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Motivation is: I would like to be able to reproduce the part of this graph for circular orbits from first-principles.

decay times

The critical information to do this is the gas density as a function of altitude, which includes the range of about 100 km to 2000 km for all objects that will decay anytime within the next few eons. This covers some parts of the Thermosphere in the lower range, and the Exosphere in the higher part.

My question is: what is a good expression for the density in these two regions? I want to avoid over-complicating, by going for the most simple non-trivial approach. If there's a first level and second level approximation, let's not bother with anything but the first one. But saying the exosphere is "approximately zero" is not helpful because that would predict infinite lifetimes.

Wikipedia provides some guidance in the form of equations, but there's still some way to go. For the Exosphere, they give a particle number per unit volume, but this volume is a variable volume, so you would have to change that, and then convert to density. I don't even have a good starting point for the Thermosphere.

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    $\begingroup$ Have you looked at various models (an online service is here: omniweb.gsfc.nasa.gov/vitmo/msis_vitmo.html)? $\endgroup$
    – Deer Hunter
    Aug 13, 2013 at 16:19
  • $\begingroup$ The above link gives nice vertical profiles, with total mass density, and other parameters, up to 1000 km. $\endgroup$
    – Deer Hunter
    Aug 13, 2013 at 16:25
  • $\begingroup$ @DeerHunter I didn't understand it at first, but now I've been able to get output from that site. This should be the answer. You can write it as an answer, but if not, I'll try to post it as community wiki later. $\endgroup$
    – AlanSE
    Nov 7, 2013 at 20:13
  • $\begingroup$ Alan, please feel free to write up an answer. I'm kind of burnt out. $\endgroup$
    – Deer Hunter
    Nov 8, 2013 at 16:39

2 Answers 2

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Since you're looking for the simplest non-trivial approach, I'd recommend the exponential model detailed here, which is very simplistic but is sometimes used as a first (zeroth?) order approximation in things like trade studies. Alternatively, the 1976 Standard Atmosphere model is also fairly simple to use.

Beyond these, you're looking at models like the classic Jacchia-70, MSIS, and the related NRLMSISE models, which are significantly higher-fidelity and, as such, more difficult to implement on your own.

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  • $\begingroup$ I thought the 1976 Standard Atmosphere would be useful, but it ends at 85 km, which doesn't even overlap with the graph I posted for orbital decay. So I looked into the Jacchia model. That gives some useful information, such as temperature and molecular weights from 0 to 1000 km. I think it gives number density too, but if I use that number in the way I think it should be, I wind up with density values that are obviously too high. $\endgroup$
    – AlanSE
    Oct 31, 2013 at 21:23
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This link (which is posted in the comments) covers the question:

http://omniweb.gsfc.nasa.gov/vitmo/msis_vitmo.html

Apparently the thermosphere exhibits more variation due to a host of factors. These include season, location on Earth, solar activity, and so on. Because of that, anyone applying a single formula for this must realize that it might be up to an order of magnitude off. However, the density profile varies by many orders of magnitude, so I still think it's meaningful to talk about.

Here is a sample output.

   km      O/cm3     N2/cm3     g/cm3
    1          2          3         4
  0.0  0.000E+00  2.120E+19 1.304E-03
 50.0  0.000E+00  1.361E+16 8.373E-07
100.0  3.995E+11  8.467E+12 5.173E-10
150.0  1.907E+10  3.236E+10 2.190E-12
200.0  4.918E+09  3.538E+09 3.100E-13
250.0  1.696E+09  5.676E+08 7.348E-14
300.0  6.293E+08  1.009E+08 2.181E-14
350.0  2.402E+08  1.873E+07 7.388E-15
400.0  9.331E+07  3.582E+06 2.723E-15
450.0  3.679E+07  7.029E+05 1.065E-15
500.0  1.471E+07  1.413E+05 4.394E-16
550.0  5.959E+06  2.907E+04 1.926E-16
600.0  2.446E+06  6.118E+03 9.136E-17
650.0  1.017E+06  1.317E+03 4.793E-17
700.0  4.279E+05  2.896E+02 2.815E-17
750.0  1.823E+05  6.506E+01 1.840E-17
800.0  7.861E+04  1.492E+01 1.308E-17
850.0  3.429E+04  3.494E+00 9.863E-18
900.0  1.513E+04  8.346E-01 7.717E-18
950.0  6.750E+03  2.033E-01 6.177E-18
1000.0  3.045E+03  5.048E-02 5.015E-18

It's also meaningful to note that the atmosphere becomes highly differentiated at high altitudes. So basically, the other elements fall to nearer to the surface and almost only Hydrogen is left at super high altitudes. This is what the numerical models spend a lot of their time on.

Here is a plot of the density.

atmosphere density

Note that I had to give different units of kg/m3 here. Just because Excel would muck up the formatting with smaller numbers.

It's actually rather interesting that an exponential trend just won't fit this data. A power fit makes a decent approximation. I don't know why. I can't explain why the atmosphere would better fit a power law than an exponential trend, since the exponential trend comes from the ideal gas state equation and the force balance. That's actually a rather intriguing statement. Of course, even the power law isn't fantastic, and I'm sure a point or two are off by a factor of 2.

Nevertheless, this is usable for the query in the question. The density figure could be combined with orbital mechanics to ballpark the orbit's lifetime.

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