Question

Consider a very luminous star. In the outer layers of this star (the photosphere) the pressure gradient will be dominated by radiation pressure Prad=31aT4=cFrad where Frad is the radiative flux and the other terms have their usual meaning.

(a) Using the equation for radiative transfer dF=−κrhoFdr, show that dPrad/dr= κrhoL /4πcr2
where κ is the opacity, rho is the density and L is the luminosity.
(b) If this radiation pressure gradient is larger than the limit for hydrostatic equilibrium, then the photosphere of the star will be blown away by the radiation pressure. Show that the maximum luminosity for a star is given by
Lmax=4πGMc/k
(c) What mechanism will dominate the opacity in the photosphere of a star? (d) What is the maximum mass a star can have if its luminosity cannot be larger than Lmax. Assume that the luminosity scales as L∝M3.3 for massive stars.

Answer 1

(a) To show that dPrad/dr = κρL/4πcr^2, we'll start with the equation for radiative transfer, which is dF = -κρFdr. Here, dF represents the change in radiative flux, κ is the opacity, ρ is the density, and dr is the change in radius.

We can rewrite this equation as dF/dr = -κρF.

Now, let's consider the pressure gradient due to radiation pressure, Prad = 31aT^4 = cFrad, where a is the radiation constant and T is the temperature.

Taking the derivative of both sides with respect to r, we have dPrad/dr = c(dFrad/dr).

Substituting -κρF for dF/dr, we get dPrad/dr = -cκρFrad.

Since Frad = F, we can rewrite this as dPrad/dr = -cκρF.

Using the equation dF/dr = -κρF, we can substitute dF/dr into the previous equation, giving us dPrad/dr = κρL/4πcr^2, where L is the luminosity.

(b) If the radiation pressure gradient is larger than the limit for hydrostatic equilibrium, the photosphere of the star will be blown away by the radiation pressure. The maximum luminosity for a star can be obtained by setting the radiation pressure gradient equal to the limit for hydrostatic equilibrium.

The limit for hydrostatic equilibrium is given by dP/dm = -GMρ/r^2, where G is the gravitational constant, M is the mass of the star, ρ is the density, and r is the radius.

Setting dPrad/dr = -GMρ/r^2, we can solve for L. Rearranging the equation, we have dPrad/dr = κρL/4πcr^2 = -GMρ/r^2.

Simplifying, we find L = 4πGMc/κ.

Therefore, the maximum luminosity for a star is given by Lmax = 4πGMc/κ.

(c) The mechanism that dominates the opacity in the photosphere of a star is typically electron scattering. In this process, photons interact with free electrons in the stellar plasma, causing the light to scatter. This scattering process contributes to the overall opacity of the star's photosphere.

(d) To determine the maximum mass a star can have if its luminosity cannot be larger than Lmax, we assume that the luminosity scales as L ∝ M^3.3 for massive stars.

Since L ∝ M^3.3, we can rewrite this as L = kM^3.3, where k is a constant.

Substituting this expression for L into the equation for Lmax, we have kM^3.3 = 4πGMc/κ.

Rearranging the equation to solve for M, we find M = (4πGMc/κk)^(1/3.3).

Therefore, the maximum mass a star can have is given by Mmax = (4πGMc/κk)^(1/3.3).