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Introduction to Astrophysics
Lecture 5: Orbits – Kepler and Newton

Kepler’s Laws The realization that planets do not move on perfectly circular orbits was the first step to understanding the law of gravity. Kepler realized that planets move on ellipses. Two example ellipses

Ellipses To make an ellipse, we take two points, called the foci.
The ellipse is the set of all points where the sum of the distances to each focus is the same. It looks like a squashed circle. A circle is a special case of an ellipse where the two foci are at the same point.

Ellipses The major axis is the longest diameter (or semi-major axis for longest radius). The minor axis is the shortest diameter (or semi-minor axis for shortest radius). The eccentricity is the ratio of the separation of the foci to the major axis. It measures how squashed the circle is. A circle is the special case of an ellipse with eccentricity zero. The major and minor axes of a circle are equal (and equal the diameter).

Orbits Using Tycho Brahe’s observations, Kepler discovered that planets move on ellipses, with the Sun located at one focus. The other focus is empty. The orbital eccentricities turn out to be very small (with the exception of Pluto), which is why it was hard to discover this. For examples, Mars’s orbit has an eccentricity of about 0.1; the Earth’s is only 0.02.

Kepler’s Laws of Planetary Motion
Kepler formulated three laws of planetary motion. Planets orbit the Sun in ellipses with the Sun at one focus. The line joining the Sun and a planet sweeps out equals areas in equal times. The square of the period of the orbit T is proportional to the cube of the major axis d : d 3 ∝ T 2. These were empirical laws; Kepler had no explanation of why they are true.

Kepler’s Laws of Planetary Motion
Planets orbit the Sun in ellipses with the Sun at one focus. The line joining the Sun and a planet sweeps out equals areas in equal times. The square of the period of the orbit T is proportional to the cube of the major axis d. The first law tells us the shape of a planet’s orbit. The second law tells us how fast the planet moves around its orbit. The third law tells us how to compare the periods of different orbits.

Kepler’s Laws of Planetary Motion
Second Law: The line joining the Sun and a planet sweeps out equals areas in equal times. A The second law tells us that a planet moves faster when it is closer to the Sun. The two shaded areas in the plot are equal, and so the planet takes the same time to get from A to B as from C to D. D C B (The course WWW site links to animations.)

Kepler’s Laws of Planetary Motion
The third law lets us work out the orbital periods of different planets. For instance, the average distance of Venus from the Sun is 108 million kilometres, as compared to 150 million kilometres for the Earth. As T 2∝d3 we have a period for Venus which is

Newton and the Law of Gravity
m M r Newton’s Law of Gravity explains Kepler’s Laws. It states that the force F between two bodies of masses M and m at separation r is given by where G = 6.67 x N m2 kg-2 is Newton’s constant. It is an inverse square law.

Newton and the Law of Gravity
The acceleration of the mass m is given by F=ma so and is independent of the mass of the body. Hence the orbit taken by a planet does not depend on its mass (in the limit where the masses are much less than the Sun’s mass).

Important points Objects do not necessarily move in the direction of the force acting upon them. Kepler’s Laws follow as a consequence of Newton’s Law of Gravity. Newton gives us deeper understanding of why planets move as they do, and why ellipses are preferred over circles. In fact, objects revolve around their common centre of mass, which is a distance dM from the mass M given by

Some changes to schedule.
This Friday (21st) will be a lecture, not a workshop. The class on Monday 24th is cancelled. Remember that Example Sheet 1 is due in by the end of Thursday’s lecture.

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