"If you obey all the rules you miss all the fun." - Katharine Hepburn
Question #91463 posted on 06/03/2019 12:12 a.m.
Q:

Dear astronomers and physicists of the 100 Hour Board,

The artistic renditions of how the other planets in the solar system would look from earth if they were as close as the moon in this article are super cool. But I have a question. How much light (and, even, maybe, heat) would reflect onto the earth from each of these planets? The artist has kept the level of light in the surrounding earth the same, as if each planet gave off the same amount of light as the moon. But Jupiter would take up half the sky, so would it in fact make the nighttime on Earth significantly brighter? Do different planets made of different materials reflect different wavelengths and quantities of light back into the atmosphere around them? If you want a fun hypothetical, I'd love for you to pick a planet or two, pretend earth is actually a moon to that planet, and try to (a) calculate how much it would illuminate the night sky and (b) speculate how life might be different on a planet that gets that much more light.

-Not Kepler

A:

Dear Nepler,

This is a great question and it requires a few calculations that I'll reproduce here. First off, we are going to assume that planets radiate very little (which is not always the case) and that all of the light that they give comes from reflecting solar radiation. The percentage of light reflected off a planet relative to the light incident upon it can be expressed as a decimal between 0 and 1 and is known as a planet's albedo. For example, Mars has an albedo of 0.25 which means that it reflects 25% of the light incident upon its surface back into space. Here is a table of known planetary albedos:

Planet Albedo
Mercury 0.088
Venus 0.76
Mars 0.25
Jupiter 0.503
Saturn 0.342
Uranus 0.3
Neptune 0.29
Moon 0.12

(Source)

Ok, that's great, but how much energy are the planets getting from the sun in their orbits? To figure this out, we have to look at the luminosity of the Sun, which puts out 3.83×1026 W (or, that many joules of energy per second). That energy spreads out as a sphere which gets larger the farther out you go, so the amount of energy that hits a given square meter of area decreases as you get farther from the Sun. To calculate how much of the Sun's energy hits a given planet, you have to imagine a sphere of radiation with the Sun at the center that has a radius of the orbital distance of the planet. Then imagine the cross-sectional area of the planet on that sphere. You can calculate the area of that planet and express it as a percentage of the total area of the solar radiation sphere. 

For example, Mercury is 5.79E+10 meters away from the Sun and has a radius of 2.44E+6 m. A sphere with the Sun at the center would have a surface area of 4π(5.79E+10)2 = 4.21E+22 m2. The planet itself only has a cross-sectional area of π(2.44E+6)2 m2 = 1.87E+13. Taking the ratio of these two will tell us how much of the Sun's energy the planet gets at that distance. In the case of Mercury, that's 4.44E-10 (or 0.0000000444% which is, of course, REALLY small). Taking this percentage of the Sun's luminosity would give you the energy incident upon the surface of that planet every second. Multiply that by the albedo and BAM, you've got yourself the total reflected energy by the planet. I'll spare you the remainder of the details and just give a final rendering of this here in lunar luminosities (that is, 1 = the amount of energy reflected by the moon).

Planet Reflected Energy
Mercury 9.6
Venus 146.5
Mars 3.4
Jupiter 261.2
Saturn 37.2
Uranus 14.6
Neptune 5.4
Moon 1.0

Now, you could do all that math again, assuming that the reflected energy is radiated in a half-sphere (because only the half of the planet exposed to the sun reflects energy) and that the Earth would intersect a portion of that hemisphere at the lunar orbital distance, but that would only give you the actual number of watts that the Earth picks up at that distance which is specificity we don't need. The ratios are exactly the same in both cases so we can look at the above numbers as being an expression of how much brighter each planet would appear than the Moon. In other words, if we were orbiting Jupiter at the distance the moon is from the Earth, Jupiter would be 261 times brighter than the Moon!

[Side note: in actuality, Jupiter's Roche limit is much larger than the lunar orbital distance. The Roche limit is the point at which external gravitational forces are greater than the internal gravitational forces of the body itself and the body crumbles apart. If we were at the lunar orbital distance around Jupiter, the Earth would disintegrate and we would all die. But let's ignore that and assume that some great celestial superglue has been applied to the planet (chemical bonds could potentially overcome the problems associated with being inside the Roche limit) and that the Earth can handle the massive tidal forces involved here.]

Of course, this would all be different if, instead of us being transported to orbit other planets, they were transported to us to be our moon (again, impossible since the gravitational disruptions would in many cases destroy the Earth, but again, pretending). Then, the planets would each be brighter or dimmer for being closer to or farther away from the Sun. I re-did the calculations in this case and came up with these numbers:

Planet Reflected Energy
Mercury 1.4
Venus 76.6
Mars 7.9
Jupiter 7,076.3
Saturn 3,419.2
Uranus 539.4
Neptune 489.5
Moon 1.0

Now, all of this feels very catastrophic, like somehow we'd all be destroyed by Jupiter-shine if it were our moon (assuming, again, that we weren't destroyed by its gravity. Which we totally would be.). But, remember that the Sun is about 400,000 times brighter than the moon. A measly 7000 times brighter wouldn't scorch us or anything. It would be very bright; perhaps bright enough to cast significant shadows. We might feel warmth from it on full-Jupiter nights. We'd see many fewer stars (perhaps only the 2 or 3 brightest in the sky), and Jupiter would take up a massive 20-degree swath of sky above us (and, not to belabor the point, we'd be dead and the Earth would be crumbled like one of those granola bars in the green packages, you know, that produce more crumbs than is physically possible? Yeah, like that.).

Now back to us orbiting planets in their current orbits about the Sun: we'd be getting different amount of sunlight if we were orbiting those planets. I'll again spare you the details (it's the same basic calculations we did before, just putting the Earth in different solar radiation spheres) and just give you the amount of sunlight we'd be getting in terms of the amount of light we get normally (i.e. 1 = the amount of sunlight we get in our own orbit). For your convenience, I'm adding a column for what would probably happen to us if we got that much light.

Planet Sunlight Status
Mercury 6.7 Dead
Venus 1.9 Dead
Mars 0.43 Dead
Jupiter 0.04 Dead
Saturn 0.01 Dead
Uranus ~0 Dead
Neptune ~0 Dead
Earth 1 Alive

In each case, the amount of sunlight we would be getting would be MUCH greater than the amount of light we'd be getting from planet-shine. Jupiter-shine tops out at 0.87% of the sunlight we'd get there. Saturn comes in at a little under 0.5%. Every other planet is negligible.

So, yeah, depending on the planet that we're orbiting (or that is orbiting us), we'd get anywhere from a few times to many thousands of times brighter light than we get from the full moon. It might give us a little noticeable heat, but the bright ones would just wash out stars, cast moon shadows, and generally make our nights brighter without much other difference. The real difference, in case you didn't pick this up from one of my many parenthetical statements, is that gravity would totally hose us and we'd all die. Space is scary and Earth is in a really lucky spot (or, alternatively, the way we've evolved to live is highly specific to this watery rock that we live on). I love our watery rock.

Best,

The Man with a Mustache  

(For those interested, here is the spreadsheet I used to do my calculations. Feel free to comment if I missed something.)

A:

Dear Kepler,

As you well know, you asked this question far in advance of Alumni Week. I picked it up, ran with it a little, got severely sidetracked by life, picked it up and ran with it a little more, then got bust with other things once more until MwaM swooped in and saved the day. So, to him, I offer my sincere thanks, and to you, my remorseful apology. 

Really cool question, though. Thanks for asking it!

-Frère Rubik