Project 47: Using Jupiter as a Slingshot
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In order to provide a spacecraft with sufficient energy to escape the solar system, Jupiter can be used to "boost" its orbit. We want to determine the necessary relationship between the spacecraft's original trajectory and Jupiter's orbit, so that Jupiter will boost the spacecraft out of the solar system. This requires that they pass close enough for Jupiter to exert a definite effect, while not close enough for Jupiter to "catch" the spacecraft. To determine the relationship, we must find the general equations of motion.

We take the origin of the coordinate system at the sun. In order to simplify matters, the solar system is represented in two dimensions, with the coordinate axes in the plane of the orbits of the planets. Figure 47.1 is a diagram of the system. In the diagram, the vector X points from the sun to the spacecraft and the vector XJ points from the sun to Jupiter (the inner circle represents the earth's orbit, while the outer circle represent Jupiter's orbit).


Figure 47.1: Solar System in Two Dimensions

47.1 Setting up the Problem: Scaling and Units

It is convenient (even necessary for accuracy) to use units of measurement that simplify the calculations. The general unit of astronomical length is the AMU (astronomical unit), where 1 AMU = the approximate distance from the earth to the sun. The distance from Jupiter to the sun is 5.2 AMU. Mass is measured in a system based on the sun as the unit; the sun is defined as having mass 1. In comparison, then, the mass of Jupiter is 0.001, and the mass of the spacecraft is tiny. As an additional convenience, time is defined in a manner so that the gravitational constant G=1 in Newton's law of gravity,


One time unit is earth days.

VARIABLES

  • The important quantities r and d are secondary variables. Use your geometry-algebra lexicon to show that

    Hints: What is r in terms of the vector X? What is r in terms of the components of X? Sketch the vector X-XJ. What are the components of this vector? What is its length in components?

    Kepler's third law states that the square of the period of a planet is proportional to the cube of its mean distance to the sun, specifically,


    The period and frequency of Jupiter's orbit about the sun are given in these units by


    where mJ=0.001 is the mass of Jupiter. We shall assume that the spacecraft is so small that it does not disturb the circular orbit of Jupiter. It is a simple matter to obtain the following parametric representation for the orbit of Jupiter:

    where t0 depends on the initial position of Jupiter.

  • Review the parametric equations of a circle from Section 16.2 of the main text and verify that these equations describe Jupiter's circular orbit.

    When t=0, where is Jupiter?

    How long does it take for Jupiter to complete one revolution about the sun?

    47.2 Newton's Law of Gravity

    Newton's law of gravity says that an object of mass m and a second object of mass M attract each other with a force of magnitude


    where r is the distance between the objects. The force acts along a line connecting the centers of the objects.

  • Let ms denote the mass of the spacecraft and let mJ denote the mass of Jupiter. Recall that X is the vector position of the spacecraft (as a function of time) and XJ is the vector position of Jupiter.