Calculating solar irradiance at each planet?

[bbbl67]

So I'm getting somewhat weird numbers when trying to calculate the solar irradiance at each planet. Starting with a baseline of irradiance at Earth of 1376 W/m^2, I use the inverse square law against distance. I find the values for Mercury and Mars are really off, while Venus & Neptune are almost right on the money! What's wrong with my method?

1. Planet: calculated value (W/m²), actual value (W/m²), percent diff
   
2. Mercury: 9183, 6283, 46.15%
   
3. Venus: 2630, 2600, 1.15%
   
4. Mars: 592.7, 710.6, 19.89%
   
5. Jupiter: 50.82, 47.42, 6.69%
   
6. Saturn: 15.128245, 13.51, 10.70%
   
7. Uranus: 3.736, 3.465, 7.25%
   
8. Neptune: 1.522, 1.526, 0.27%

I'm taking the mean orbital radius and solar flux as stated in Wolfram Alpha, for example:
mars average orbital radius - Wolfram|Alpha
solar flux at uranus - Wolfram|Alpha

 

[mathman]

Possible problem for Mars, orbit is fairly elliptical, so average radius may not be accurate enough.

 

[Charles Link]

Please show your calculations, especially for mercury. I think your arithmetic must be off, or else their number is incorrect. $\\$ Edit: See: https://nssdc.gsfc.nasa.gov/planetary/factsheet/mercuryfact.html They give 9082.7 for Solar irradiance on about line 15.

 

[bbbl67]

Possible problem for Mars, orbit is fairly elliptical, so average radius may not be accurate enough.

And Mercury has that issue with General Relativity.

 

[bbbl67]

Please show your calculations, especially for mercury. I think your arithmetic must be off, or else their number is incorrect. $\\$ Edit: See: https://nssdc.gsfc.nasa.gov/planetary/factsheet/mercuryfact.html They give 9082.7 for Solar irradiance on about line 15.

Well, I'm just using Wolfram Alpha to do the calculations:
Mercury: 1376 W/m^2 * (Earth average orbit radius)^2 / (Mercury average orbit radius)^2 - Wolfram|Alpha
Mars: 1376 W/m^2 * (Earth average orbit radius)^2 / (Mars average orbit radius)^2 - Wolfram|Alpha

So you're saying my calculations are more right than the values in Wolfram-Alpha?

 

[Charles Link]

Well, I'm just using Wolfram Alpha to do the calculations:
Mercury: 1376 W/m^2 * (Earth average orbit radius)^2 / (Mercury average orbit radius)^2 - Wolfram|Alpha
Mars: 1376 W/m^2 * (Earth average orbit radius)^2 / (Mars average orbit radius)^2 - Wolfram|Alpha

So you're saying my calculations are more right than the values in Wolfram-Alpha?

Who got 6283 for mercury? Where did that come from? You are unclear in your post which one is yours.

 

[Bandersnatch]

The values reported by W-A when you type in e.g. 'Mercury solar irradiance' are (as noted) based on time-dependent distance from the Sun. I.e., these are not the average values, but values at this particular moment.
This simply means that for planets whose current position in their elliptical orbits is significantly different than the average distance, the current solar irradiance will also differ significantly from what you get from calculations which assume circular orbit.

If you scroll down the results page, you'll see an entry along the lines of X * average solar irradiance of Y. Those should be closer to what you're calculating.

 

[OmCheeto]

These are the values I get, along with bbbl67's:

values are watts/meter^2
\begin{matrix}
Planet & peri & ave & aph & current & wolf & bbbl67\\
Mercury & 14600 & 9187 & 6309 & 6452 & 9183 & 6283\\
Venus & 2669 & 2632 & 2596 & 2629 & 2630 & 2600\\
Earth & 1424 & 1376 & 1323 & 1392 \\
Mars & 720 & 591 & 493 & 718 & 593 & 711 \\
Ceres & 210 & 179 & 155 & 204 \\
Jupiter & 56.2 & 50.8 & 46.2 & 47.9 & 50.8 & 47.4\\
Saturn & 16.8 & 15.0 & 13.5 & 13.7 & 15.1 & 13.5\\
Uranus & 4.11 & 3.73 & 3.41 & 3.50 & 3.74 & 3.47\\
Neptune & 1.55 & 1.52 & 1.50 & 1.54 & 1.52 & 1.53\\
Pluto & 1.56 & 0.88 & 0.57 & 1.22
\end{matrix}

"current" values are based on data from 3 different sources. All of them were a tad bit different.
Sources:

NASA Eyes application
NASA Heliocentric Trajectories for Selected Planets
Distance- Brightness- and Size of Planets​

 

[Charles Link]

@OmCheeto Thank you=that explains it. Mercury must have a very elliptical orbit.

 

[OmCheeto]

@OmCheeto Thank you=that explains it. Mercury must have a very elliptical orbit.

Almost as elliptical as Pluto.

distances in AU
\begin{matrix}
planet & aph & perih & aph/perih \\
Mercury & 0.467 & 0.307 & 1.52 \\
Venus & 0.728 & 0.718 & 1.01 \\
Earth & 1.02 & 0.983 & 1.04 \\
Mars & 1.67 & 1.382 & 1.21 \\
Ceres & 2.98 & 2.56 & 1.16 \\
Jupiter & 5.46 & 4.95 & 1.10 \\
Saturn & 10.1 & 9.04 & 1.12 \\
Uranus & 20.1 & 18.3 & 1.10 \\
Neptune & 30.3 & 29.8 & 1.02 \\
Pluto & 49.3 & 29.7 & 1.66
\end{matrix}

Btw, I had everything locked and loaded from a HW problem from last December.
When I looked at the equation for "watts/m² = 1376 / r²", it really confused me. It was missing all manner of values, π's and squares. What kind of voodoo math was I up to that day? Then I realized those all dropped out because Earth is at 1 AU.

 

[bbbl67]

The values reported by W-A when you type in e.g. 'Mercury solar irradiance' are (as noted) based on time-dependent distance from the Sun. I.e., these are not the average values, but values at this particular moment.
This simply means that for planets whose current position in their elliptical orbits is significantly different than the average distance, the current solar irradiance will also differ significantly from what you get from calculations which assume circular orbit.

If you scroll down the results page, you'll see an entry along the lines of X * average solar irradiance of Y. Those should be closer to what you're calculating.

Ah, I didn't realize that Wolfram-Alpha used real-time values for those!