Show that the angular radius of the star is given by....

[FaraDazed]

Homework Statement

A star has an effective temperature $T_{eff}$ and is observed to have a flux $F$. Show that the angular radius in arcseconds of the star (as seen from Earth) is given by

$\theta = (\frac{2.06 \times 10^5}{T_{eff}^2}) \sqrt{\frac{F}{\sigma}}$

Homework Equations

$L = 4 \pi R^2 \sigma T_{eff}^4 \\<br /> T_{eff} = (\frac{L}{4 \pi R^2 \sigma})^{\frac{1}{4}} \\<br /> F = \frac{L}{4 \pi d^2}$

And probably..
[tex] \omega = \frac{A}{R^2}<br /> [/itex] ?<br /> <br /> <h2>The Attempt at a Solution</h2> <br /> [/B]<br /> I am a bit lost here! I missed several of the lecture containing this material, but from looking at what I have go already I should be able to do it?<br /> <br /> First I started by rearranging the eqn for Flux for L<br /> $F = \frac{L}{4 \pi d^2} \\<br /> L = 4 \pi d^2 F$<br /> And then subsituted that into the eqn for T_eff<br /> $T_{eff} = (\frac{L}{4 \pi R^2 \sigma})^{\frac{1}{4}} \\<br /> T_{eff} = (\frac{4 \pi d^2 F}{4 \pi R^2 \sigma})^{\frac{1}{4}} \\<br /> T_{eff}^4 = (\frac{ d^2 F}{R^2 \sigma}) = (\frac{d}{R})^2 (\frac{F}{\sigma}) \\<br /> (\frac{R}{d})^2 = \frac{1}{T_{eff}^4}(\frac{F}{\sigma}) \\<br /> (\frac{R}{d}) = \sqrt{\frac{1}{T_{eff}^4}(\frac{F}{\sigma})} \\<br /> \frac{R}{d} = \frac{1}{T_{eff}^2} \sqrt{\frac{F}{\sigma}} \\$<br /> <br /> That is as far as I have got. I assume in mine, R/d is the angle in radians, so I hope that it relates to the angle in arcsecond by that 2.06 * 10^5 factor? Or I am way off. I'd appreciate some help/advie with this please.<br /> <br /> EDIT: yes It does and I can see how now. My bad.[/tex]

 

[gneill]

Yep. Looks good.