### Orbital lifetime

Posted:

**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):

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