Interchanging planets
Published on Nov 4, 2006 at 2:15 pm.
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Filed under amusement, planets, skywatching.
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Here’s just a fun post. It isn’t really all that scientific or realistic. It is just something that I wondered about. You see, when you look at the different planets from Earth, they appear different brightnesses. Venus blazes away at magnitude -4.5 at its brightest. Mars can get as bright as -2.9, somewhat brighter than the brightest star, though normally it is far dimmer than that. Mercury, at its brightest can be nearly -2.1 magnitude.
One of my students wrote something this past week that got me to thinking about what would happen if the planets were in a different order. How would these brightesses appear. Now, the gas giants, we’ll leave alone, since moving them around would wreck havoc with the orbits of the rest. And, we don’t want to move Earth, since that would make it uninhabitable. But, what if Mars and Venus were interchanged? Now, in real life, if Venus were where Mars is today, it would probably not have the coulds of sulfuric acid around it, and Mars located where Venus is would be quite different from the Mars that we have today. But, ignoring all that, lets assume that you could just pick up the planets and interchange them, without changing the characteristics of the planets. So, what we are talking about here is simply a thought experiment, nothing more, and not even a particularly realistic one. This is just for fun, so I wanted to keep it simple to keep the math easy.
So, first of all, moving Mars to Venus shouldn’t affect Earth’s orbit, since Mars is much smaller than Venus. Oh, but being smaller means that it doesn’t have as much surface area to reflect light. That means less light would come to us, and that would make Mars appear dimmer. By my back-of-the-envelope figuring, the smaller size would make it about 0.315 times as bright. Oh, but Mars is rocky, rather than covered in reflective clouds (and likely still would be even if you did move it to Venus’ location). That means that Mars reflects only 15% of the light that hits it, rather than the 65% that Venus does. That makes Mars only 0.231 times as bright. Putting this together, we find that Mars will appear only 7.27% as bright as Venus if you were to swap the two. Doing a little algebra to the equations relating brightness to magnitude, you find that
Δ m = log(L1/L2 ) / log( 2.512)
which means that Mars will appear 2.85 magnitudes dimmer than Venus if you were to simply interchange them. If all other things were equal, that would also mean that Venus would look 2.85 magnitudes brigher than Mars if you swapped the two. That would make Mars peak out at magnitude -1.6 and Venus at -5.8.
But, you might wonder at this. Because even though it would be closer to us in Venus’ orbit, Mars would actually appear dimmer than it currently does if you move it to Venus’ orbit! What is going on? Well, this has to do with the fact that superior planets (those farther from the Sun than Earth) appear brightest at opposition, at which time they are closest to us and are also fully lit by the Sun. Inferior planets (those closer to the Sun than Earth), however, have the far side of them lit at closest approach, and they are very far away when they are fully, or nearly fully lit. So, they appear brightest when the effects of the combination of their distance and percentage lit add up to be the brightest. That happens when they appear as a crescent. So, Mars would have far less of itself lit up at this stage, and it is far less reflective than Venus. But, Venus would be reflecting light from its entire disk if it were at Mars’ orbit, so even though it was farther from the Sun (with dimmer sunlight) and farther from Earth, it would still appear much brighter. In fact, it would appear even brighter from Mars’ orbit than it does from its own orbit!
But, this leaves out an interesting point. If Venus really were at Mars’ distance from the Sun, it likely would not have the dense layer of highly reflective sulfuric acid clouds that it currently has. In fact, it is not at all clear what Venus would be like. If it were just rocky like Mars, then it would be only 1.26 magnitudes brigher than Mars, or magnitude -4.2. That still would make it nearly as bright as it currently is. Most likely, though, Venus would be covered in snow and glaciers. That would make its albedo (how much it reflects) about 0.4. Using this figure, I work out the difference in magnitude to be about 2.3, so Venus would appear as about magnitude -5.2, still brighter than it appears today.
But, what if Mercury and Mars were to switch places? Well, this turns out to be much easier. Mercury has an albedo of 0.11 (meaning that it reflects 11% of the light that hits it), compared to Mars’ albedo of 0.15, and a surface area of about 52% of Mars’ surface area. Using these data, I compute a magnitude difference of 1.1 from interchanging the two planets. This means that Mars would appear mangitude -3.1 from Mercury’s orbit, brighter than it currently is, mostly due to the increased brightness of the Sun that close to it. Mercury, though, would appear to be magnitude -1.8 at its brightest if it were in Mars’ orbit, slightly brighter than Sirius. But, this ignores the fact that the planets would change if at those distances. Mars would llikely have its rocks altered a bit by the UV from the Sun, and its atmosphere would be lost into space, and Mercury would be able to hold onto a thin atmosphere at Mars’ distance from the Sun, so the albedos would likely be reversed if the positions were reversed. Taking that into account, you find the magnitude difference to be only about 0.7 magnitudes if you interchange them.
But, of course, none of this is real. I was just having some fun with the calculator.
-Astroprof
(Images courtesy of NASA)
