Different timescales
At different stages of evolution,
different timescales will be
important
The characteristic timescales
There are three characteristic timescales that aid concepts in stellar evolution
The dynamical timescale
For the Sun 𝑡 𝑑~2000 s
The thermal timescale
Time for a star to emit its entire reserve of thermal energy upon contraction
provided it maintains constant luminosity (Kelvin-Helmholtz timescale)
For the Sun 𝑡𝑡ℎ~30 Myr
The nuclear timescale
Time for star to consume all its available nuclear energy, 𝜀 is the typical
nucleon binding energy/nucleon rest mass energy
For the Sun, 𝑡 𝑛𝑢𝑐 is larger than the age of Universe
1
3 22
d
r
t
GM
 
=  
 
2
~th
GM
t
Lr
2
~nuc
Mc
t
L

d th nuct t t  
The equation of radiative transport
We assume for the moment that the condition for convection is not satisfied, and
we will derive an expression relating the change in temperature with radius in a
star assuming all energy is transported by radiation. Hence we ignore the effects
of convection and conduction.
The energy carried by radiation per square meter per second, i.e. the flux 𝐹𝑟𝑎𝑑,
can be expressed in terms of the temperature gradient and a coefficient of
radiative conductivity, 𝜆 𝑟𝑎𝑑, as follows:
𝐹𝑟𝑎𝑑= − 𝜆 𝑟𝑎𝑑
𝑑𝑇
𝑑𝑟
where the minus sign indicates that heat flows down the temperature gradient.
Assuming that all energy is transported by radiation, we will now drop the
subscript 𝑟𝑎𝑑 from the remainder of this discussion.
The radiative conductivity measures the readiness of heat to flow. Astronomers
generally prefer to work with an inverse of the conductivity, known as the opacity
𝜅, which measures the resistance of material to the flow of heat.
The equation of radiative transport
It can be show that the opacity, 𝜅, is defined by the relation
𝜅(𝑟) =
4𝑎𝑐𝑇(𝑟)3
3𝜌(𝑟)𝜆
where 𝑎 is the radiation density constant and 𝑐 is the speed of light. Combining
the above equations we obtain:
𝐹(𝑟) = −
4𝑎𝑐𝑇(𝑟)3
3𝜌(𝑟)𝜅(𝑟)
𝑑𝑇
𝑑𝑟
Recalling that flux and luminosity are related by
𝐿(𝑟) = 4𝜋𝑟2 𝐹(𝑟)
The equation of radiative transport
we can write:
𝐿(𝑟) = −
16𝑎𝑐𝑟2 𝑇(𝑟)3
3
𝑑𝑇
𝑑𝑟
On rearranging, we obtain:
𝑑𝑇
𝑑𝑟
= −
3𝜌 𝑟 𝜅 𝑟
16𝑎𝑐𝑟2 𝑇 𝑟 3
𝐿(𝑟)
This is known as the equation of radiative transport and is the temperature
gradient that would arise in a star if all the energy were transported by radiation.
It should be noted that the above equation also holds if a significant fraction of
energy transport is due to conduction, but in this case 𝐿 refers to the luminosity
due to radiative and conductive energy transport and refers to the opacity to
heat flow via radiation and conduction.
Solving the equations of stellar structure
Hence we now have four differential equations, which govern the structure of
stars (note – in the absence of convection).
2 3
( ) 3 ( ) ( )
( )
16 ( )
dT r r r
L r
dr acr T r
 

= −
2
( ) ( ) ( )dP r GM r r
dr r

= −
2( )
4 ( )
dM r
r r
dr
 =
2( )
4 ( ) ( )
dL r
r r r
dr
  =
𝑟 … radius
𝑃… pressure at 𝑟
𝑀… mass of material within 𝑟
𝜌 … density at 𝑟
𝐿 … luminosity at r (rate of energy flow across
sphere of radius 𝑟)
𝑇 … temperature at 𝑟
𝜅 … Rosseland opacity at 𝑟
𝜀 … energy release per unit mass per unit time
We will consider the quantities:
𝑃 = 𝑃(𝜌, 𝑇, chemical composition) The equation of state
𝜅 = 𝜅(𝜌, 𝑇, chemical composition)
𝜀 = 𝜀(𝜌, 𝑇, chemical composition)
Boundary conditions
Two of the boundary conditions are fairly obvious, at the center of
the star 𝑀 = 0, 𝐿 = 0 at 𝑟 = 0.
At the surface of the star its not so clear. There is no sharp edge to
the star, but for the Sun 𝜌(surface)~10-4 kg m-3. Much smaller than
the mean density 𝜌(mean)~1.4103 kg m-3 (which we derived). We
know the surface temperature (5780K) is much smaller than its
minimum mean temperature (2106 K).
Thus, we make two approximations for the surface boundary
conditions: 𝜌 = 𝑇 = 0 at 𝑟 = 𝑟𝑠
i.e. that the star does have a sharp boundary with the surrounding
vacuum
Use of mass as the independent variable
The above formulae would (in principle) allow theoretical models of stars with a
given radius. However from a theoretical point of view it is the mass of the star
which is chosen, the stellar structure equations solved, then the radius and other
parameters are determined. We observe stellar radii to change by orders of
magnitude during stellar evolution, whereas mass appears to remain constant.
Hence it is much more useful to rewrite the equations in terms of 𝑀 rather than 𝑟.
If we divide the other three equations by the equation of mass conservation, and
invert the latter:
2
1
4
dr
dM r 
=
4
4
dP GM
dM r
= −
dL
dM
= With boundary conditions:
𝑟 = 0, 𝐿 = 0 at 𝑀 = 0
𝜌 = 𝑇 = 0 at 𝑀 = 𝑀𝑠
We specify 𝑀𝑠 and the chemical composition and now have a well defined set of
relations to solve. It is possible to do this analytically if simplifying assumptions
are made, but in general these need to be solved numerically.
2 4 3
3
16
dT L
dM r acT


= −
Stellar evolution
We have a set of equations that will allow the complete structure
of a star to be determined, given a specified mass and chemical
composition. However what do these equations not provide us
with?
In deriving the equation for hydrostatic support, we have seen that
provided the evolution of star is occurring slowly compared to the
dynamical time, we can ignore temporal changes (e.g. pulsations)
1
3 22
d
r
t
GM
 
=  
 
And for the Sun for example, this is 𝑡 𝑑 ~ 2000s,
hence this is certainly true
And we have also made the assumption that time dependence can
be omitted from the equation of energy generation, if the nuclear
timescale (the time for which nuclear reactions can supply the stars
energy) is greatly in excess of 𝑡𝑡ℎ
Stellar evolution
If there are no bulk motions in the interior of the star, then any
changes of chemical composition are localised in the element of
material in which the nuclear reactions occurred. So the star would
have a chemical composition which is a function of mass 𝑀.
In the case of no bulk motions, the set of equations we derived
must be supplemented by equations describing the rate of change
of abundances of the different chemical elements. Let 𝐶 𝑋,𝑌,𝑍 be the
chemical composition of stellar material in terms of mass fractions
of hydrogen (X), helium, (Y) and metals (Z)
Now lets consider how we could evolve a model
, ,
, ,
( )
( , , )X Y Z M
X Y Z
C
f T C
t



=
0 0
, ,
, , , , , ,
( )
( ) ( ) X Y Z M
X Y Z M t t X Y Z M t
C
C C
t


+ = +
Ideally we would like to know exactly how much energy is transported by
convection – but lack of a good theory makes it difficult to predict exactly.
Heat is convected by rising elements which are hotter than their surroundings
and falling elements which are cooler. Suppose the element differs by 𝛿𝑇 from
its surroundings, because an element is always in pressure balance with its
surroundings, it has energy content per kg which differs from surrounding kg
of medium of 𝑐 𝜌 𝛿𝑇 (where 𝑐 𝜌 is the specific heat at constant pressure).
If material is mono-atomic ideal gas then 𝑐 𝜌 = Τ5𝑘 2𝑚
Where 𝑚 is the average mass of particles in the gas
Assuming a fraction 𝛼 (1) of the material is in the rising and falling columns
and that they are both moving at speed 𝑣 [ms-1 ] then the rate at which excess
energy is carried across radius is:
𝐿 𝑐𝑜𝑛𝑣 = surface area of sphere x rate of transport x excess energy
Influence of convection
2
2 5 10
4
2
conv
k T r vk T
L r v
m m
   
 = =
Hence putting in known solar values, at a radius halfway between
surface and center, we get
𝐿 𝑐𝑜𝑛𝑣 = 1026
𝛼𝑣𝛿𝑇 [W]
The surface luminosity of the Sun is 𝐿 𝑆𝑢𝑛 = 3.86 × 1026 W, and
at no point in the Sun can the luminosity exceed this value (see
equation of energy production).
What can we conclude from this?
As the 𝛿𝑇 and 𝑣 of the rising elements are determined by the
difference between the actual temperature gradient and adiabatic
gradient, this suggests that the actual gradient is not greatly in
excess of the adiabatic gradient. To a reasonable degree of accuracy
we can assume that the temperature gradient has exactly the
adiabatic value in a convective region in the interior of a star and
hence can rewrite the condition of occurrence of convection in the
form
1P dT
T dP


−
=
Thus in a convective region we must solve the four differential equations, together
with equations for 𝜀 and 𝑃
2 4 3
16
3
rad
r acT dT
L
dM


=
2
1
4
dr
dM r 
=
4
4
dP GM
dM r
= −
dL
dM
=
1P dT
T dP


−
=
The equation for luminosity due to radiative transport is still true:
And once the other equations have been solved, 𝐿 𝑟𝑎𝑑 can be calculated. This can
be compared with 𝐿 (from Τ𝑑𝐿 𝑑𝑀 = 𝜀) and the difference gives the value of
luminosity due to convective transport 𝐿 𝑐𝑜𝑛𝑣 = 𝐿 − 𝐿 𝑟𝑎𝑑
In solving the equations of stellar structure the equations appropriate to a
convective region must be switched on whenever the temperature gradient
reaches the adiabatic value, and switched off when all energy can be transported
by radiation.