Introduction
• Stars are held together by gravitation – attraction exerted on each
part of the star by all other parts
• Collapse is resisted by internal thermal pressure.
• These two forces play the principal role in determining stellar
structure – they must be (at least almost) in balance
• Thermal properties of stars – continually radiating into space. If
thermal properties are constant, continual energy source must exist
• Theory must describe - origin of energy and transport to surface
We make two fundamental assumptions :
1) Neglect the rate of change of properties – assume constant with time
2) All stars are spherical and symmetric about their centres
We will start with these assumptions and later reconsider their validity
What are the main physical processes which determine the structure of stars ?
For our stars – which are isolated, static, and spherically
symmetric – there are four basic equations to describe
structure. All physical quantities depend on the distance
from the centre of the star alone
1) Equation of hydrostatic equilibrium: at each radius,
forces due to pressure differences balance gravity
2) Conservation of mass
3) Conservation of energy: at each radius, the change in
the energy flux = local rate of energy release
4) Equation of energy transport: relation between the
energy flux and the local gradient of temperature
These basic equations supplemented with
• Equation of state: pressure of a gas as a function of its
density and temperature
• Opacity: how opaque the gas is to the radiation field
• Core nuclear energy generation rate
Balance between gravity and internal pressure is known as
hydrostatic equilibrium
Mass of element
Consider forces acting in radial direction
1. Outward force: pressure exerted by stellar
material on the lower face:
2. Inward force: pressure exerted by stellar material
on the upper face, and gravitational attraction of all
stellar material lying within r
Equation of hydrostatic support
𝛿𝑚 = 𝜌 𝑟 𝛿𝑠𝛿𝑚 where 𝜌 𝑟 is density at r
𝑃(𝑟)𝛿𝑠
𝑃 𝑟 + 𝛿𝑟 𝛿𝑠 +
𝐺𝑀(𝑟)
𝑟2
𝛿𝑚 = 𝑃 𝑟 + 𝛿𝑟 𝛿𝑠 +
𝐺𝑀 𝑟
𝑟2
𝜌(𝑟)𝛿𝑠𝛿𝑟
In hydrostatic equilibrium:
If we consider an infinitesimal element, we write
for 𝛿𝑟 → 0
Hence rearranging above we get
The equation of hydrostatic support
𝑃 𝑟 𝛿𝑠 = 𝑃 𝑟 + 𝛿𝑟 𝛿𝑠 +
𝐺𝑀 𝑟
𝑟2
𝛿𝑚
⇒ 𝑃 𝑟 + 𝛿𝑟 − 𝑃 𝑟 = −
𝐺𝑀 𝑟
𝑟2 𝜌(𝑟)𝛿𝑟
𝑃 𝑟 + 𝛿𝑟 − 𝑃(𝑟)
𝛿𝑟
=
𝑑𝑃(𝑟)
𝑑𝑟
𝑑𝑃(𝑟)
𝑑𝑟
= −
𝐺𝑀(𝑟)(𝑟)
𝑟2
Now acceleration due to gravity is 𝑔 =
𝐺𝑀(𝑟)
𝑟2
⇒
𝑑𝑃(𝑟)
𝑑𝑟
+ 𝑔𝜌 = 𝜌 𝑟 𝑎
Which is the generalised form of the equation of hydrostatic
support
We have assumed that the gravity and pressure forces are
balanced - how valid is that?
Consider the case where the outward and inward forces are not
equal, there will be a resultant force acting on the element
which will give rise to an acceleration 𝑎
Accuracy of hydrostatic assumption
𝑃 𝑟 + 𝛿𝑟 𝛿𝑠 +
𝐺𝑀 𝑟
𝑟2
𝛿𝑠𝛿𝑟 − 𝑃 𝑟 𝛿 = 𝜌 𝑟 𝛿𝑠𝛿
⇒
𝑑𝑃(𝑟)
𝑑𝑟
+
𝐺𝑀 𝑟
𝑟2
𝜌 𝑟 = 𝜌 𝑟 𝑎
Now suppose there is a resultant force on the element (LHS  0).
Suppose their sum is small fraction of gravitational term (𝛽)
𝛽𝜌 𝑟 𝑔 = 𝜌 𝑟 𝑎
Hence there is an inward acceleration of 𝑎 = 𝛽𝑔
Assuming it begins at rest, the spatial displacement 𝑑 after time
𝑡 is 𝑑 =
1
2
𝑎𝑡2 =
1
2
𝛽𝑔𝑡2
If we allowed the star to collapse i.e. set 𝑑 = 𝑟 and substitute
𝑔 =
𝐺𝑀
𝑟2 ⇒ 𝑡 =
1
𝛽
2𝑟3
𝐺𝑀
1
2
assumming 𝛽 ≈ 1 ⇒ 𝑡 𝑑 =
2𝑟3
𝐺𝑀
1
2
𝑡 𝑑 is known as the dynamical time
Accuracy of hydrostatic assumption
Equation of mass conservation
Mass 𝑀 𝑟 contained within a star of radius r is determined
by the density of the gas 𝜌(𝑟)
Consider a thin shell inside the
star with radius 𝑟 and outer
radius 𝑟 + 𝛿𝑟
In the limit where δ𝑟 → 0
This is the equation of mass conservation
𝛿𝑉 = 4𝜋𝑟2 𝛿𝑟
δ𝑀 = 𝛿𝑉𝜌 𝑟 = 4𝜋𝑟2 𝛿𝑟𝜌(𝑟)
𝑑𝑀(𝑟)
𝑑𝑟
= 4𝜋𝑟2 𝜌(𝑟)
Stars are rotating gaseous bodies – to
what extent are they flattened at the
poles ?
If so, departures from spherical
symmetry must be accounted for
Consider mass 𝛿𝑚 near the surface of
star of mass 𝑀 and radius 𝑟
Element will be acted on by additional
inwardly acting force to provide
circular motion
Accuracy of spherical symmetry assumption
Centripetal force is given as 𝑚𝜔2 𝑟 where 𝜔 is the angular velocity
of the star. There will be no departure from spherical symmetry
provided that
𝑚𝜔2
𝑟
Τ𝐺𝑀𝑚 𝑟2
≪ 1 or 𝜔2 ≪
𝐺𝑀
𝑟3
Accuracy of spherical symmetry assumption
Note the RHS of this equation is similar to 𝑡 𝑑
𝑡 𝑑 =
2𝑟3
𝐺𝑀
1
2
or
𝐺𝑀
𝑟3 =
2
𝑡2 ⇒ 𝜔2
≪
2
𝑡2
and 𝜔 =
2𝜋
𝑃
where 𝑃 is the rotation period
Spherical symmetry is met if 𝑃 ≫ 𝑡 𝑑
For example 𝑡 𝑑(Sun) ~ 2000s and 𝑃~1 month
 For the majority of stars, departures from spherical symmetry
can be ignored
Some stars do rotate rapidly and rotational effects must be
included in the structure equations - can change the output of
models
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