In orbital mechanics and aerospace engineering, a gravitational slingshot is the use of the motion of a planet to alter the path and speed of an interplanetary spacecraft. It is a commonly used maneuver for visiting the outer planets, which would otherwise be prohibitively expensive, if not impossible, to reach with current technologies.
Consider a spacecraft on a trajectory that will take it close to a planet, say Jupiter. As the spacecraft approaches the planet, Jupiter's gravity will pull on the spacecraft, speeding it up. After passing the planet, the gravity will continue pulling on the spacecraft, slowing it down. The net effect on the speed is zero, although the direction may have changed in the process.
So where is the slingshot? The key is to remember that the planets are not standing still, they are moving in their orbits around the Sun. Thus while the speed of the spacecraft has remained the same as measured with reference to Jupiter, the initial and final speeds may be quite different as measured in the Sun's frame of reference. Depending on the direction of the outbound leg of the trajectory, the spacecraft can gain a significant fraction of the orbital speed of the planet. In the case of Jupiter, this is over 13 km/s. A slingshot may be simulated by rolling a steel ball past a magnet in one hand that is then moved away. Because both masses must not cross paths, the acceleration is oblique to the field and thus is similar to a sail vehicle tacking to work against the force.
It is important to understand how spacecraft move from planet to planet. The simplest way to solve this problem is to use a Hohmann transfer orbit, an elliptical orbit with the Earth at perigee and Mars at apogee. If you arrange the timing correctly the spacecraft will arrive at the outer end of its orbit right as Mars is passing by. These types of transfers are commonly used, e.g., for moving between orbits over the Earth, Earth-Moon and Earth-Mars transfers.
A Hohmann transfer to the outer planets requires long times and considerable "delta V", the sum of the changes in velocity needed at either end of the transfer orbit. This is where the slingshot finds its most common applications. Instead of the Hohmann trajectory directly to, say Saturn, the spacecraft is instead sent in a path that is aimed only as far as Jupiter, and the slingshot is then used to accelerate the spacecraft on towards Saturn. In doing so, even small amounts of fuel spent in positioning and accelerating the spacecraft on its way to Jupiter will be magnified many times once it arrives. Such missions require careful timing, which is why you often hear references to a launch window when discussing them.
A Hohmann transfer to Saturn would require a total of 15.7 km/s delta V, which is not within the capabilities of our current spacecraft boosters. A trip using multiple gravitational assists may take longer, but will use considerably less delta V, allowing a much larger spacecraft to be sent. Such a strategy was used on the Cassini probe, which was sent past Venus, Venus again, Earth, and finally Jupiter on the way to Saturn. The 6.7-year transit is slightly longer than the six years needed for a Hohmann transfer, but cut the total amount of delta V needed to about 2 km/s, so much that the large and heavy Cassini was able to reach Saturn even with the small boosters available.
Another example is Ulysses, the ESA spacecraft which studied the polar regions of the Sun. All the planets orbit more or less in a plane aligned with the equator of the Sun: to move to an orbit passing over the poles of the Sun, the spacecraft would have to change its 30 km/s of the Earth's orbit to another trajectory at right angles to the plane of the Earth's orbit, a task impossible with current spacecraft propulsion systems. Instead the craft was sent towards Jupiter, aimed to arrive at a point in space just "in front" and "below" the planet.
As it passed the planet, the probe 'fell' through Jupiter's gravity field, borrowing a minute amount of momentum from the planet; after it had passed Jupiter, the velocity change had bent the probe's trajectory up out of the plane of the planetary orbits, placing it in an orbit that passed over the poles of the Sun, rendering that region visible to the probe. All this required was the amount of fuel needed to send Ulysses to a point near Jupiter, well within current technologies.
Powered slingshots
A well-established way to get more energy from a slingshot is to fire a rocket engine near the pericentron to increase the spacecraft's speed. At the bottom of a slingshot, when a vehicle is moving its fastest, firing the rocket increases the velocity by the same amount as if the spacecraft were not near the planet. However, an object's kinetic energy is m(v2)/2. where m is mass, and v is velocity. Energy is defined as force times distance, or newtons (force) times meters which equals joules. Since force equals mass times acceleration, a newton equals one kilogram times an acceleration of one meter per second every second. So, since the energy grows as the square of the velocity, a small increase of a larger velocity at the bottom of a gravity well reduces the total energy of the spacecraft by removing fuel mass, but gives the spacecraft substantially more energy per kilogram than an equal-sized increase in the speed of a slower spacecraft.
For example, say a 2 kg spacecraft is moving at 1 meter per second. If it fires a rocket using 1 kg of fuel, and goes to 2 m/s its energy increases from one to two joules, and its mass decreases from 2 kg to 1 kg for a net gain of 1.5 joule of energy per kilogram of mass (increasing from 0.5 J/kg to 2 J/kg). (Use the energy equation above.)
However, say the same spacecraft is in a slinghot, moving 1 kilometer per second while close to a planet. At the bottom, the speed of passage gives the spacecraft 1,000,000 joules of kinetic energy, about 500,000 J/kg. The same 1 m/s burn, using 1 kg of fuel, changes the total kinetic energy of the spacecraft from 1,000,000 J in a 2 kg spacecraft to 501,000 J in a 1 kg spacecraft. This is a loss of total energy, but a gain of 1000 joules per kilogram. Since energy is conserved, when the spacecraft leaves the planet's gravity well, the spacecraft loses the 500,000 J/kg of the gravity well, but keeps the 1,000 additional J/kg of energy. Although the spacecraft is now only one kilogram, it is going about 32 m/s faster, added to whatever velocity it gained from the orbital speed of the planet.
The example's slingshot therefore multiplied the efficiency of the (rather poor) imaginary rocket more than thirty-fold. The additional energy does not come from the gravitational potential energy of the fuel, because the increase of energy is proportional to the efficiency of the rocket. The multiplication of a rocket's efficiency increases with the gravity, because the speed achieved at closest approach is more in greater gravity.
See also specific energy change of rockets:
where ε is the specific energy of the rocket (potential plus kinetic energy) and Δv is a separate variable, not just the change in v.
Limits to slingshot use
The main practical limit to the use of a slingshot is the size of the available masses in the mission.
Another limit is caused by the atmosphere of the available planet. The closer the craft can get, the more boost it gets, because gravity falls with the square of distance. If a craft gets too far into the atmosphere, the energy lost to friction can exceed that gained from the planet.
Slingshots using the Sun are certainly possible, but limited by a spacecraft's ability to resist the heat.
There's also another, theoretical limit based on general relativity. If a spacecraft gets close to the Schwarzschild radius of a black hole (the ultimate gravity well), space becomes so curved that slingshot orbits require more energy to escape than the energy that could be added by the black hole's motion.
The proof is that the path of light is always a geodesic, an "inertial straight line". A mass following the local geodesic experiences no forces. At the Schwarzschild radius, light travels in a circle, so an inertial straight line is a circle. At the Schwarzschild radius the inertia of the "orbiting" spacecraft's mass therefore does not oppose the centripetal force of gravity: A stable orbit is impossible. As orbits approach the Schwarzschild radius, the orbiting spacecraft would require increasing amounts of energy to escape. At some point, the needed escape energy will be greater than the energy that could be added by a slingshot.
On the other hand, a spinning mass produces frame-dragging. A spinning black hole actually is surrounded by a region of space, called the ergosphere, within which standing still (with respect to the black hole's spin) is impossible, as space itself slips in the same direction as the black hole's spin at the speed of light. Suffice it to say that there is a subtle relativistic effect which can transfer angular momentum between any spinning mass and a passing object.
See also
External links
Last updated: 05-07-2005 16:55:45
Last updated: 05-13-2005 07:56:04
