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Given a satellite in GEO or a graveyard orbit, what characteristics of a passing NEO would be required to perturb said satellite out of Earth's influence and into a sun-bound or sun-centered highly eccentric orbit? Or would the required mass be so high that it would itself be disrupted by the moon or be required to impact the earth?

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    $\begingroup$ For a more violent interpretation of "perturb", a collision with an object larger than the satellite would by conservation of momentum result in at least some fragments moving above escape velocity. $\endgroup$ Commented Sep 10, 2021 at 16:14

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I think the maximum velocity change from a flyby would help quantify this.

$$\Delta v \leq \sqrt{\frac{GM}{r_P}}$$

That is, with perfect relative velocity and angle, the change in velocity from such a flyby perturbation is limited by the mass ($M$) of the asteroid, and the distance at closest encounter ($r_P$), which in the best case is to barely miss collision.

So how much of a velocity change is needed from GEO to escape Earth orbit? As a lower bound, By nudging the apoapsis up to the orbital radius of the Moon, successive lunar flybys will sooner or later by able to sling the satellite out of the system. This apoapsis raising comes at a cost of $\Delta v =1053m/s$, not too much lower than a direct escape at $\Delta v = 1270m/s$

So about a kilometre per second of velocity change.

No near-Earth asteroid is capable of this. Take Ceres (not a NEO) for example, the greatest asteroid of them all:

$$\sqrt{\frac{GM_{ceres}}{r_{ceres}}} = 365 m/s$$

We can rewrite the inequality in terms of density to gen an idea of how large the asteroid has to be:

$$\Delta v \leq \sqrt{\frac{4\pi G\rho r^2}{3}}$$

For a high density M-type asteroid at 5.3 g/cm³, the required radius would have to be about 870km. Such an object is definitely large enough to cause a spectacular collision if it hit the Moon on its way through the system, and at geostationary distance the tidal effects would be sizeable, but we could get away with just some earthquakes, tsunamis and some non-catastrophic changes to the Moon's orbit.

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    $\begingroup$ So to restate (to make sure I understand): There is no known asteroid in the solar system that could generate enough delta-v to expel an artificial satellite out of earth orbit. It would take something very dense (and therefore small) to do what I am describing and even that would cause problems for both the Moon and Earth. $\endgroup$
    – McKenning
    Commented Sep 10, 2021 at 17:03
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    $\begingroup$ @McKenning Yes, that's a clear and accurate summary. (Even, say, osmium density would need to be be 400+km in radius) $\endgroup$ Commented Sep 10, 2021 at 17:51
  • $\begingroup$ I wonder what the minimum mass of a black hole would need to be to deflect the satellite without damaging it. I think this will depend significantly on the size, shape, and durability of the satellite. $\endgroup$ Commented Sep 11, 2021 at 17:31
  • $\begingroup$ @CharlesStaats Black hole bad! A black hole small enough to perturb a satellite, but not gravitationally disrupt the Earth's orbit, would be so small that it would be evaporating, rapidly, due to Hawking radiation. It would be a source of radiation of so much energy that it would fry the satellite( and the Earth!) to a crisp. $\endgroup$ Commented Sep 12, 2021 at 9:15
  • $\begingroup$ @PcMan I tried putting relevant mass values into the hawking radiation formula, and got >10^40 years of evaporation time. Not sure if that's "rapid". $\endgroup$ Commented Sep 12, 2021 at 19:00

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