Astronomy 161:
An Introduction to Solar System Astronomy

Lecture 19: Orbits

Key Ideas:

Newton generalized Kepler's laws to apply to any two bodies orbiting each other Orbital Speed determines the orbit shape:

Kepler's Laws Revisited

Kepler's Laws of Planetary Motion are as follows:

First Law:
Planets orbit on ellipses with the Sun at one focus.

Second Law:
Planet sweeps out equal areas in equal times.

Third Law:
Period squared is proportional to the size of the semi-major axis cubed.
Expressed Mathematically as: P2=a3, for P in years and a in AUs.

Newton's Generalization

Newton showed that Kepler's Laws can be derived from

Further, Newton generalized the laws to apply to any two bodies moving under the influence of their mutual gravitation. For example, these laws apply equally to

and so on...

First Law of Orbital Motion

The shape of an orbit is a conic section with the center of mass at one focus.
There are two parts to Newton's formulation of Kepler's First Law:

Shapes of Orbits are Conic Sections:

The Center of Mass is at the Focus:

Because the Earth is so much smaller than the Sun, their mutual center of mass is inside the Sun, so the difference is not immediately apparent.

Conic Section Curves

These are curves formed by the intersection of a cone and a plane cutting it at various angles.

Conic curves come in two families:

Closed Curves:

Open Curves:

Which of these orbits you will be in is determined by your orbital speed. There are two special speeds of particular interest...

Circular Speed

Speed needed to sustain a circular orbit at a given radius, r, around a massive body, M:
circular speed
This formula is only strictly valid in the case that the orbiting object is much less massive than the central massive body. For example, it is correct for a satellite or astronaut, but not for the Moon.

Escape Speed

This is the minimum speed required to escape the gravity of a body of mass M starting at a distance, r, away from it:
escape speed
Like with circular speed, this formula is only strictly valid in the case that the moving object is much less massive than the central massive body it is "escaping from". Also, strictly speaking an object never reall "escapes" or "breaks free" from another object's gravity, it just becomes unbound and is on a trajectory that will never turn back on the bigger object like a closed or "bound" orbit.

At the Earth's surface:

Orbit Families

With these two speeds (circular and escape), we can see how the various types of closed and open conic-section orbits around a large central massive body come into being. All of the orbits shown below start from a common point, P, marked on the right (where all 5 curves intersect), but with different speeds:

Newtonian Orbit Families
[Click on the image to view a full-size version (9k GIF)] (Graphic by R. Pogge)
Which orbit the object has is determined by the orbital speed at that point relative to the circular and escape speed at that point:

Center of Mass

Two objects orbit about their center of mass:

Center-of-Mass Schematic
[Click on the image to view a full-size version (5Kb)](Graphic by R. Pogge)

Example: Earth and Sun
Msun = 2 x 1030 kg
Mearth = 6 x 1024 kg

From the balance relation, the distances of the Sun and Earth from their mutual center of mass are related to the size of the semi-major axis of the Earth's orbit (a) and the ratio of the masses:

asun + aearth = 1 AU = 1.5 x 108 km
asun/aearth = Mearth/Msun = 3 x 10-6

After some simple algebra (do it!), we find:

asun = 450 km

Since the radius of the Sun is 700,000 km, this means that the center-of-mass of the Earth-Sun system is deep inside the Sun.

Second Law of Orbital Motion

Orbital motions conserve angular momentum.

This doesn't sound much like "equal areas in equal times", but in fact it is the same thing.

Angular Momentum:

L = mvr = constant

m = mass,
v = speed,
r = distance from the center of mass.
This formula is a simplification for the case of a circular orbit, but it illustrates the basic effect. In general, the angular momentum is a vector, equal to the mass times the vector cross-product of the radial vector and the velocity vector in 3 dimensions.

Angular Momentum & Equal Areas

Angular Momentum is conserved, which means that L is a constant.

If the distance changes, the speed must change to compensate so as to keep L constant:

Near Perihelion:

Near Aphelion:

A familiar example of the same principle at work is a figure skater doing a spin. In an "upright spin", the skater stands on one leg with arms outstretched and spins about an up/down axis. The spin is accelerated by the skater drawing in his/her arms. By drawing in his/her arms, they are moving mass closer to the center of their body, and conservation of angular momentum demands that they spin faster.

Third Law of Orbital Motion

Newton's Generalization of Kepler's 3rd Law:
Generalized Kepler third law

P = period of the orbit
a = semi-major axis of the orbit
M1 = mass of the first body
M2 = mass of the second body

A Third Law for Every Body

The proportionality between the square of the period and the cube of the semi-major axis now depends on the masses of the two bodies.

For planets orbiting the Sun, Msun is so much bigger than any planet (even Jupiter, at 1/1000th Msun), that we recover Kepler's version of the Third Law from Newton's more general form:

Solar System version of the Third Law

Note that the constant of proportionality is the same for all planets (to a good approximation, certainly to within Kepler's data from Tycho).

In Kepler's version, the constant of proportionality works out to be 1.0 if we use units of years for P and AUs for the semi-major axis, a. While computationally convenient, it hides the underlying dependence on the mass of the Sun from sight. The problem with empirical laws, like Kepler's formulation, is that they only show the surface, not the important details underlying them.

Measuring Masses

Newton's generalized form of Kepler's 3rd law gives us a way to measure masses from orbital motions!

For exampl, we can derive the mass of the Sun by using the period and size of the Earth's orbit:

Pearth = 1 year = 3.156 x 107 seconds
aearth = 1 AU = 1.496 x 1011 meters

Using Newton's Form of Kepler's 3rd law for the solar system above, we see that once we know P and a (G and pi are constants), the only unknown is the Mass of the Sun, which can be solved for easily after a little light algebra:

Mass of the Sun from the Third Law
(You can verify the numbers for yourself by using G=6.67 x 10-11 Newton m2/kg2, and the values of P and a for the Earth given above in seconds and meters. Do it!).

A Universal Method for Measuring Masses

The generalized form of Kepler's Third law gives us a powerful tool for measuring the masses of objects by measuring the periods and sizes of their orbits. For example:

We can measure the mass of Jupiter from the orbits of the Galilean moons, since MJupiter>>Mmoons

We can measure the total mass of the Earth and Moon system.

We can measure the masses of binary stars using the full formula and by observing their orbit parameters (you will see this done in Astronomy 162).

The Why of Planetary Motions

Kepler's Laws are descriptions of the motion:

Newton provides the explanation:

By addressing the why, Newton gave his laws predictive power that allows us to use them as tools to explore the Universe, both figuratively and literally. We can predict new phenomena or understand oddities in the motions (they laws give us a framework in which to interpret data), and we can literally use them to fly spacecraft through the solar system.

Updated: 2010 October 10, Todd A. Thompson
Copyright Richard W. Pogge, All Rights Reserved.