- #1

fasterthanjoao

- 731

- 1

First year astronomy question, goes as follows:

"A closely exact description of the cooling described above is to consider the differential equation which says that rate of charge of thermal energy equals the rate of radiative

output. That is:

[tex] \frac{d}{dt} \frac{\left(3MkT\right)}{m_p} = -4*pi*R²*T^4[/tex] -eqtn1

where M, R are constant. Show that the resulting time evolution of the temperature is then given by:

T(t)```````````````1

--- = ---------------

T(0) ``` (1+3t/t_cool)^(1/3)

(kept getting problems in my latex for that part, so i just typed it out..., i'd like to get this question sorted before i have to go to math)

anyway, its also said that:

````````````3Mk

t_cool = -----------

``````````4*pi*R²*T³*sigma

everything has its usual meaning, sigma is the Stefan-Boltzmann constant, k is the Boltzmann constant.

------------

i'm not really that sure what to do, I tried rearranging eqtn one after taking dT/dt out, "multiplying" each side by dt then integrating. after that, I'm really not much closer to the point. any guides on where to go appreciated. thanks.

"A closely exact description of the cooling described above is to consider the differential equation which says that rate of charge of thermal energy equals the rate of radiative

output. That is:

[tex] \frac{d}{dt} \frac{\left(3MkT\right)}{m_p} = -4*pi*R²*T^4[/tex] -eqtn1

where M, R are constant. Show that the resulting time evolution of the temperature is then given by:

T(t)```````````````1

--- = ---------------

T(0) ``` (1+3t/t_cool)^(1/3)

(kept getting problems in my latex for that part, so i just typed it out..., i'd like to get this question sorted before i have to go to math)

anyway, its also said that:

````````````3Mk

t_cool = -----------

``````````4*pi*R²*T³*sigma

everything has its usual meaning, sigma is the Stefan-Boltzmann constant, k is the Boltzmann constant.

------------

i'm not really that sure what to do, I tried rearranging eqtn one after taking dT/dt out, "multiplying" each side by dt then integrating. after that, I'm really not much closer to the point. any guides on where to go appreciated. thanks.

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