Astrophysics - Energy radiated from a star through contraction

[sxc656]

Hi

Homework Statement

As the sun evolved to the main sequence it contracted under gravity and its internal temperature changed from 3E4 K to 6E6 K. Assuming the sun is fully ionised at these temperatures estimate the energy radiated during this phase, stating your assumptions.

Homework Equations

I think this involves using the equation for gravitational potential energy and possible the scaling relation; Temp proportional to Mass/Radius for stars.

The Attempt at a Solution

I am not sure how to attempt this, my thinking is the temperature increase corresponds to a pressure decrease over which the radius of the sun decreases, thus releasing gravitational potential energy. But how would i relate the temperature change to a radius change?

Thanks

 

[mark726]

for bringing up this interesting topic! I would approach this problem by first making some assumptions and simplifications.

First, let's assume that the sun's contraction is isothermal, meaning that its internal temperature remains constant as it contracts. This is a reasonable approximation for the early stages of the sun's evolution to the main sequence.

Next, we can use the ideal gas law to relate the temperature and pressure of the sun's interior. Assuming the sun is fully ionized, we can use the equation of state for an ideal gas of fully ionized particles, which is given by P=nkT, where P is the pressure, n is the number density of particles, k is the Boltzmann constant, and T is the temperature.

Now, we can use the scaling relation you mentioned, which states that for a star in hydrostatic equilibrium, the temperature is proportional to the mass divided by the radius. This can be written as T~M/R, where M is the mass of the sun and R is its radius.

Combining these two equations, we can express the pressure as P~nM/R.

Finally, we can use the equation for gravitational potential energy, U=-GMm/R, where G is the gravitational constant, M is the mass of the sun, m is the mass of a particle in the sun's interior, and R is the sun's radius.

Putting everything together, we can write the energy radiated during the sun's contraction as:

E=∫PdV=∫nM/RdV=∫M^2/R^2dV=∫(GMm/R^2)dV=GM^2/R

Where the integral is taken over the entire volume of the sun.

Of course, this is a simplified calculation and there are many other factors that could affect the energy radiated during the sun's contraction. But it gives us a rough estimate and a starting point for further investigation. I hope this helps!