Astronomy 161: An Introduction to Solar System Astronomy Prof. Richard Pogge, MTWThF 2:30

Lecture 19: Orbits

Key Ideas:

• First Law: Orbits are conic sections with the center-of-mass of the two bodies at the focus.
• Second Law: angular momentum conservation.
• Generalized Third Law that depends on the masses of the two bodies.

• Circular Speed
• Escape Speed

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

• The Three Laws of Motion
• The Law of Universal Gravitation.

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

• The Moon orbiting the Earth.
• A space probe orbiting the Moon.
• Two stars orbiting each other.

First Law of Orbital Motion

The shape of an orbit is a conic section with the center of mass at one focus.

Shapes of Orbits are Conic Sections:

• Curves found by cutting a cone with a plane.
• Circles, Ellipses, Parabolas, and Hyperbolas

The Center of Mass is at the Focus:

• Strictly speaking, it is not just the Earth orbitting the Sun. The Earth and Sun orbit each other about their mutual Center of Mass.

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:

• Ellipses
• Circles, which are a special case of an ellipse with e=0
• These orbits are bound: objects will orbit forever around the parent body.

Open Curves:

• Hyperbolas
• Parabolas, which are a special case of a hyperbola
• These orbits are unbound: objects will pass by the parent body only once and then escape from the parent body's gravity.

Circular Speed

• If your speed is less than v_C at your current distance, your orbit will be an ellipse smaller than the circular orbit.
• If your speed is a little greater than v_C at your current distance, your orbit will be an ellipse larger than the circular orbit,
• At some point, however, your speed can become large enough that you break free and escape...

Escape Speed

At the Earth's surface:

• v_C = 7.9 km/sec (28,400 km/hr)
• v_E = 11.2 km/sec (40,300 km/hr)

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:

[Click on the image to view a full-size version (9k GIF)]

• If the orbital speed is exactly the circular speed at P (V_C), the orbit will be a Circle passing through P, with the center on the central body (red curve).

• If the orbital speed is slower than V_C at P, the orbit will be an Ellipse smaller than the circle that passes through P, with the central body at the far focus (blue curve).

• If the orbital speed is faster than V_C at P, but less than the escape speed at P (V_E), then the orbit will be an Ellipse larger than the circular orbit that passes through P, with the central body at the near focus (green curve).

• If the orbital speed is exactly the escape speed at P (V_E), the orbit will be an open Parabola (magenta curve).

• If the orbital speed is greater than the escape speed from P, the orbit will be a Hyperbola (black curve). The greater the speed, the "flatter" (more open) the hyperbolic orbit.

Center of Mass

Two objects orbit about their center of mass:

• Balance point between the two masses

• Semi-Major axis: a = a₁ + a₂
• Relative positions: a₂ / a₁ = M₁ / M₂

M_sun = 2 x 10³⁰ kg

M_earth = 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:

a_sun + a_earth = 1 AU = 1.5 x 10⁸ km

a_sun/a_earth = M_earth/M_sun = 3 x 10^-6

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

a_sun = 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

Where:

m = mass,

v = speed,

r = distance from the center of mass.

Angular Momentum & Equal Areas

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

Near Perihelion:

• Planet is closer to the Sun, so r is smaller.
• The speed v must be proportionally faster to compensate.

Near Aphelion:

• Planet is farther to the Sun, so r is larger.
• The speed v must be proportionally slower to compensate.

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

Where:

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

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

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:

P_earth = 1 year = 3.156 x 10⁷ seconds

a_earth = 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:

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 M_Jupiter>>M_moons

• Find M_Jupiter = 300 M_earth

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

• Earth is only ~81x more massive than the Moon, so you have to use the full formula.
• Get the mass of the Earth independently (e.g., our falling bodies experiment from yesterday's lecture).

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:

• Arrived at by trial and error, and some vague notions about celestial harmonies
• Only describe the motions, without explaining why they move that way.

Newton provides the explanation:

• Kepler's Laws are a natural consequence of Newton's Three Laws of Motion and his law of universal gravitation.