Temperature of body in space

I'll try to work it out from here.In summary, the conversation discusses determining the temperature of a body in space exposed to the Sun as it varies with time. The equation for this is given, but the method of finding the temperature is not clear. Suggestions for finding the temperature include evaluating the integral numerically or using approximations, but it is noted that these methods may not work for all temperatures.
  • #1
sid_galt
502
1
I want to determine the temperature of a body in space exposed to the Sun as it varies with time.
I tried this.
Let A be the area exposed to the Sun and 2A the total area of the body. Let 1380 W/m2 be the power of the sunlight falling on the body. Let [tex]\sigma[/tex] be the Boltzmann constant, [tex]\epsilon[/tex] the emissivity of the body and T the temperature at a a particular instant of time, m the mass of the body and C the specific heat constant. Then
[tex]
\displaystyle\frac{dT}{dt} = \displaystyle\frac{1380A - 2\sigma\epsilon A T^4}{mC}[/tex]
[tex]
\displaystyle\frac{mC}{A}\int\displaystyle\frac{1}{1380 - 2\sigma\epsilon T^4}dT = \int dt
[/tex]
I tried to integrate it on integrals.wolfram.com taking Boltzmann constant as 5.6E-8 and emissivity as 0.7. The result was
[tex]\displaystyle\frac{0.156942mC}{A}(\arctan[0.00230862T]+arctanh[0.00230862T]) + C' = t[/tex]

C' is here the integration constant

I don't know how to proceed further. Can anyone help please?
Thank you
 
Last edited:
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  • #2
Can anyone help please?
 
  • #3
Are you sure it's T^4? My thermodynamics knowledge is limited but I thought the rate of heat transmition was proportional to deltaT, I might be wrong though. Otherwise, there is no analytic solution to the lower equation for T.
Edit - nevermind, it's blackbody radiation. Yeah, sorry, can't figure out what's wrong with either your physics or math. The last equation is not solvable for T.
 
Last edited:
  • #4
But there must be someway to find temperature as a function of time for a body in space exposed to the Sun.
 
  • #5
sid_galt said:
But there must be someway to find temperature as a function of time for a body in space exposed to the Sun.

Yes, there is. You can evaluate your integral numerically.
 
  • #6
But how do I integrate high temperatures into the equation, say if I want to evaluate it for a body with an initial temperature of 1000 K.
My Arctanh would give an unreal value for all temperatures higher than 434.78 K.
 
  • #7
if you notice that ArcTan[x]+ArcTanh[x] for x > 1 always give some value a - 1.5708i , well , make it so your integration constant takes out the imaginary part.
Edit- more specifically
for x>1
ArcTanh[x] = ArcTanh[1/x] -1/2pi*i

Edit: Another approximation:
for small x
ArcTanh[x] = ArcTan[x]

another nice identity:
(you might be able to solve for this actually)
[tex]tan^{-1}(x) +tan^{-1}(y) = tan^{-1}(\frac{x+y}{1-xy})[/tex]

Last edit - I tried all the above, it doesn't work (not even with shoddy approximations).
 
Last edited:
  • #8
µ³ said:
if you notice that ArcTan[x]+ArcTanh[x] for x > 1 always give some value a - 1.5708i , well , make it so your integration constant takes out the imaginary part.

I didn't notice that before. Thanks for the help.
 

1. What happens to the temperature of the body in space?

The temperature of the body in space can vary depending on a few factors. If the body is in direct sunlight, it can become very hot due to the lack of atmosphere to protect it from the sun's rays. However, if the body is in the shadow of a spacecraft or any other object, it can become very cold due to the lack of atmosphere to trap heat.

2. How does the lack of atmosphere affect body temperature in space?

The lack of atmosphere in space means there is no medium for heat to transfer through. This means that the body cannot lose or gain heat through convection or conduction, and can only do so through radiation.

3. Can the body survive extreme temperatures in space?

Yes, the body can survive extreme temperatures in space for short periods of time. However, prolonged exposure to extreme temperatures can be dangerous and even fatal. This is why astronauts wear specialized suits that regulate their body temperature during spacewalks.

4. What is the average temperature of the human body in space?

The average temperature of the human body in space is around 98.6°F (37°C). This is because the body's internal temperature is regulated by its own mechanisms, such as sweating and shivering, and is not affected by the external temperature.

5. How does the temperature of the body in space affect bodily functions?

The extreme temperatures in space can have a significant impact on bodily functions. For example, the lack of gravity can cause changes in blood flow and the distribution of fluids in the body, which can affect the body's ability to regulate its temperature. Additionally, extreme cold temperatures can also cause numbness and reduced dexterity, making it more difficult for astronauts to perform tasks.

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