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Math Help - PLEASE help. Maths problem (astronomy/physics)

  1. #1
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    Question PLEASE help. Maths problem (astronomy/physics)

    The albedos of Saturn and Uranus are 0.50 and 0.65 respectively while the ratio of the planets radii is 2.50, Satur's being the larger. Their heliocentric distances (orbits being assumed circular and coplanar) are 9.5 and 19.2 AU respectively. Neglecting the effect of Saturn's rings, calculate the magnitude difference in the brightness of the two planets when both are observed at opposition.
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  2. #2
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    Quote Originally Posted by emmaastronomy View Post
    The albedos of Saturn and Uranus are 0.50 and 0.65 respectively while the ratio of the planets radii is 2.50, Satur's being the larger. Their heliocentric distances (orbits being assumed circular and coplanar) are 9.5 and 19.2 AU respectively. Neglecting the effect of Saturn's rings, calculate the magnitude difference in the brightness of the two planets when both are observed at opposition.
    What part of this are you having problems with.

    CB
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    Quote Originally Posted by CaptainBlack View Post
    What part of this are you having problems with.

    CB
    The whole the thing, i have no idea how to do it.
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    Quote Originally Posted by emmaastronomy View Post
    The whole the thing, i have no idea how to do it.
    1. The difference in magnitudes of two sources giving fluxes F_1 and F_2 is:

    \Delta m= 2.5 \log_{10}(F_1/F_2).

    2. If the two planets have mean distances from the Sun R_1 and R_2 (in AU) then their distances from the Earth at opposition are (R_1-1) and (R_2-1) AU (assuming circular orbits anyway).

    3. The Solar flux at the planets are k/R_1^2 and k/R_2^2 respectivly, where k is a constant who's value does not matter for this calculation. So the fluxs at the Earth due to reflected sunlight are:

    F_1=\frac{K \alpha_1 r_1^2}{R_1^2(R_1-1)^2},

     <br />
F_2=\frac{K \alpha_2 r_2^2}{R_2^2(R_2-1)^2}<br />
,

    where \alpha_1 and \alpha_2 are the albedos of the two planets and again K is a constant who's value does not matter and r_1 and r_2 are the planets radii.

    CB
    Last edited by CaptainBlack; February 27th 2009 at 03:27 AM. Reason: missed planetary radii
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    Quote Originally Posted by CaptainBlack View Post
    1. The difference in magnitudes of two sources giving fluxes F_1 and F_2 is:

    \Delta m= 2.5 \log_{10}(F_1/F_2).

    2. If the two planets have mean distances from the Sun R_1 and R_2 (in AU) then their distances from the Earth at opposition are (R_1-1) and (R_2-1) AU (assuming circular orbits anyway).

    3. The Solar flux at the planets are k/R_1^2 and k/R_2^2 respectivly, where k is a constant who's value does not matter for this calculation. So the fluxs at the Earth due to reflected sunlight are:

    F_1=\frac{K \alpha_1}{R_1^2(R_1-1)^2},

     <br />
F_2=\frac{K \alpha_2}{R_2^2(R_2-1)^2}<br />
,

    where \alpha_1 and \alpha_2 are the albedos of the two planets and again K is a constant who's value does not matter.

    CB
    I have tried this but the answer i get is 2.5log10 (7.67x10*-5/5.3x10*-6) = 36.2. I know this is incorrect because i have already been given the answer which is 4.89. I am probably doing the calculation wrong but i am not sure, i have repeated it many times. Maybe i am putting it into the calculator wrong, but i doubt it. The question refers to the planets radii, is this unimportant? Thanks for the help so far, i'd have given up entirely otherwise.
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    Quote Originally Posted by emmaastronomy View Post
    I have tried this but the answer i get is 2.5log10 (7.67x10*-5/5.3x10*-6) = 36.2. I know this is incorrect because i have already been given the answer which is 4.89. I am probably doing the calculation wrong but i am not sure, i have repeated it many times. Maybe i am putting it into the calculator wrong, but i doubt it. The question refers to the planets radii, is this unimportant? Thanks for the help so far, i'd have given up entirely otherwise.
    Note I have just changed the earlier post as I missed off the planetary radii (but should only have altered things by about 2 magnitudes).

    CB
    Last edited by CaptainBlack; February 27th 2009 at 07:06 AM. Reason: spelling
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  7. #7
    Grand Panjandrum
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    Quote Originally Posted by emmaastronomy View Post
    I have tried this but the answer i get is 2.5log10 (7.67x10*-5/5.3x10*-6) = 36.2. I know this is incorrect because i have already been given the answer which is 4.89. I am probably doing the calculation wrong but i am not sure, i have repeated it many times. Maybe i am putting it into the calculator wrong, but i doubt it. The question refers to the planets radii, is this unimportant? Thanks for the help so far, i'd have given up entirely otherwise.
    My calculations give assuming the radius of Uranus is 1 (which we can do as we are working with ratios):

    Code:
    >F1=(0.5*2.5^2)/(9.5^2*8.5^2)
      0.000479253 
    >F2=0.65/(19.2^2*18.2^2)
     5.32314e-006 
    >
    >
    >2.5*log10(F1/F2)
          4.88599 
    >
    Also you have an error in your arithmetic, your result should be: 2.9 magnitude (the difference between this and the above is the approx. 2 magnitudes from my having forgotten the planetary radii (the way that you have written your answere suggests you handled the powers of 10 incorrectly)).

    Reality check: Maximum brightness of Saturn 0.7 Mag, maximum brightness of Uranus 5.5 Mag.

    CB
    Last edited by CaptainBlack; February 27th 2009 at 04:13 AM.
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    Quote Originally Posted by CaptainBlack View Post
    My calculations give assuming the radius of Uranus is 1 (which we can do as we are working with ratios):

    Code:
    >F1=(0.5*2.5^2)/(9.5^2*8.5^2)
      0.000479253 
    >F2=0.65/(19.2^2*18.2^2)
     5.32314e-006 
    >
    >
    >2.5*log10(F1/F2)
          4.88599 
    >
    Also you have an error in your arithmetic, your result should be: 2.9 magnitude (the difference between this and the above is the approx. 2 magnitudes from my having forgotten the planetary radii (the way that you have written your answere suggests you handled the powers of 10 incorrectly)).

    Reality check: Maximum brightness of Saturn 0.7 Mag, maximum brightness of Uranus 5.5 Mag.

    CB
    Thanks, i am a moron.
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