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

  1. Oct 18, 2015 #1
    1. The problem statement, all variables and given/known data
    A star has an effective temperature [itex]T_{eff}[/itex] and is observed to have a flux [itex]F[/itex]. Show that the angular radius in arcseconds of the star (as seen from Earth) is given by

    [itex]
    \theta = (\frac{2.06 \times 10^5}{T_{eff}^2}) \sqrt{\frac{F}{\sigma}}
    [/itex]

    2. Relevant equations
    [itex]
    L = 4 \pi R^2 \sigma T_{eff}^4 \\
    T_{eff} = (\frac{L}{4 \pi R^2 \sigma})^{\frac{1}{4}} \\
    F = \frac{L}{4 \pi d^2}
    [/itex]

    And probably..
    [tex]
    \omega = \frac{A}{R^2}
    [/itex] ?

    3. The attempt at a solution

    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?

    First I started by rearranging the eqn for Flux for L
    [itex]
    F = \frac{L}{4 \pi d^2} \\
    L = 4 \pi d^2 F
    [/itex]
    And then subsituted that into the eqn for T_eff
    [itex]
    T_{eff} = (\frac{L}{4 \pi R^2 \sigma})^{\frac{1}{4}} \\
    T_{eff} = (\frac{4 \pi d^2 F}{4 \pi R^2 \sigma})^{\frac{1}{4}} \\
    T_{eff}^4 = (\frac{ d^2 F}{R^2 \sigma}) = (\frac{d}{R})^2 (\frac{F}{\sigma}) \\
    (\frac{R}{d})^2 = \frac{1}{T_{eff}^4}(\frac{F}{\sigma}) \\
    (\frac{R}{d}) = \sqrt{\frac{1}{T_{eff}^4}(\frac{F}{\sigma})} \\
    \frac{R}{d} = \frac{1}{T_{eff}^2} \sqrt{\frac{F}{\sigma}} \\
    [/itex]

    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.

    EDIT: yes It does and I can see how now. My bad.
     
    Last edited: Oct 18, 2015
  2. jcsd
  3. Oct 18, 2015 #2

    gneill

    User Avatar

    Staff: Mentor

    Yep. Looks good.
     
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