Radiation pressure and hydrostatic equilibrium in giant stars: recall that the m
ID: 3162887 • Letter: R
Question
Radiation pressure and hydrostatic equilibrium in giant stars: recall that the momentum of a photon is p = E/c. Derive an expression for the radiation pressure, defined as the outward momentum unit time per unit area, delivered to the outer layers of a star with luminosity L, assuming that all the photons from the interior are absorbed in those layers. Calculate the total radiation force on the outer layers of a star with L = L_circleddotoperator. How does the racially outward force from radiation compare to the force of gravity on the layers the layers lie R = 100 R_circleddotoperator, have mass m = 10^-6 M_circleddotoperator, and the rest of the star has a M = 1 M_circleddotoperator ? Repeat b for a typical AGB star: L = 7000 L_circleddotoperator, T_e = 3000 K, same masses.Explanation / Answer
We know that
luminosity = energy emmitted by the star per unit time
a) Luminosity = L
Assume energy emmitted by the star = E
let consider some time step t such that E/t = L
now, momentum of photons, outwards, assuming all the photons are absorbed by the outher layers of the star
p = E/c = Lt/c
now, radiation pressure = p/A*t [ A is the surface area of the outer layers]
radiation pressure, P = Lt/c*A*t = L/cA [ where A is the surface area of the pouther layers of a star]
for a star with radisu r
A = 4*pi*r^2
P = L/4*pi*c*r^2
b) Radiation force = radiation pressure * Surface Area = P*A = L/c
now L = L. = 3.828*10^26 W
so, Fr = 3.828*10^26/3*10^8 = 1.276*10^18 N
at R = 100 R.
force of gravity = GM.*(10^-6*M.)/(100R.)^2
For sun
M. = 1.989*10^30 kg
R. = 695,700,000 m
Fg = 5.45194*10^22 N
Fr/Fg = 2.34045*10^-5
c) L = 7000L.
Te = 3000 K
NOw for every star
k* Radius^2 * temperature^4 = Luminosity ( k is some constant)
so, R. ^2 * T. ^4 / L. = R ^2 * 3000^4 / 7000L. [ here T. is temperatuer of sun = 5778 K, R. is radius of sun 695,700,000m , L. is luminosity of sun 3.828*10^26 W, R is radisu of the star in question]
R. ^2 * 5778 ^4 = R ^2 * 3000^4 / 7000
R = 96321.485R.
Radiation force = radiation pressure * surface area = L/c = 7000*3.828*10^26/3*10^8 = 8.932*10^21 N
force of gravity = GM.*(10^-6M.)/(96321.485R.)^2 = 5.87631*10^16 N
Fr/Fg = 152,000.1392
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