PLEASE help. Maths problem (astronomy/physics)

**emmaastronomy** · February 25th 2009, 03:32 AM

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.

**CaptainBlack** · February 25th 2009, 12:22 PM

Originally Posted by emmaastronomy

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

**emmaastronomy** · February 25th 2009, 10:45 PM

Originally Posted by CaptainBlack

What part of this are you having problems with.

CB

The whole the thing, i have no idea how to do it.

**CaptainBlack** · February 25th 2009, 11:02 PM

Originally Posted by emmaastronomy

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

**emmaastronomy** · February 27th 2009, 01:32 AM

Originally Posted by CaptainBlack

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.

**CaptainBlack** · February 27th 2009, 03:29 AM

Originally Posted by emmaastronomy

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

**CaptainBlack** · February 27th 2009, 03:59 AM

Originally Posted by emmaastronomy

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

**emmaastronomy** · February 27th 2009, 05:23 AM

Originally Posted by CaptainBlack

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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