .
–3–
When the stars are plotted in a log T s – log L plane we thus expect them to lie within a line with
slope (1/12) ≈ 0.08. The observed slope is ∼ 0.13.
We saw above that stars are gravitationally bound systems in which self-sustaining
nuclear reactions are taking place in the center. The process of combining four protons into
a helium nuclei releases ∼ 0.03m p c2 of energy, which is ∼ 0.7% of the original energy,
4m p c2 . Assuming that a fraction ≈ 0.01 of the rest-mass energy can be made available for
nuclear reactions, we find that the lifetime of the nuclear burning phase of the star will be
tstar = M/L ≈ 3 × 109 yr(/0.01)(M/M∗ )−2 if the opacity is due to Thomson scattering. This
defines the characteristic time scale in