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In 2001, the Russian space station Mir was deorbited and burned up in the atmosphere, after $4.2 billion in expenditures. As it was orbiting within the thermosphere, it was encountering perpetual drag and would have deorbited eventually on its own anyway.

It weighed 129,700 kg in 2001. It's orbit was 354 km x 374 km (this is an elliptical orbit with a perigee of 354 km, the closest approach to Earth as a distance above mean sea level, and an apogee of 374 km, the furthest it will reach from Earth as a distance above mean sea level).

What I'm wondering is:

1) What is the delta V required to boost it to the minimum stable orbit?

2) Could this have been done in a single launch of, say, a Saturn V or Delta IV heavy or a Proton-M (or any of the existent rocket launch platforms available to us)?

3) If not, how many launches would it take?

4) What is the delta V required to boost it to a lunar transfer orbit and perform lunar orbital insertion?

I'm guessing the answers to these questions will make it immediately clear why it was deorbited and not further utilized, but I'd like to see the comparison. Note that a Proton M rocket with a Progress M1-5 was launched to carry out the deorbiting operation, so it would be interesting (to me, at least) to know whether or not it could have placed Mir in a stable orbit for future use.

Edit:

I posted my own answer after some extensive research (and no answers came). This isn't any sort of homework; though it is an exercise of sorts (just my curiosity). I could still use a thorough review.

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  • $\begingroup$ Would Space Exploration be a better home for this question? $\endgroup$ – Qmechanic Dec 9 '14 at 8:01
  • $\begingroup$ This is a valid physics question, and phys.SE has a lot more users (potentially submitting answers) than spaceexp.SE. $\endgroup$ – Ehryk Dec 9 '14 at 8:10
  • $\begingroup$ On a side note, it's extremely frustrating that there are more calculators and information available online for doing this in Kerbal Space Program then there are for doing it in our solar system or even Earth. $\endgroup$ – Ehryk Dec 9 '14 at 8:39
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    $\begingroup$ Questions 1 and 4 are probably on topic here, if you give the parameters of the relevant orbits and show your own attempts to calculate the result. (You'll probably have to explain something like "354km x 374km" since most of us aren't familiar with rocket science conventions.) Questions 2 and 3 are off topic here, but would presumably be on topic at Space Exploration. $\endgroup$ – David Z Dec 9 '14 at 8:54
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    $\begingroup$ @Ehryk The existence of another site doesn't make these questions off topic, it's just the fact that they're not about physics. We don't deal with the details of specific models of rockets. $\endgroup$ – David Z Dec 12 '14 at 5:31
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The exobase is defined as the effective end of the atmosphere, and is a gray area between 500 km and 1,000 km. Presumably, once a craft's orbit is outside of the exobase, drag is negligible and stationkeeping basically not a necessity (as in won't return to earth for a century or more, until we can repark it or utilize it).

From my time in Kerbal, I would suspect that the most fuel efficient maneuver would be a Hohmann Transfer from 354km x 374 km to 354km x (exobase, 500 km - 1000 km according to wikipedia, performed at 354km to maximize the Oberth effect) and then another maneuver to circularize to exobase x exobase.

It appears the low end of exobase is not sufficient; Hubble Space Telescope orbits at 559 km (circular orbit) and requires some orbital maintenance (though far less frequent than the ISS). I'll use 1,000 km until further research can find a better value. Help?

1) Delta V to a Stable Orbit

If a 1,000km circular orbit is indeed stable, then by using a Hohmann Transfer orbit:

$\Delta V_1 = \sqrt { \frac \mu {r_1} } * ( \sqrt { \frac {2r_2} {r_1 + r_2} } - 1) = 171.6 m/s$

$\Delta V_2 = \sqrt { \frac \mu {r_2} } * ( 1 - \sqrt { \frac {2r_1} {r_1 + r_2} } ) = 170.5 m/s$

$\Delta V_{total} = \Delta V_1 + \Delta V_2 = 342.1 m/s$

Using the Tsiolkovsky Rocket Equation, Mir's weight of 129,700 kg and the average rocket exhaust velocity of 4,500 m/s, the mass of a Progress M1-5 being 7,150 kg with 1,950 kg of maximum propellant,

$\Delta V_{single M1-5} = 4,500 m/s * ln ( \frac {129,700 + 7,150} {129,700 + 7,150 - 1,950}) = 64.58 m/s$

If this is accurate, and the launch vehicle could deliver fully fueled Progress M1-5s to Mir and they docked, then it would require ~5.3 of them (6 to account for some losses due to non instantaneous thrust, etc.)

However, this then begs the question, why bother with the M1-5s at all? A single Delta IV Heavy can deliver 25,980 kg to ISS orbit at 407 km, which is above Mir's orbit. Thus it could take a second second-stage, with the same engine/structure (burnout mass of 2522 kg, 18,516 kg propellant) as the Delta IV Heavy with a docking node and some gyros.

This docked to Mir would give:

$\Delta V_{deltaIV} = 4,500 m/s * ln ( \frac {129,700 + 2,522 + 18,156} {129,700 + 2,522}) = 579 m/s$

Unless I've made some errors, then it would have been possible to move Mir to a 1,000km circular orbit with a single Delta IV Heavy launch, with some delta V to spare for things like guidance and communications and orientation and such if they couldn't be utilized on Mir (which some certainly could).

2) Capable in a single launch?

Delta IV Heavy, yes (25,000 kg to ISS LEO). Proton M, yes (22,000 kg to LEO), Saturn V, definitely (118,000 kg to LEO).

3) How many launches?

A single launch of the above vehicles carrying what basically amounts to an engine, lots of fuel, and a docking port.

4) Delta V to Lunar Transfer Orbit?

Way, way too much. Just to get into the absolute minimum LTO is to bring the apoapsis up to 326,364 km (Earth-Moon L1 Lagrange Point), which would take 3,079 m/s delta V.

$\Delta V_{LTO} = \sqrt { \frac \mu {r_1} } * ( \sqrt { \frac {2r_1} {r_1 + r_2} } - 1) = 3,079 m/s$

This MIGHT be just barely doable with a Saturn V. With the same, single engine structure (the Delta IV heavy second stage) that magically has the rest of the payload as usable fuel:

$\Delta V_{SaturnV} = 4,500 m/s * ln ( \frac {129,700 + 118,000} {129,700 + 2,522}) = 2,825 m/s$

So, not even a Saturn V and some magicry can get Mir to the moon, but a stable parking orbit (or at least a 1,000km circular orbit, presumably stable) is doable with a custom payload and a single Delta IV Heavy / Proton-M launch. And they did a Proton-M launch to deorbit Mir.

What a waste.

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  • $\begingroup$ Could anyone please help to validate this? $\endgroup$ – Ehryk Dec 11 '14 at 1:06

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