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| Fig. 1: Groundtrack of an example of a Molniya orbit. The satellite spends the majority of its time over the northern hemisphere, and very little time over the southern hemisphere. Orbit: semimajor axis 26,554 km, eccentricity 0.722, inclination 63.3 degrees, right ascension of the ascending node 308 degrees, and argument of perigee of 270 degrees. |
In 2014, Alfonso Cuaron dazzled audiences with a cinematic masterpiece called Gravity. Sandra Bullock and George Clooney are a pair of star-crossed astronauts that float across the screen amidst dramatic views of the earth, and through an even more dramatic series of unfortunate events. While the movie's breathtaking clips have been praised for their authenticity, the underlying physics behind the main plot points is substantially less praise-worthy. Pesky bits of space debris keep appearing where the astronauts are located, and they make a daring sequence of attempts to escape to safety. In reality, the threats to the heros' well-being are highly improbable, and the escape methods are beyond the capabilities of modern technology.
The drama begins when the Russians are said to have blown up one of their own spy-satellites in an anti-spacecraft test demonstration. While the details are not specified in the film, a likely candidate for the spy satellite's orbit would be a Molniya orbit. Molniya orbits are precisely shaped and timed such that they have a long loiter time above the northern hemisphere, twice per day. Molniya orbits that have long loiter times over North America also happen to have long loiter times over Russia. They have been commonly used for Russian communication satellites, but are also ideal for Russian satellites to spy on the US, and vise-versa. A groundtrack for a sample Molniya orbit is depicted in Fig. 1.
According to the film, debris from the satellite explosion begins to threaten other space objects. It begins its destruction with a series of communications satellites, and then proceeds to continue on towards the astronauts where they are currently repairing the Hubble Space Telescope. But what sort of energy would be required for this threat to be feasible? The best case energy transfer scenario would involve the debris from the original Russian spy satellite explosion extending all the way to the Hubble Space Telescope orbit. To put a lower bound on the energy required, we will perform an analysis based on the energy required to lift an explosive to a location within a Molniya orbit, and the difference between a Molniya orbit energy level and a transfer orbit that intersects both the Molniya orbit and the Hubble orbit.
Because the Hubble Space Telescope is in a roughly circular orbit at an altitude of 563 km above Earth, the debris from the Russian satellite must, with a single blast, go into an orbit that both intercepts the Molniya orbit and reaches down to an altitude of 563 km. [1] There will be a certain amount of energy required to lift the bomb from the Earth to a given altitude, and then a certain amount of energy required from the bomb to initiate the orbit transfer. Lifting energy will be lower bounded by the potential energy increase of the eplosives from ground to orbital altitude, and bomb energy will be lower bounded by the difference between a Molniya orbit's specific orbital energy and the transfer orbit's specific orbital energy. Note that both of these assumptions are under-approximations of the actual energy that would be required. The specific orbital energy of an object in orbit is given by
where μ is the gravitational parameter of the Earth, ra is the apoapsis (highest point in the orbit), and rp is the periapsis (lowest point in the orbit). For a Molniya orbit with a periapsis of 7,378 km and an apoapsis of 45,730 km, the specific orbital energy equates to -10.39 MJ/kg. Applying the bomb directly at the apoapsis of the Molniya orbit would require less of energy to meet the requirements than it would at any other point along the orbit; such an orbit would have a periapsis of 563 km, and an apopasis of 45,730 km, resulting in a -8.825 MJ/kg energy orbit. Thus, the most efficient possible way for a piece of debris to make its way from the spy satellite to the Hubble would require a blast that transferred at least 1.568 MJ/kg to it. If we assume that the satellite was originally 1000 kg, and that the energy transfer per unit mass is the same for every debris particle, an explosion of that magnitude would require 1,568 MJ, the equivalent of 374.8 kg of TNT. [2] Alternatively, if we assume that the bomb was made out of a high energy density gas, such as methane, and that all of the stored chemical energy can be released at once, then the analysis still shows that an explosion at apoapsis is most energy efficient, and requires 28.15 kg of methane. [3]
Modern-day rockets are powerful enough to accomplish this feat. Russia has a Proton-M rocket that can send over 6,000 kg to geostationary transfer orbit, an orbit that is of an even higher energy level than a Molniya orbit. [4] While the Russians are capable of executing the scenario portrayed in Gravity, it is not necessarily sensible. A Molniya orbit passes down to an altitude of 7,378 km. It would be far easier to shoot down a satellite at a lower altitude than waiting until it reached an altitude of 45,730 km. On the other hand, it is possible that since the movie claims that this was a training exercise, the more challenging route might have been selected intentionally. Even if this was the case, it would take far less than 375 kg of TNT or 28 kg of methane to decommission a satellite; they wouldn't need to send that much in the first place. Because every kilogram of added weight makes a significant impact on launch expenses, it is doubtful that such an excess of firepower would be utilized.
Once the Space Shuttle has been destroyed, George Clooney and Sandra Bullock are the only remaining survivors. Since communications with the ground station have been cut off, they decide to head to the International Space Station (ISS). They do this by using a jet pack that Clooney has been testing. Transfer from the Hubble to the ISS is far more difficult than depicted in the film. In particular, it requires two types of maneuvers: an altitude change and an inclination change.
First, the Hubble and ISS are in circular orbits of 563 km in altitude and 350 km in altitude, respectively. [1,5] This constitutes a specific orbital energy difference of 0.9107 MJ/kg. If we assume that the pair of astronauts and their gear weighs a total of 400kg, then the change in orbits alone translates to 364.3 MJ if the transfer is done 100% efficiently -- and both assumptions are under-approximations. Additionally, the two orbits are at orbital inclinations of 28.5 degrees and 51.3 degrees, respectively. [1,5] The most efficient way to make an orbital inclination change is to fire thrusters as an object passes through the equatorial plane, changing the orbit's orientation instantaneously. In reality, this is impossible to do with the magnitudes of thrust that a personal jet pack would be capable of. Even so, it provides a lower bound for the amount of energy that the pack would need to expend to complete the inclination change over the course of the entire orbital transfer. The energy change required is calculated as follows:
| dV | = | 2 v sin(dI/2) |
| E | = | - m dV2 |
Where dV is the magnitude of the change of velocity required, v is the current velocity of the satellite, dI is the inclination angle change, E is the energy needed, and m is the mass of the satellite. The inclination change by itself requires 1,836 MJ. Thus, in total, the jet pack would have to be capable of storing over 2,218 MJ. The jet pack in the movie is based off of the Manned Maneuvering Unit (MMU) that was demonstrated on multiple space shuttle missions. The real-life MMU was equipped with 18 kg of gaseous nitrogen compressed to 4500 psi at 70°F. [6] This equates to a stored energy of 3.967 MJ. George Clooney and Sandra Bullock would need three orders of magnitude more propellant than available on a cold gas thruster. If, instead, the jet pack were to consist of liquid rocket thrusters, the mass required becomes more reasonable, even though the safety issues become less reasonable. Two commonly used rocket propellants are hydrazine and kerosene, which have chemical potential energies of 19.5 MJ/kg and 42.8 MJ/kg, respectively. [7,8] Assuming perfect efficiency conversion from chemical potential energy into kinetic energy, this would mean that the astronauts would need 113.7 kg of hydrazine, or 51.8 kg of kerosene for fuel. While the weight requirements for such fuel sources are far more reasonable than those for compressed gas, the safety of the system would be in question. Rocket exhaust is not something that you want in close proximity to astronauts, especially when they are in spacesuits that contain highly oxygen-rich air.
It remains to be seen whether Gravity will inspire a new generation to out-of- this-world careers, or frighten would-be astronauts out of aspirations for fear of becoming stranded in space. Future spacefarers can rest assured, however, that nearly every single life-threatening turn of events in the movie were not physically realizable. Additionally, though the escape attempts are equally infeasible, in a real space mission, hundreds of engineers are working continually to ensure that there are actual safe backup plans for every forseeable malfunction.
© Ashley Clark. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] "Space Shuttle Mission STS-125: The Final Visit to Hubble," U.S. National Aeronautics and Space Administration, May 2009.
[2] A. Thompson and B. N. Taylor, "Guide for the Use of the International System of Units (SI)," U.S. National Institute of Standards and Technology, Special Publication 811, March 2008.
[3] H. Dorin, P. Demmin, and D. L. Gabriel, Chemistry: The Study of Matter, 4th Ed. (Prentice Hall, 1992).
[4] "Proton Launch System Mission Planner's Guide," International Launch Services, LKEB-9812-1990," Rev. 7, July 2009, Chapter 2.
[5] "Reference Guide to the International Space Station," U.S. National Aeronautics and Space Administration, NP-2010-09-682-HQ, November 2010.
[6] J. A. Lenda, "Manned Maneuvering Unit," Martin Marietta Corp., MCR-78-617, May 1978.
[7] N. Wiberg and A. F. Holleman, Inorganic Chemistry (Academic Press, 2001).
[8] "Aircraft Fuels" in Encyclopedia of EnergyTechnology and the Environment (Wiley, 1995), Vol 1, pp. 257-259.