Interaction of an astronaut with the CMB flux

[astroYEEET]

Homework Statement

Suppose the body of an astronaut is a perfect spherical absorber with mass 𝑚 = 60 kg and the average density and specific thermal conductivity are d = 1000 kg m-3 and 𝐶 = 4200 J kg-1 K-1
a) What is the approximate rate of energy absorption of the
astronaut due to cosmic microwave radiation (CMB)
The spectral energy distribution of CMB can be approximated by T(CMB) = 2,728 K
b) Approximately how many CMB photons per second will the astronaut absorb?
c) In how much time will CMB radiation increase the temperature of the astronaut by Δ𝑇 = 1 K?

Relevant Equations

E=σΤ^4 where c=5,67×10^−8 W/m^2K^4, Maximum frequency distribution h=3KT where K = 1.38x10^-23 j / K, Q = m 𝐶 ΔT where
Q the heat absorbed by a body of mass m causing an increase in its temperature by ΔΤ and 𝐶 constant of the material

From the density and the mass we can find the volume using d=m/v <=> v=0.06 m^3. Since we consider the astronaut a sphere we find his radius using V(sphere)=4/3*π*R^3 =>R=0.242m. Now we can calculate the surface area with the formula A=4πR^2=0.735m^2.
The energy absorbed will be i suppose equal to E=σΤ^4=3.14*10^-6 J.
Could you offer me any ideas and hints on how to continue the solution. Thanks in advance for your help. (If i have done something wrong please inform me)

 

[TSny]

Welcome to PF!

Homework Statement:: Suppose the body of an astronaut is a perfect spherical absorber with mass 𝑚 = 60 kg and the average density and specific thermal conductivity are d = 1000 kg m-3 and 𝐶 = 4200 J kg-1 K-1

$c$ is the specific heat capacity, not the thermal conductivity.

Relevant Equations::

E=σΤ^4 where c=5,67×10^−8 W/m^2K^4

Note that E here cannot represent energy. To see this, what are the overall dimensions (or units) of σΤ⁴?

Maximum frequency distribution h=3KT where K = 1.38x10^-23 j / K

I'm not sure what you mean by this. Does h represent Planck's constant? If so, the equation h = 3kT cannot be correct since the overall units for each side do not match.

Q = m 𝐶 ΔT where
Q the heat absorbed by a body of mass m causing an increase in its temperature by ΔΤ and 𝐶 constant of the material

Yes. Here, $c$ is the specific heat capacity of the material.

From the density and the mass we can find the volume using d=m/v <=> v=0.06 m^3. Since we consider the astronaut a sphere we find his radius using V(sphere)=4/3*π*R^3 =>R=0.242m. Now we can calculate the surface area with the formula A=4πR^2=0.735m^2.

Looks good (although I get A = .741 m²)

The energy absorbed will be i suppose equal to E=σΤ^4=3.14*10^-6 J.

As noted above, σΤ⁴ does not give energy.

Could you offer me any ideas and hints on how to continue the solution. Thanks in advance for your help. (If i have done something wrong please inform me)

Once you have the correct physical interpretation of σΤ⁴, you might see how to proceed. What are the overall units of σΤ⁴?

 

[TSny]

It looks like part (c) wants you to completely ignore any thermal radiation emitted by the astronaut's body. If the astronaut is alive, then he will emit much more radiant energy than he will absorb. So, his body would cool rather than heat up. But, I guess you are to assume that the only energy change of the body is due to CMB radiation absorbed.