Desktop Project Part 23: What are the odds of a satellite re-entering over water?

By Phil Plait | April 17, 2012 2:00 pm

[I’m approaching the Desktop Project endgame here; I’m almost out of pictures to post. I’ve done this every day for weeks, and my computer desktop is almost clean! Of course, more stuff keeps coming in, so I could do this forever. But that would be cheating. Sweet, sweet cheating.]

I’ve got something different for you today. Over the past few weeks I’ve posted an illustration, and a couple of dozen pictures, but no graphs! That’ll change now, and I think this particular set of plots is nifty.

Whenever a big satellite is about to re-enter Earth’s atmosphere — like UARS, or ROSAT, or Phobos-Grunt — the media freak out. You start seeing numbers being thrown about of the odds of getting hit by a chunk of flaming debris, and I get lots of panicked email and tweets. Then I have to point out to people that the Earth has a lot of real estate for a satellite to come down on, and of that, 3/4 is water. And most of that is Pacific Ocean. So really, the most likely scenario is a re-entry into the Pacific, or some other ocean, and that’s that.

But is that really true? After all, satellites can have different orbits, inclined with respect to the Earth’s equator. So the odds of getting dumped in the ocean might be different for a satellite that’s over the equator versus one in a polar orbit (that is, orbits almost completely in a north/south direction).

Happily, orbital debris specialist Mark Matney did the math! In a paper published in the Orbital Debris Quarterly Newsletter (bet you didn’t know that existed!) he calculated those odds. He created two graphs for the paper, and both are really cool if you’re a graph nerd like I am.
Here’s the first one:

That plot shows the fraction of the total area of the Earth covered by land versus latitude. It’s easy to read: at 0° latitude — the equator — the amount of land is 23%. In other words, if you flew a plane around the world at the equator, you’d be over land 23% of the time.


I live in Boulder, which is at 40° north latitude, and at that latitude there is a lot more land, about 45%. The big peak at +60° is due to Canada and Russia. Now look at +90°, the north pole. There’s no land at all! That makes sense; the north pole is covered by the sea, and the only solid surface is ice, which doesn’t count.

Now look at -90°, the south pole: it’s 100% land! That too makes sense, since Antarctica is a continent. Even at -85° or so you’re entirely over Antarctica.

That graph is totally cool. And armed with that, and some orbital physics (including the fact that the Earth rotates under a satellite as it orbits), Mark was able to calculate how long a satellite spends over land depending on how tilty its orbit is. He came up with this graph:

This plots how long a satellite spends over land as a fraction of its orbit, versus the inclination of that orbit — 0° if it stays over the equator, and 90° if it’s on a perfect north/south path.

Check the axis on the left: no matter what the tilt of an orbit, a satellite spends about 22 – 33% of its time over land! The big maximum in the middle is for polar orbits, but that’s heavily weighted by Antarctica. Since almost no one lives there, it doesn’t really count. That’s why he subtracted it (dashed line) in the plot, to give you a better idea of how much time a satellite spends over inhabitable land.

The bottom line: saying the chance of a satellite dropping in the ocean is about 75% is actually pretty close to being right! And remember, people occupy a very teeny percentage of the planet’s land area. That’s why you’ve never heard of anyone getting killed by falling satellite debris, and it’s actually not even very common to have photos of one coming back down! They tend to fall far from where people are, due to the sheer amount of empty land on Earth.

Disney might think it’s a small world after all, but 500,000,000 square kilometers (200 million square miles) is actually a whole lot of room, even when the sky is falling.

Tip o’ the deflector shield to PhG_Reentry. Orbit diagram from the Canadian Space Agency.

CATEGORIZED UNDER: Cool stuff, DeathfromtheSkies!, Space
MORE ABOUT: orbit, re-entry, satellite

Comments (31)

Links to this Post

  1. Earth « Earthpages.ca | April 18, 2012
  2. Post-Easter link catch-up « The Outer Hoard | April 22, 2012
  1. SweetChuck

    I always like to remind people that when Columbia broke up over the southern US, even though it was much more massive than most satellites, and broke up much lower in the atmosphere than a satellite would, and over a fairly densely populated area (compared to the population density of the entire Earth), no one on the ground was killed by debris.

  2. Ohio Mike

    Simple math:
    200 million square miles = 5.7568 quadrillion square feet. Say a person is 1′ x 3′ and, it seems to me, you’ve got a 1 in 1.85856 quadrillion chance of getting hit (assuming a piece of debris smaller than 3 square feet).

    You’ll win the lotto way before you get hit with re-entry debris.

  3. SocraticMammal

    Very cool graphs indeed. FYI I think you meant “habitable land”.

  4. Blargh

    Nice! I’ve wondered about this myself. :)
    I spent the weekend of UARS’s reentry roughing it outdoors, and its probability of hitting land/someone was something even the “normal” people around me talked about around the campfire. It would have been nice to have actual data back then rather than just mental calculations based on half-remembered geography. :)
    (It made me happy though that said people’s reactions were mostly “Oh, wouldn’t it be neat if we could catch a glimpse of it tonight?” rather than “My God we’re doomed!”, despite the media scaremongering)

    And the article is freely available too. Thanks for the link!

  5. @1

    Space Shuttle Columbia was purposely drove into land area because it was going to actually land on the surface, if it wasn’t for the catastrophic failure of the wing! You cannot compare it with a satellite falling, for example UARS, ROSAT or Phobus-Grunt, because these take on an essentially random path…

  6. sailor

    As a sailor, I dotn not find this particularly comforting…

  7. Roman

    Something fishy about the size of Antarctica here…did someone forget to normalize by cos(lat)?

  8. lepton

    Phil, I wonder if you can refine the estimate with population density:
    http://en.wikipedia.org/wiki/Population_density

    We probably also need to consider the atmosphere density variation at different latitude, that might be another important variable.

  9. Andrei

    Of course – you cannot have a satellite orbiting at a fixed, non zero latitude. The mean latitude must always be 0 degrees, otherwise you’ll need a lot of fuel to keep the satellite in position. So I find a bit missleading the first graph that says 100% land for -90 degrees. But that could be also because of the general context – i.e. orbiting satellites and not a geography lessons…

  10. Chinahand

    I am a little confused – the Earth is made up of about 30% land (the most detailed figure Google can give me is 29.22% – defining land is probably debateable – how small a lake can be ignored etc.) If you are a circumpolar satellite you’ll eventually pass over the entire earth’s surface (excepting some strange resonance orbits which interact with the earth’s orbit to have a constant track) – but the Earth is not made up of over 33% land as the second graph seems to imply.

    As far as I can see it is impossible for the Satellite to have a probability of re-entering over land higher than the proportion of land on the earth’s surface – but that seems to be what the graph is telling us.

    Am I missing something?

  11. Victor Bogado

    @5 (Gonçalo Aguiar)

    I think that’s the whole point of SweetChuck, Columbia was over a densely populated area, had less altitude to “disintegrate” (in fact it was projected to do not disintegrate in reentry) and even with that much higher probability it didn’t kill anyone so in a satellite that falls we have even lower probability of fatality. But people do over-react much more with those.

    I would like to have more data, for instances how many airplane accidents killed people on the floor? I would bet that the majority of such accidents were near dense-populated airports.

  12. Slim

    @3

    “Don’t worry! This planet is INhabitable!”
    later: “‘Inhabitable’ means ‘habitable’? What a country!”

  13. Andrei

    @9 (Victor) – good point with the plane accidents! I belive the Amsterdam plane crash was one of the worst with respect to ground fatalities (it hit an apartments building)

  14. Peter B

    Roman @ #7 said: “Something fishy about the size of Antarctica here…did someone forget to normalize by cos(lat)?”

    What’s fishy about Antarctica’s size? Are you talking about the first graph or the second?

  15. Don ROgers

    Great math lesson here for my high school students. If only Phil would have used the correct term of probability instead of odds.

  16. Jon

    I thought I saw somewhere that if you took the ice cap away, the South Pole would be under water. I could be wrong…

    J.

  17. RobT

    @3 – Habitable means the same as Inhabitable, like Flammable is the same as Inflammable. But unlike Secure and Insecure – English is definitely a messed up language.

    @Phil and @2 Ohio Mike – What’s a mile? Isn’t that an archaic way of measuring distance? I never got why people measure things in Imperial, you know, because 5,280 feet is such a nice round number.

  18. Barry

    John (17): If the Antarctic ice cap melted, the sea level would rise, possibly even enough to cover Antarctica (the continent underneath). In that case, the other continents would also be measurably smaller.

    But if the ice cap is simply removed from consideration in determining land vs sea, the underlying continent is still there, just like the other 6.

  19. Tell the Shire of Esperance it did not get hit.

  20. Adrock
  21. @17… Jon, Antarctic glaciers slide into the ocean from the land mass, so while there’s deep ice over much/most of the continent, it’s sitting on dry land except for the small portion extending into the sea surrounding it.

  22. jt

    @21… Only parts of Antarctica are above sea level. Approximately half is below sea level, covered in ice down to the bedrock. See the maps here: http://www.zegarkus.com/2007/11/antarctica-with-no-ice.html

    Another map from wikipedia:
    http://en.wikipedia.org/wiki/File:AntarcticBedrock.jpg
    “Map is not corrected for sea level rise or isostatic rebound, which would occur if the Antarctic ice sheet completely melted to expose the bedrock surface.”

  23. ceramicfundamentalist

    “We probably also need to consider the atmosphere density variation at different latitude, that might be another important variable.”

    @ 8 lepton:
    according to wikipedia, the most extreme atmospheric pressures recorded on earth are 108.6kPa in mongolia and 87kPa in during typhoon tip in the north pacific ocean. since these extreme variations in air pressure during extreme weather phenomenon only amount to about a 20% difference, it’s probably safe to say there is no major difference in air pressure at various latitudes.

    but i don’t bring this up to critique your idea. actually, it gave me the most amazing mental image of a spacecraft plunging to earth through the eye of a hurricane. anyone want to guess on the odds of that mega-disaster?

  24. @11 Chinahand: I think the polar regions get overrepresented when you do the calculations. Think about it: a polar satellite in a low orbit will go over each pole every 90 minutes, or 16 times a day, while it will only hit 32 spots on the equator. So land at the poles counts for more than land at the equator, and there’s land at the poles about 50% of the time.

    Another way of thinking about it is that for polar satellites the Earth is in a equirectangular projection: http://en.wikipedia.org/wiki/Equirectangular_projection . Look at that to see how large Antarctica looms.

  25. vince charles

    24. ceramicfundamentalist Said:

    “according to wikipedia, the most extreme atmospheric pressures recorded on earth are 108.6kPa in mongolia and 87kPa in during typhoon tip in the north pacific ocean. since these extreme variations in air pressure during extreme weather phenomenon only amount to about a 20% difference, it’s probably safe to say there is no major difference in air pressure at various latitudes.”

    ceramicfundamentalist, I think lepton’s talking about the _higher_ atmosphere in the tropics, not the pressure at the surface. Due to various effects like insolation, Hadley circulation, different composition, etc., the tropical stratosphere is higher than the polar stratosphere, and the mesosphere, and so forth, so reentry would be a bit different too.

  26. Nathan Buck

    Phil, I really appreciate this article, it is a perfect accompaniment to discussions I have with some of my more panic-inclined and logical-thinking-deficient co-workers and friends.

    A corollary to this that I frequently find myself addressing is the question of “Why, if asteroids and meteors are such a danger to the planet, does space debris not leave large impact craters?” in regards to some of the cases where debris does impact land.

    I have explained the relative speed components and mass components repeatedly, but I suspect that a well put together article with similar graphs would be much more helpful than my white board doodles.

  27. Infophile

    @11 Chinahand:

    #25 ToSeek gives a good explanation, but in case that doesn’t do it for you, here’s another way of looking at it. Let’s imagine two hypothetical planets:

    Planet #1 “Ring”: A thin strip of land around the equator, accounting for 5% of the planet’s surface, and no other land. If a satellite around this planet is in a 0 degree inclination orbit, it will be over the equator at all times, and hence over land at all times. So, despite the fact that the planet is only 5% land, a satellite can be over land 100% of the time.

    Planet #2 “Cap”: All land in the planet is in a single cap over the south pole, from 45 degrees south to 90 degrees south. After a bit of math, this comes to representing 14.6% of the planet’s surface being covered by land. Now, imagine a satellite in a polar orbit. It will be over land when it’s between -45 and -90 degrees, and over water the rest of the time. Since it travels through an equal number of degrees in an equal amount of time, and there are 180 degrees of latitude total (mirrored to give 360), this implies that the satellite spends 25% of its time over land. This is similar to the case with Antarctica – for a polar orbit, a satellite spends a disproportionate amount of time over the small amount of area near the poles.

  28. icemith

    @17 Jon, @19 Barry, @ Bill, @23 jt …….

    I seem to recall that if all the ice was removed from the Antarctic continent, (melted, realistically), then the continental bedrock would “rise again”, alluded to in the quote by jt @23, in the second Wiki link, “… Map is not corrected for sea level rise or isostatic rebound …”, the word “isostatic” being the key word.

    The land would rise up, but whether it rises above the new general mean Sea Level, is another matter. I think it would, as the extra water from the melted ice would spread across the whole Earth’s oceans. We are only looking at hundreds of meters of increase in sea level worldwide, (which definitely swamps my house – by hundreds of meters!).

    Phil, this has been a very enlightening topic, as it involves Cause and Effect problems that the world faces, and has faced before. I live in Sydney, Australia, where eons ago the eastern shore line has alternated between the sandstone laid down in seabeds, but now form the Blue Mountains west of the city, (OK, that could have also been uplifted tectonically), and out to sea to the edge of the continental shelf, not very far, as it drops away steeply to great depths in the Pacific. Check out any Google Earth map on the global scale where the seabed is depicted.

    True, there would be less “dry” land if all the ice melted, but there would still be a significant amount for the remaining population to continue on its merry way. Of course, if the Antarctic melted, then the other great repositories of mass ice (Greenland, Iceland etc.), would also melt, contributing considerably to the sea rise, but excepting already floating sea ice, and icebergs, as they have already displaced their maximum amount of seawater.

    Ivan.

  29. icemith

    Oh, and I forgot to mention that the second graph in BA’s original topic has a symmetrical outcome. I wondered why, as the land distribution was anything but symmetrical.

    I finally realised that the result is from the many passes over the Earth’s surface to complete the “scan”. It occurred to me that eventually the same land would be traversed twice, once on a north/south direction and then on a south/ north direction. This would devolve to a symmetric pattern of coverage.

    Ivan.

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