Lecture
5 Polytropic models
The
equation of hydrostatic equilibrium, discussed in the previous lecture, can
be solved in the case where density ρ is a known function of
pressure P. A particular example of this is a relation of
the form
(5.1)
where
K and γ are constants; this is called a polytropic
relation, and the resulting models are called polytropic models.
Models
of this nature have played a very important role in the development of the
subject; they are still very useful as simple examples which are,
nevertheless, not too dissimilar from realistic models. More importantly,
there are cases where the polytropic relation (5.1) is a very good
approximation to reality. An example is where pressure and density are
related adiabatically, as in equation (3.34).
To
obtain the equation satisfied by polytropic models, we note that from
equations (4.5) and (4.7) we have
(5.2)
Hence,
using equation (5.1), we obtain
(5.3)
It
is convenient to replace γ by the polytropic index n,
defined by
(5.4)
Furthermore,
we introduce a dimentionless measure θ of density ρ by
(5.5)
where
ρ_{c}
is central density. Then equation (5.3) becomes
or
(5.6)
To
simplify the equation further, we introduce a new measure ξ for the distance to the centre by
(5.7)
Then
the equation finally becomes
(5.8)
This
equation is called the LaneEmden equation, and the solution θ=θ(ξ) is called
the LaneEmden function. From equation (5.5) it follows that θ must satisfy the boundary condition
(5.9)
The
surface of the model is defined by the point ξ=ξ_{1},
where θ(ξ_{1})=0.
Given
the solution θ(ξ), we
can obtain relations between the various quantities characterizing the model.
It follows immediately from equation (5.7) that the surface radius of the
model is
(5.10)
The
mass m(ξ) interior to ξ may be obtained by integrating equation
(4.7), using equations (5.5, 5.7, 5.8) as
(5.11)
Using
the expression (5.7) for α, we finally obtain
(5.12)
In
particular, the total mass is given by
(5.13)
From
equations (5.10) and (5.13), by eliminating ρ_{c},
we may find a relation between M, R and K.
The result is
(5.14)
There
are two different interpretations of this relation. If the constant K in
equation (5.1) is given in terms of basic physical constants and hence is
known, equation (5.14) defines a relation between the mass and the radius of
the star. If, on the other hand, equation (5.1) just expresses
proportionality, the constant K being essentially arbitrary, then
equation (5.14) may be used to
determine K for a star with a given mass and radius; as shown
below one may then determine other quantities for the star. In the former
case, therefore, there is a unique polytropic model for a given mass, whereas
in the latter case a model can be constructed for any value of M and
R.

Exercise
5.1. Verify equation
(5.14).

From
the last of equations (5.11) we find that the mean density of the star is
(5.15)
and
hence the central density is determined by the mass and radius as
(5.16)
where
the last equation defines constant a_{n} which depends on the polytropic index
n only. Finally, using that from equation (5.1)
(5.17)
and
using equations (5.14) and (5.16), we find that
(5.18)
where
c_{n} depends on the polytropic index n
only. The pressure throughout the model is then determined by
(5.19)

Exercise
5.2. Fill in the
missing details in the derivation of equations (5.15), (5.16), and (5.18).

In
the case where the temperature is related to pressure and density through the
ideal gas law (3.13), it may be determined from equations (5.5) and (5.19) as
(5.20)
where
(5.21)
where
b_{n} depends on the polytropic index n
only. In the case when the star is composed of an ideal gas, therefore, θ is a measure of the temperature.
To
determine the structure of a polytropic star completely, we only need to find
the solution to the LaneEmden equation (5.8). Unfortunately, in general no
analytical solution is possible. The only exceptions are n=0, 1 and
5 where the solutions are
(5.22)
(5.23)
(5.24)

Exercise
5.3. Verify that
these solutions satisfy the LaneEmden equation (5.8) and the boundary
condition (5.9).

The
solution for n=5 is evidently peculiar, in that it has infinite
radius. On the other hand, since
(5.25)
is
finite, so is the mass of the model. It may be shown that only for n<5
does the LaneEmden equation have solutions corresponding to finite radius.
For
values of n other than 0 and 1 ,
the LaneEmden equation must be solved numerically. Extensive tables of the
solution exist; in any case, with modern computational facilities the
solution of the equation is a simple numerical problem. Table 5.1 lists a
number of useful quantities, which enter into the expressions given above,
for a selection of polytropic models.
n

ξ_{1}

a_{n}

b_{n}

c_{n}

0

2.449

1.00

0.5

0.12

1

3.142

3.29

0.5

0.39

1.5

3.654

5.99

0.54

0.77

2

4.353

11.40

0.60

1.64

3

6.897

54.18

0.85

11.05

4

14.97

662.4

1.67

247.6

Table
5.1. Properties of polytropic models. Constants a_{n},
b_{n} and c_{n} specify the central density, central
temperature and central pressure as given by equations (5.16), (5.18) and (5.21).
The
next table, Table 5.2, presents the solution for two particular cases n=1.5
and n=3, at selected values of ξ.
ξ

θ

dθ/dξ


ξ

θ

dθ/dξ

0

1

0


0

1

0

0.5

0.96

0.16


0.5

0.96

0.16

1.0

0.85

0.29


1.0

0.86

0.25

1.5

0.68

0.36


1.5

0.72

0.28

2.0

0.50

0.37


2.0

0.58

0.26

2.5

0.32

0.34


3.0

0.36

0.18

3.0

0.16

0.28


4.0

0.21

0.12

3.5

0.03

0.22


6.0

0.04

0.06

3.654

0

0.22


6.897

0

0.04

Table
5.2. Properties of polytropes of indices n=1.5 and n=3.
From
Table 5.1 it follows that the properties of polytropic models vary widely
with n. This is true in particular of the degree of central condensation,
as measured by a_{n}, the ratio between central and mean density. For
n=0 it is obvious from equation (5.5) that density ρ is constant, and hence a_{1}=1 , whereas the value of a_{n} tends to infinity as n→5. For stars on the main sequence the
central condensation is typically around 10^{2}, corresponding to a
polytrope of index around 3.3.
It
should be noticed also that equation (5.18) for the central pressure and, in
the ideal gas case, equation (5.21) for the central temperature, confirm the
simple scaling derived in the previous Lecture (section 4.2). Now, however,
the polytropic relations contain the additional numerical constants b_{n} and c_{n} . It is obvious from Table 5.1 that that c_{n} varies strongly with n; hence the estimate in equation (4.9)
of the central pressure is at most a rough approximation. On the other hand,
the range of variation of b_{n} is much more modest, except when n is
very close to the critical case n=5 . Thus equation (4.10) gives a
reasonable estimate for the central temperature for a wide range of models.

Exercise
5.4. Find ρ_{c}
, P_{c} and T_{c} in a polytrope of index 3 with solar
mass (2.00×10^{30}kg)
and radius (6.96×10^{8}m)
and chemical composition X=0.7 , Z=0.02 ,
where the ideal gas equation of state is assumed to be valid. Find also ρ , P and T at the
point where r=R/2. (Use the data in Tables 5.1 and 5.2).

