# Michael Minovitch's Great Discovery

/Most Physics and Mathematics students would not be aware of the American mathematician Michael Minovitch (1936- ). While a graduate student at UCLA in 1961 he used the IBM 7090 computer, the most powerful computer at that time, to numerically solve the three body problem, this problem being the movement of three bodies due to their mutual gravitational attraction. This is a very difficult problem, so difficult that Isaac Newton said that it "made his head ache". Minovitch applied his technique to the case of a spacecraft approaching from behind close to a planet orbiting the Sun and found that after the close approach the speed of the spacecraft was increased relative to the Sun. This encounter did not require the use of any extra fuel and this procedure is now known as the sling shot effect or a gravity assist manoeuvre.

## Pioneer 10

This technique was first used in December 1973 to increase the heliocentric speed of the Pioneer 10 spacecraft as it passed by Jupiter so that it could escape from the solar system. James A van Allen in his article "Gravitational assist in celestial mechanics-a tutorial" in the *American Journal of Physics* (**71**, May 2003, page 448) stated that during the encounter the heliocentric speed of Pioneer 10 was increased from 9.8 km/s to 22.4 km/s, increasing the kinetic energy by a factor of 5.2 and transforming the original elliptical orbit into a hyperbolic orbit. During the interaction Jupiter's speed decreased by 2.1x10-24 km/s.....an exceptionally small amount!

## Where does the Extra Spacecraft Kinetic Energy Come From?

During the gravitational interaction of the spacecraft, Jupiter and the Sun, the total linear momentum and energy of this three body system are both constant. This means that if we add the momentum vectors of each object before and after the interaction we obtain the same vector and if we add all of the kinetic energies and mutual gravitational potential energies before and after the interaction we obtain the same number. Due to the very small relative mass of the spacecraft to Jupiter and the small value of the ratio of the close approach distance of the spacecraft to Jupiter to the distance of Jupiter from the Sun the interaction effectively rotates the velocity vector of the spacecraft relative to Jupiter slowing Jupiter down very, very slightly and adding Jupiters lost momentum to the spacecraft, so that the speed of the spacecraft relative to the Sun increases by a large amount.

## Some Student Misconceptions

- The speed of the spacecraft increases because it is accelerated by the gravitational field of the planet. This is incorrect. If the planet was stationary (not moving about the Sun) the interaction would only change the speed of the spacecraft by a very small amount, depending on the relative mass of the two objects. To perform a gravity assist manoeuvre it is essential that the planet is moving.
- The speed of the spacecraft increases if it approaches the planet from behind or from in front. This is incorrect. If the spacecraft is aimed in front of the planet the speed of the spacecraft will be decreased during the encounter. If it is aimed such that the point of closest approach is behind the moving planet the gravitational force turns the velocity vector of the planet giving it an extra component in the direction of the planets movement.
- If the mass of the spacecraft is very small compared to the planet the gravity assist manoeuvre does not occur. This is incorrect. A very small spacecraft can extract energy and momentum from a very large planet provided the planet is moving and the approach is from behind the planets movement.
- A spacecraft orbiting the Moon fires its engines and uses the sling-shot effect to escape the Moon's gravitational field. This is incorrect. The student should say that the force exerted by the exhaust gases produced by the engines changes the momentum vector of the spacecraft so that it may reach a point where the Earth's gravitational pull on the spacecraft is greater than the Moon's gravitational pull and so the spacecraft will be captured by the Earth's gravitational field.