\' These questions are all short and related: Please answer all of them. For a s

Question

' These questions are all short and related: Please answer all of them.

For a solar composition u ~=0.6. How does one compute u? What is u for a pure H plasma (fully ionized)? What is u for pure He?

what is the pressure scale height? How is it computed?

What is the characteristic timescale for the star to maintain hydrostatic equilibrium? What is the relation between the timescale, the free-fall timescale, and the sound crossing timescale?

In thermal equilibrium, why is the next flux zero? Topic of Radiation Transport

What is the mean free path and how is it computed?

How is the opacity related to the cross-section? To the mean free path of a photon? How does the photon mean free path compare to the radius of the sun at the center? at the edge?

What happens to a ray of light traversing an absorbed/scattering medium?

Thank you for your help! Please answer all and give as much detail as possible!

Explanation / Answer

To find the u we need to know the mass fractions xj of a gas that is made up of element j. Xj for hydrogen is usually written as X and Xj for helium written as Y. When Xj and all other elements are combined written as Z known as metallicity

X+Y+Z=1

Number densities as calculated from mass fractions

nj=p/mH*Xj/Aj

For fully ionised gases

u=(summission z( Xj/Aj)^-1

=(2X+3Y/4+Z/2)^-1 ( assume 1+Zj/Aj=1/2)

u for hydrogen=0.7

u for helium=0.28

Pressure scale height is the height at the pressure at which the pressure 1/e times its value on the surface.

P(z)=P(0)e^-mgz/KT

pressure scale height=z=kT/mg

Where Pz= pressure at height z

m=molecular mass

g= accleration due to gravity

k=boltzman constant

T= temperature

Characteristic timescale of a star is the approximate time taken by a star to radiate away its total kinetic energy at its current luminosity rate also known as kelvin-helmholtz time scale given by

Tth=total kinetic energy/rate of energy loss

=GM^2/RL

= G= gravitational constant

R= radius of star

M= mass of star

L=luminosity of star

For Free fall time scale:

consider a uniform sphere with density p, a particle which is initially at radius R

equation of motion of particle=-GM/R^2=-4piGp/3R

then all the particles withing the sphere will fall from rest to origin within 1/4 or quarter of the oscillation period

Then free fall time is given by tff=T/4

=1/4*2pi/w

=square root(3pi/16Gp) proportional to (GP)^-1/2

Now sound crossing time scale

Ts=R/vs

vs=(v2)^1/2=(KT/mp)^1/2

Thermal equilibrium is a condition in which in there is no heat flows between the two systems. It occurs when the system does not allow the heat to either enter or pass through it or the system is isolated. So the net heat flux is zero

The mean free path is the mean distance travelled by the moving particle like atom, photon or molecule by the successive impacts or collisions that changes the direction, energy or other particle properties.

Mean free path is calculated by length of path divided by number of collisions.

When a ray of light is absorbed by the medium there is reduction in its intensity due to absorption of part of its photon known as atteunation. Also when the light ray get scattered it deviates from its original direction to some other direction

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