# Homework Help: Radiation in space (Thermodynamics)

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1. Sep 22, 2015

### alivedude

1. The problem statement, all variables and given/known data

A space probe carry an instrument that detects radiation in space, the area of the probe is $5000 \, cm^2$ and its collecting data for 10 minutes. The information sent back to earth is given from this image below, how much radiation energy is it for 10 minutes?

(frequency on the x-axis)

2. Relevant equations

3. The attempt at a solution

I have no idea how to approach this problem, the answer is $10-50 \, MJ$ but I have no hints or anything so I'm lost. Been playing around with the dimensions a bit but its pretty much just guessing. Any hints?

2. Sep 22, 2015

### haruspex

Yes, please quote Planck's radiation law. Does its general shape match the graph? If so what values for the unknowns in the equation would make it match the graph exactly?

3. Sep 22, 2015

### rude man

The y axis units don't make sense to me. Especially the m-3 factor.

4. Sep 23, 2015

### alivedude

$u(\nu , T)=\frac{8 \pi h \nu^3}{c^3} \cdot \frac{1}{e^{\frac{h \nu}{kT}}-1}$

I have really no idea to be honest.

5. Sep 23, 2015

### alivedude

exactly, clueless

6. Sep 23, 2015

### haruspex

7. Sep 23, 2015

### haruspex

Compare your graph with the family of curves at https://en.m.wikipedia.org/wiki/Planck's_law. Looks promising to me.
What variables in the general law can you tune to try to match your given curve?

8. Sep 23, 2015

### alivedude

Im thinking like this:

integrate $u(\nu) d\nu$ and then multiply by the volume given by $A \cdot c \cdot t$ where A is the arean of the probe, $c$ is the speed of radiation(light) and $t$ is the time (10 mins)

Is this a crazy idea? I think that integrating that #u(\nu)# is beyond the course but maybe i can approx the area under the curve?

9. Sep 23, 2015

### haruspex

Yes indeed, you need the area under the curve. But the method I was leading you to is to identify the source temperature that gives rise to the observed spectrum. Armed with that, no approximation is necessary. The integral has an analytic solution (which you can look up at the page I linked to).
However, on closer inspection I'm beginning to suspect there is an error in the question. It seems to me that the frequencies on the given graph should multiples of 1015, not 1014.

If you feel happy just approximating the area by hand, feel free.

10. Sep 23, 2015

### alivedude

Im going to check this out, but I think that I understand now. Thanks a lot!

11. Sep 23, 2015

### rude man

The data from the spacecraft has to be a spectral energy density, which would be J - s which, when integrated over the frequencies gives the required units which is J. That's what the problem states: "how much radiation energy is it for 10 minutes?"

The fact that the integration time is over 10 minutes and over an area of 5000 cm2 is of no computational consequence.
So the units on the given chart, which was represented as the acquired data, should have been J Hz-1 or J-s.

Of course, an on-board computer might have taken area and integration time into account, sending out B which would have the units J m-2 s-1 s = J m-2. Again, though, B ⋅ 5000 cm2 ⋅ 10 minutes integrated over frequency would give J. This B has the same units as in the wikipedia article: "The SI units of are W·sr−1·m−2·Hz−1".
Dimensionally:
W = J s-1
Hz-1 = s
The graph given has units for $B_{\lambda}$ on the Y axis, not $B_{\nu}$. Unless that is a mistake, the area under the curve will not give the right answer. You would need to convert the graph to have the X axis in terms of wavelength first.