# Time evolution of temperature help

• fasterthanjoao
In summary, the conversation is discussing a question related to the time evolution of temperature in a cooling process. The conversation involves equations and constants such as the Stefan-Boltzmann constant and the Boltzmann constant. The equation for the time evolution of temperature is shown and the conversation involves rearranging and solving for the temperature. Ultimately, the conversation concludes with the solution being found and the problem being resolved.
fasterthanjoao
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:

$$\frac{d}{dt} \frac{\left(3MkT\right)}{m_p} = -4*pi*R²*T^4$$ -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.

Last edited:
I don't see the problem. This equation is separable:

$$\frac{dT}{T^4}=\frac{-4\pi m_pR^2\sigma}{3Mk}dt$$

yeah i see that, I've tried to take it some steps further but I'm getting in a bit of a mess. could you try the next step or two, not sure what it is I'm having the problem with.

$$\frac{1}{3}(T_0^{-3}-T^{-3})=\frac{-4\pi m_pR^2\sigma t}{3Mk}$$

$$T^{-3}=T_0^{-3}+\frac{4\pi m_pR^2\sigma t}{Mk}$$

$$\frac{T_0^3}{T^3}=1+\frac{4\pi m_pR^2T_0^3\sigma t}{Mk}$$

That help?

Last edited:
very much so. thanks. glad to lay that to rest

## 1. How does temperature change over time?

The temperature of a system can change over time due to various factors such as external heat sources, internal energy transfer, and changes in the surrounding environment. The specific changes in temperature over time will depend on the specific conditions and properties of the system.

## 2. What is the relationship between time and temperature?

The relationship between time and temperature is highly dependent on the specific system being studied. In general, as time increases, the temperature of a system may increase, decrease, or remain constant. This is determined by the specific processes and mechanisms affecting the temperature at that particular time.

## 3. How can we measure the time evolution of temperature?

The time evolution of temperature can be measured through various methods such as using temperature sensors, thermometers, or thermal imaging cameras. These tools allow for the continuous monitoring of temperature over a specific period of time, providing valuable data on its evolution.

## 4. What factors influence the time evolution of temperature?

Several factors can influence the time evolution of temperature, including the properties of the system, the surrounding environment, and any external heat sources. Additionally, the rate of energy transfer within the system and the specific processes occurring can also impact the temperature change over time.

## 5. How can we predict the future temperature of a system based on its time evolution?

Predicting the future temperature of a system based on its time evolution can be challenging as it depends on many variables and factors. However, by analyzing the data obtained from temperature measurements over time and understanding the underlying processes and mechanisms, scientists can make informed predictions about the future temperature of a system.

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