Chapter 4Chapter 4Gravitation and the Waltz of the Gravitation and the Waltz of the

PlanetsPlanets

The important concepts of Chapter 4 pertain to orbital motion of two (or more) bodies, central forces, and the nature of orbits.1. What we see in the sky results from the rotation of the Earth on its axis, the orbital motion of the Earth about the Sun, the orbital motion of the Moon about Earth, and, to a small extent, the gravitational effect of the Sun and the Moon on the Earth’s axis of rotation.2. The motions of the Earth produce a fundamental frame of reference for stellar observations.

Two great observers:Hipparchus Tycho Brahe

(2nd century BC) (1546-1601 AD)

Angle measuring devices: mural quadrantcross staff

Nicholas Copernicus (1473-1543) revived the heliocentric modelfor the solar system, where planetary orbits are envisaged as circular for simplicity. Even circular orbits are sufficient for understanding the difference between sidereal (star) period of a planet, Psid = time to orbit the Sun, and its synodic period, Psyn = time to complete a cycle of phases as viewed from Earth.The relationship between the two is best demonstrated by considering the amount by which two planets A and B advance in their orbits over the course of one day.

Over the course of one day, planet A advances throughthe angle

Planet B advances throughthe angle

The difference in the angles is the amountby which planet Ahas gained on planetB, which is relatedto its synodic period,i.e.:

In other words,

Or:

If A is Earth, and B a superior planet (orbits outside Earth’s orbit), then:

For Earth P = 365.256363 days (a little more than 365¼ days), i.e.:

Given two values, the third can be found !

The same technique can beused to relate a planet’srotation rate and orbitalperiod to the length of itsday. The arrow is a fixedfeature on the planet.

For example, the planet Mars has a synodic period of 780 days, which means it returns to opposition from the Sun every 2.14 years. But its true orbital period is 687 days, or 1.88 years, which means it returns to the same point in its orbit about the Sun every 1.88 years.

Some further consequences:

Mercury: Prot = 58d.67, Psid = 88d.0, Pday = 176d.

Venus: Prot = 243d (retrograde rotation), Psid = 224d.7, Pday = 117d.

Moon: Prot = 27d.3215, Psid = 365d.2564, Pday = 29d.5306.

Earth: Prot = 23h 56m, Psid = 365d.2564, Pday = 24h.

Which planet has the longest “day”?

Johannes Kepler (1571-1630)

How Kepler triangulated the orbit of Mars. He took Tycho’s observations of Mars relative to the Sun separated by the planet’s 687d orbital period (with Earth at different parts of its orbit) and used them to triangulate the location of Mars, which was at the same point of its orbit.

Kepler’s study of the orbit of Mars resulted in his three laws of planetary motion:1. The orbits of the planets are ellipses with the Sun at one focus. Actually they are conic sections.2. The line from the Sun to a planet sweeps out equal areas of orbit in equal time periods. Angular momentum is conserved, i.e. mvr = constant.3. The orbital period of a planet is related to the semi-major axis of its orbit by P2 = a3 (Harmonic Law).

2a

2b

centre distance, c = ae

c2 + b2 = a2

aeb

a = ½ string length 2a

e = (1 ˗ b2/ a2) ½

Isaac Newton formulated Kepler’s Laws into a model of gravitation, in which: a mass attracts another mass with force inversely proportional to the square of the distance between the two, i.e. F ~ 1/d2. Forces produce acceleration of an object proportional to its mass, i.e. F = m×a, and objects stay at rest or in constant motion in one direction unless acted upon by a force.

Objects in orbit around Earth are constantly falling towards the Earth. They are acted upon by gravity, but are in free-fall towards Earth. They will not “fall” to Earth if their transverse speed is large enough.

The importance of Kepler’s 3rd Law is that, as shown by Newton, the constant of proportionality for a3 = P2 contains two constants, π (pi) and G (the gravitational constant), plus the sum of the masses of the two co-orbiting bodies. If one can determine orbital periods P and semi-major axes a, then one can derive the masses of the objects in the system: either planets or stars !For example: Jupiter’s mass from the Galilean satellites.

Astronomers try to keep the calculations simple, so they usually omit π and G. Thus, the Newtonian version of Kepler’s 3rd Law is usually written as:

where the sum of the masses of the two co-orbiting objects, “M1” and “M2”, is calculated in terms of the Sun’s mass, the orbital semi-major axis (~radius) “a” is calculated in terms of the Earth’s distance from the Sun, the Astronomical Unit, and the orbital period “P” is expressed in Earth years.

The point to be emphasized is that a measurement of two of the parameters permits one to calculate a value for the third parameter. Astronomers use the relationship to measure the masses of planets and stars.

Vis-Viva Equation.There is a useful relationship for orbital speed that can be obtained from the energy equation:

Solving the equation for the velocity v gives:

which is the vis-viva equation, where a is the semi-major axis of the orbit, r is any point in the orbit, and v is the speed in the orbit at r. Escape velocity is attained whena → ∞, i.e.:

Circular orbits apply when r = a everywhere, i.e.:

Earth’s orbital velocity.

And escape velocity from Earth orbit is:

The maximum encounter velocity for a solar system object is the sum of the previous two values, i.e.:

which is close to the encounter velocity of meteoroids associated with Halley’s Comet.

Sending a satellite to the Sun.Here, the situation is pictured atright, where an orbit from Earth to the Sun will have asemi-major axis of ½a. ByKepler’s 3rd Law the orbitalperiod is calculated as:

But aphelion to perihelion constitutes exactly half an orbit,so the time to reach the Sun is:

Astronomical TerminologyRotation. The act of spinning on an axis.Revolution. The act of orbiting another object.Geocentric. = Earth-centred.Heliocentric. = Sun-centred.Opposition. When a planet is opposite (180° from) the

Sun.Conjunction. When a planet is in the same direction as.

Typically refers to conjunction with the Sun.Inferior planet. A planet orbiting inside Earth’s orbit.Superior planet. A planet orbiting outside Earth’s orbit.Prograde motion. When a planet’s RA increases nightly.Retrograde motion. When a planet’s RA decreases

nightly.Astronomical Unit = A.U. The average distance between

Earth and the Sun.Inertia. An object’s resistance to its state of motion.Inertial reference frame. = non-accelerated frame.

Astronomical Terminology 2Eccentricity. The amount of non-circularity of an orbit,

from round (e = 0.0) to very flattened (e = 0.9).Semi-major Axis. Half the length of the long axis of an

ellipse, equivalent to the “radius” of an orbit.Orbital Period. The time taken for one object to orbit

another object.Synodic Period. The time taken for an object to cycle

through its phases as viewed from Earth.Inferior planet. A planet orbiting inside Earth’s orbit.Superior planet. A planet orbiting outside Earth’s orbit.Prograde motion. When a planet’s RA increases nightly.Retrograde motion. When a planet’s RA decreases

nightly.Gravity. The force exerted by an object on any other

object in the universe.“Zero gravity.” A fictional term referring to the

apparent weightlessness of an object in free fall.

Sample QuestionsSample Questions

1. Imagine a planet moving in a perfectly circular orbit around the Sun. Because the orbit is circular, the planet is moving at a constant speed. Is the planet experiencing acceleration? Explain your answer.

Answer: Yes, it is. The planet experiences acceleration since it is constantly falling towards the Sun.

2. Suppose that astronomers discovered a comet approaching the Sun in a hyperbolic orbit. What would that say about the origin of the planet?

Answer. Objects in hyperbolic orbits are not bound to the object they are orbiting. Astronomers would therefore conclude that the comet is not bound to the solar system and must therefore have originated from “outside” the solar system.

3. Why is the term “zero gravity” meaningless? Is there a place in the universe where no gravitational forces exist?

Answer. All objects are subject to the attractive force of every other object, in proportion to the inverse square of the separation r from the other object. For one object to experience no outside gravitational forces, i.e. zero gravity, it would have to be an infinite distance away from every other object, which is not possible. So the term “zero gravity” cannot apply anywhere in the known universe.