If we assume for the moment that the
condition for
the occurrence of convection is not satisfied,
we can write down the fourth equation of stellar
structure, which expresses the rate of change of temperature with
radius in a star assuming that all energy is transported by radiation
(i.e. ignoring the effects of convection and conduction).
The energy carried by radiation per square metre per second, i.e. the
flux F_{rad}, can be expressed
in terms of the temperature gradient and a coefficient of radiative
conductivity, _{rad}, as
follows:
F_{rad} = -
_{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,
which measures the resistance of material
to the flow of heat. Detailed arguments (see Appendix 2 of Tayler)
show that the opacity, , is defined
by the relation
= 4acT^{3} /
3_{},
where a is the radiation density constant and c
is the speed of light. Combining the above equations we obtain:
F = -
(4acT^{3} /
3_{})
(dT / dr).
Recalling that flux and luminosity are related by
L =
4r^{2}F,
we can write:
L = -
(16acr^{2}T^{3} /
3_{})
(dT / dr).
On rearranging, we obtain:
dT / dr = -
3_{}L /
16acr^{2}T^{3}.
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.
Clearly, the flow of energy by radiation can only be determined if an
expression for is available. How such an
expression can be obtained will be discussed later in the course.