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Many artificial satellites are going to do a destructive reentry in the atmosphere and some will reach the ground, constituting a significant threat (e.g. GOCE, UARS).

Often I heard that for those objects the reentry trajectory is not easy to calculate, or even impossible to predict. But why?

What are the main uncertainties and difficulties to model such a motion?

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Basically, you don't know exactly what drag the spacecraft at such low orbits will be a subject to. There are two main points to this:

  • Unpredictability of the atmospheric density that is a subject to larger changes the lower you get and space weather also playing an important role in influencing particle density, for example the Sun is now in the period of solar maximum, resulting in more ejecta hitting our upper atmosphere, increasing drag on the spacecraft that isn't constant in its orbital path. See more in this answer, but just to back this up with a nice quote from Cornell University's document on Simultaneous Orbit and Atmospheric Density Estimation for a Satellite Constellation (PDF):

    For many satellites in low earth orbit (LEO), the largest dynamic model uncertainty stems from atmospheric drag. Acceleration due to atmospheric drag $a_D$ is related to atmospheric density $p$ by the equation:

    $$a_D = - {1\over2}({C_D{A_v(t)\over{m_s}}})\ {pv_r}^2e_v$$

    where $C_D$ is a drag coefficient, $A_v(t)$ is the cross-sectional area of the satellite in the direction of travel, $m_s$ is the total spacecraft mass, $v_r$ is the velocity magnitude relative to the ambient atmosphere, and $e_v$ is a unit vector in the relative velocity direction. Uncertainty enters this equation in three ways. First, the scalar product $({C_D{A\over{m_s}}})$, known as the inverse ballistic coefficient, is generally uncertain and may be time-varying. Second, the relative velocity may be uncertain, either because it has not yet been estimated accurately or because the local wind does not rotate perfectly with the Earth. Finally, atmospheric density is very difficult to determine. Three basic paradigms exist for dealing with drag uncertainty: It can be modeled, measured directly or indirectly, or estimated in conjunction with satellite orbits.

  • Inability to predict spacecraft's flight stability, how it will react to increased drag and orient itself without a functioning Attitude Control System (ACS) as it used up all of its Xenon propellant for its ion thrusters. GOCE is aerodynamically streamlined, so it might be pretty stable as current telemetry indicates, and is not losing altitude fast even at its perigee now below the official outer space altitudes, or Kármán line of 100 km (62 mi) above the Earth's sea level. It still seems to have no problems reaching apogee of over 130 km, as it did during last orbit at the time of writing this answer. It is however not an aeroplane and eventually it will lose control, spin and fall apart due to immense atmospheric pressure (its current velocity is nearly 8 km/s) affecting front-facing side more than its tail while it spins in the atmosphere.

       GOCE ground track

       Latest GOCE ground track as it heads towards China airspace during one of its last orbits (Source: ESA GOCE Tracking)

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    $\begingroup$ As luck would have it, it turns out that the tracking printscreen I added to the answer was one of the last frames of telemetry data ESA has received from GOCE, and it has by now already fallen to Earth somewhere between East Asia and the Western Pacific. If true, that would be not even 15 minutes later. :) $\endgroup$ – TildalWave Nov 11 '13 at 1:25

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