Expert: Philip A. Stahl - 5/8/2007

Question
Hi Philip,

I'm having a little trouble with a problem that I hope you can point me in the right direction with please!

The question I'm faced with is:
A supernovae remnant has an angular diameter of 4.3arcmins. and a flux at 100MHz of 1.6 × 10−19 erg cm−2 s−1 Hz−1. Assume the emission is thermal.
(i) What is the brightness temperature T?

So far I have:

T=[Ix(3x10^10cm/s)^2]/[2*(100*10^6Hz)^2 *(1.3807*10^-16 erg K^-1)]

The reason the speed of light is in cm/s is because I have a value for flux (1.6 × 10^−19 erg cm^−2 s^−1 Hz^−1) in cm^-2. When I put all these units together I'm faced with the flux needing to be multiplied by ster^-1 (or divided by the solid angle) to create I to put into the equation.

So my question is how do I create a solid angle value from arcmins. I've looked all over the internet and can't seem to find a clear explanation of the conversion... please help!

Thankyou very much... it's much appreciated :-)

Answer
Hello,

I think all you really need to solve your problem is a nudge in the right direction with respect to the units.

First, remember that:

1 steradian = the degrees in one radian squared

or 1 Sr =   (57.3 deg)^2  =  3, 283 deg^2

Recall also, that 1 arcmin = 1/60 deg

Thus, you can obtain the relation for Sr and (arcmin)^2

since:

57.3 deg =  [57.3 deg x 60 arcmin/deg] = 3438 arcmin

So 1 Sr =  (3438 arcmin)^2 =  11819844 arcmin^2

You should be able to get the solution from here. If you're still having problems after this 'nudge', let me know!