1.3 Oppenheimer–Volkoff spherical equilibrium stars 17
Polytropes
Some of the simplest and most useful families of equilibrium models are constructed from
an isentropic equation of state of the form,
P = Kρ
0
(K ,constants), (1.86)
where ρ
0
is the rest-mass density. The constant K is the polytropic gas constant and
the quantity n defined by ≡ 1 + 1/n is called the polytropic index. Stellar models
constructed from such an equation of state are called polytropes. For an equation of
stategivebyequation(1.86), we find from the first law of thermodynamics (or from
equation 1.74) that ρ
0
= P/( − 1) and ρ = ρ
0
+ P/( − 1).
There are a number of physically interesting stars that can be modeled as polytropes
in a first approximation. For example, stars supported against collapse by the pressure
of noninteracting, nonrelativistic, degenerate fermions can be modeled as n = 3/2 poly-
tropes, while stars supported by noninteracting, ultrarelativistic, degenerate fermions can
be modeled as n = 3 polytropes. In such cases, lower-mass objects are constructed from
nonrelativistic fermions, while higher-mass objects are constructed from highly relativistic
fermions. White dwarfs, which are supported by the pressure of degenerate electrons, and
neutron stars, which are supported by degenerate neutrons, are members of this class of
models.
23
When nuclear interactions are included, high-mass neutron stars are better rep-
resented by a “stiffer” equation of state, but the resulting models are often crudely modeled
as n = 1 relativistic polytropes. Another example is a star supported by thermal radiation
pressure at constant specific entropy, which can be modeled as an n = 3 polytrope.
24
When using a polytropic equation of state to construct stellar equilibrium models, it is
always possible to scale out the constant K . In gravitational units K
n/2
has units of length,
so that we can introduce a new set of nondimensional quantities, often denoted by a bar:
¯
r ≡ K
−n/2
r, ¯ρ
0
≡ K
n
ρ
0
, ¯ρ ≡ K
n
ρ,
¯
P ≡ K
n
P,
¯
M ≡ K
−n/2
M,
¯
M
0
≡ K
−n/2
M
0
. (1.87)
One can thus set K = 1 in numerical integrations and either use the above relations to scale
the results to more physical values of K , or express answers in terms of nondimensional
ratios that are independent of K (e.g., R/M, M
2
ρ
0
,etc.).
In Figure 1.2 we plot the equilibrium sequence for n = 1 polytropes as an example. The
turning point along a curve of equilibrium mass vs. central density, like the ones plotted
here, identifies the maximum mass configuration. It also marks the onset of radial dynam-
ical instability along the sequence. In particular, configurations to the left of the turning
point, where dM/dρ
c
> 0, are dynamically stable to small radial perturbations and will
23
For ultrarelativistic degenerate fermions there is a maximum mass limit, which for a white dwarf is called the
Chandrasekhar limit and is about 1.4M
. S. Chandrasekhar received the Nobel prize in 1983, in part for identifying
this important limit (Chandrasekhar 1931).
24
For a thorough discussion of polytropes and more detailed models of compact objects like white dwarfs, neutron stars
and supermassive stars and their stability properties, see Shapiro and Teukolsky (1983) and references therein.