Orbital lifetime

Orbital lifetime

Postby Xan » Sat Nov 02, 2013 8:46 pm

In a low orbit satellite loses energy due to friction of the atmosphere. Aerodynamic drag is equal to:

F = S * Cx * rho * v^2 / 2

where S — cross-section of satellite.
And acceleration is equal to:

a = F / m = (Cx * rho * v^2 / 2) / (m/S)

where m/S — surface density = mass divided by cross-section.

Microsatellite has very low surface density, for about two orders lower then "adult" one.
For example Sputnik-1 had a cross-section = 2642 cm2 (diameter = 58 cm) and weight = 83600 grams

Its surface density = 83600 / 2642 = 31.64 g/cm2

If mass of microsatellite = 19.9 g and diameter = 5 cm, then its cross-section = 1/4 * 3.14 * 5^2 = 19.63 cm2

surface density = 19.9 / 19.63 = 1.014 ~ 1 g/cm2

The lower the surface density — the shorter the lifetime being.
For example Sputnik-1 (with perigee = 228 km) made 1440 revolutions.
Microsatellite will only

1440 * 1.014 / 31.64 = 46

It is interesting how lifetime depends on orbit altitude and solar activity.
(see http://www.braeunig.us/space/atmos.htm -- MSISE-90 Model of Earth's Upper Atmosphere)

This is the result of simulation for the circular orbit and the satellite's surface density = 1 g/cm2.

Revolutions = function(altitude, activity):

Code: Select all
h[km]   low    mean   high

140       0.1    0.1    0.1
150       0.3    0.3    0.2
160       0.6    0.5    0.3
170       1.1    0.9    0.5
180       1.8    1.5    0.8
190       3.0    2.3    1.1
200       4.7    3.3    1.5
210       7.3    4.8    2.0
220      11      6.7    2.6
230      16      9.2    3.3
240      23     13      4.1
250      33     17      5.0
260      47     22      6.2
270      65     28      7.5
280      89     37      8.9
290     122     47     11
300     165     59     13
310     222     74     15
320     296     93     17
330     392    116     20
340     516    141     23
350     676    177     27
360     882    217     31
370    1145    265     36
380    1481    323     41
390    1908    391     47
400    2451    473     53
410    3138    570     61
420    4006    685     69
430    5098    821     78
440    6471    982     88
450    8191   1172     99
460   10340   1396    111
470   13012   1660    125
480   16323   1970    140
490   20413   2333    157
500   25449   2759    176
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Re: Orbital lifetime

Postby Xan » Wed Jul 16, 2014 3:58 pm

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Re: Orbital lifetime

Postby lavalamp » Sun Jul 20, 2014 8:14 pm

Xan wrote:In a low orbit satellite loses energy due to friction of the atmosphere. Aerodynamic drag is equal to:

F = S * Cx * rho * v^2 / 2

Unfortunately this formula isn't valid at such low pressures. Once you get above 50km (~100 Pa), you're transitioning into a molecular flow regime. By the time you get to 100 km (~0.1 Pa) you're all the way there.
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Re: Orbital lifetime

Postby Xan » Mon Jul 21, 2014 2:57 am

This formula is valid at all pressures.

Cx depends of pressure, but not very much.
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Re: Orbital lifetime

Postby lavalamp » Mon Jul 28, 2014 9:01 pm

Ah, I apologise. I did some reading around and it seems you are indeed correct.

What does appear to be different in molecular flow (and quite drastically so) is the coefficient of drag. However I see that you have essentially isolated that by comparing to sputnik and instead only looking at the surface density.

Since Sputnik was essentially a sphere, and I would assume n-prize satellites are more likely to be cuboid, it would be interesting to see how they compare to the GOCE satellite, which was more angular, or some low altitude cube sats.
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Re: Orbital lifetime

Postby Xan » Tue Jul 29, 2014 5:33 am

At molecular flow regime Cx weakly depends upon angle.
Approximately Cx = 1.5 for 90 degrees and Cx = 1 for 0.

Solar activity affects the braking is much stronger than the form.
It makes no sense to accurately account form, just take a cross-section.
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Re: Orbital lifetime

Postby pauldear » Tue Aug 05, 2014 10:12 am

To be honest, I'm pleasantly surprised that drag isn't worse than it is, for N-class satellites. I'd have expected the difference between Sputnik and a nanosat to have been much greater than the calculations suggest...
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Re: Orbital lifetime

Postby Jay » Tue Aug 05, 2014 12:04 pm

Thank you for writing “N-class” satellites! I’ve said it before, but it bears repeating; by creating the N-Prize Paul has effectively defined the lower limit of true spaceflight -- orbital and suborbital nanosatellites carried aloft by nanolaunchers at minimal cost. The fact that aerodynamic drag in the “vacuum” of low-Earth space is a real factor in determining whether a tiny spacecraft might achieve nine revolutions or only eight is clear evidence of this. Fortunately that drag is just a very small factor.

Using expressions such as N-class or N-type spaceflight reinforces the recognition the N-Prize deserves for setting such “low” standards for us all! Thanks again, Paul.
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Re: Orbital lifetime

Postby lavalamp » Tue Nov 03, 2015 7:05 pm

I've recently been working on implementing atmospheric models into some orbit software to take account of drag, and thought I'd share this plot.

I used the same 20g and 20 cm^2 (ie: 1 g/cm^2) satellite and the US Standard Atmosphere 1976, and I gave it the slightly (but not overly) pessimistic drag coefficient of 2.4. The initial conditions were for an equatorial and circular 260km orbit, and it completed around 9.4 orbits.

I can relatively easily run more orbits if people are interested in seeing such things, and may run some more myself. So if anyone is curious to know how a certain size/mass/orbit combo performs let me know and I'll crunch the numbers.

Note: The Earth is not to scale (because it's huge), but the atmosphere and orbit track around it are. The blue part is my representation of the atmosphere and extends to 100 km (Karman line), but the full atmospheric model goes up to 1000 km.
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Re: Orbital lifetime

Postby lavalamp » Tue Nov 03, 2015 7:31 pm

OK, here's one more. This is the same 1 g/cm^2 Cd = 2.4 Satellite, but placed into the same orbit as Sputnik was. This is an elliptical orbit with perigee around 235 km and apogee at 960 km. The n-prize sat lasted for 165 orbits.

This is basically the kind of orbit that would have been required had the rules not been revised down from a lifetime of 99 orbits, to 9. Of course, a higher circular orbit would do, but those are so much harder to achieve compared to staying relatively low and burning until the fuel runs out.
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