Online Astronomy eText: Orbital Motions
Kepler's Third Law: The Harmonic Law Link for sharing this page on Facebook
(also see Ellipses and Other Conic Sections, Kepler's First Law, and Kepler's Second Law)

The square of the period of an orbit is proportional to the cube of its semi-major axis

     This is a compilation of notes concerning orbital motions presented in various classes from 2004 - 2005. They represent portions of various lectures that were shown onscreen as a Word document, and do not fully represent the actual lectures, which included diagrams, illustrations from the website, and a considerable amount of verbal discussion. Each lecture was similar to the others, but there were considerable variations in what I showed onscreen, so all three semesters' notes are shown below. A future revision of this page will simplify and expand on the current contents of the page, and will include (in addition to any topics already covered here) the following topics:

(1) The implications of Kepler's Third Law for the average velocities of planets
(2) The implications of Kepler's Third Law for the nature of the force acting on the planets
(3) The (amazing) result that regardless of eccentricity, Kepler's Third Law works perfectly well
(4) Newton's Revision of Kepler's Third Law to include the mass of the system

Kepler’s Third Law of Planetary Motion: The Harmonic Law (First Semester)
The square of the orbital period is proportional to the cube of the semi-major axis, or
P-squared = a-cubed
      or more accurately, P-squared = (some constant times) a-cubed
      What this implies is that in bigger orbits it takes longer to go around, for TWO reasons:
      (1) you have further to go in bigger orbits
      (2) you move slower in bigger orbits
     
      Pluto’s orbit is 100 times bigger than Mercury’s
      So, you have 100 times further to go (approx)
      In addition, Pluto goes 10 times (sq rt of 100) slower than Mercury, SO 10 times slower, times 100 times further,
      Pluto takes 1000 times as long to go around
      1/4 millennium, instead of 1/4 year
     
      Saturn’s orbit is a little over 9 times bigger than ours, so it moves a little more than 3 times slower than we do (sq rt of 9), so it takes a little more than 9 times 3 (=27) times as long, to go around.
     
      You can figure out the speed by taking the square root of the orbital size (or, by dividing the size of the orbit by the period, which gives you the same result)
      The fact that things work out this way means that there is some sort of ‘harmony’ in the orbits, specified by the math:
      Exponent on left = 2
      Exponent of right = 3
      To a mathematician, this is like a symphony.
      So, Kepler called this the Harmonic Law.
     
      Now, as it turns out, this law needs to be changed, slightly, and was, by Newton, because the force that makes the planets move the way they do, GRAVITY, is a strange force, which acts on all things, equally, regardless of their mass.
Do this using Kepler’s Third Law, for the planets.
      P-squared = a-cubed
      Square of orbital period = cube of orbital ‘size’
      For all the planets going around the Sun, this works out almost exactly correctly, BUT it is WRONG, by a very small correction.
     
      Newton’s Version of Kepler’s Third Law
      Mass (times) P-squared = a-cubed
      Additional Note: This relationship does NOT depend upon the eccentricity of the orbit. Orbits of the ‘same’ size (the same semi-major axis) have exactly the same orbital period (example shown with diagram of circular and eccentric orbits).
      THIS IS THE WAY THAT WE WEIGH THE UNIVERSE.
      Things that weigh more, have more gravitational force on them. But, according to Newton’s Third Law (Law of Action and Reaction), if more force is exerted on something, then it exerts more force back on ‘you’. In the case of gravity, Jupiter has over 300 Earth masses. This means that the Sun would have to pull on Jupiter over 300 times harder than on the Earth (if they were in the same place), in order to move it the same amount (which it would do, willy or nilly, because gravity moves all things in the same way, regardless of their mass). BUT if the Sun pulls on Jupiter 300+ times harder, Jupiter must pull back on the Sun 300+ times harder, and in fact, Jupiter must pull on EVERYTHING 300+ times harder than the Earth does (given the same distance between ‘everything’ and the Earth/Jupiter).
     
      Example:
      Moon going around the Earth
      (semi-major axis) a = 250,000 miles
      (orbital period) P = 27 1/3 days
      Now, let’s make each of these ‘1’ unit
      1-squared = 1-cubed is Kepler’s 3rd Law
     
      Io going around Jupiter
      (semi-major axis) a = approx 250,000 miles
      (orbital period) P = approx 1 1/2 days
      18 times less than for our Moon, even though the orbit is the ‘same’ size.
      (1/18)-squared is NOT 1-cubed BUT, put in the mass of Jupiter (approx 18-squared)
      (18)-squared (times) (1/18)-squared IS equal 1
      That’s how we know that Jupiter is over 300 Earth masses, and exerts over 300 Earth gravities on things at the ‘same’ distance.
     
      Do the same thing with the Sun and Earth
      (semi-major axis) = 400 times Moon’s orbit
      (orbital period) = 13 times Moon’s orbit 13-squared NOT EQUAL 400-cubed
      170 NOT EQUAL 64000000
      BUT if the Sun is 333,000 Earth masses, THEN 333000 (times) 170 IS EQUAL TO 64000000
      So, that’s the mass of the Sun.
     
      NOW GIVEN THIS, we can measure the masses of all the objects in the solar system that have satellites (Sun, with the planets; planets, with their moons; and asteroids, if they have moons)
     
      Discussion of how gravity works very strangely, compared to other forces
      Normal relationship, as given by the Force Law (Newton’s Second Law)
      F = m (times) a
      Force = mass (times) acceleration
      Mass = ‘inertia’ or difficulty of moving something
      Acceleration = rate of change of motion
      Acceleration depends upon both force & mass
      EXCEPT FOR GRAVITY
      For gravity, ALL THINGS FALL AT THE SAME RATE
      Somehow, gravity knows how hard to pull on things, to make them fall the same way, DESPITE their difference in mass.
     
      If something has a small mass, m,
      weight = mass (times) acceleration of gravity
      Small w = small m (times) constant of gravity
     
      If something has a BIG mass, M,
      WEIGHT = MASS (times) acceleration of gravity
      BIG W = BIG M (times) constant of gravity
     
      Gravity is the ONLY force that works this way.
      SO, you can measure the mass of an object, by seeing how much it weighs, OR by how hard it pulls on other objects.

Orbital Motion (Second Semester)
     In Greek physics, circular orbits are presumed to require no force -- it is just the natural thing for planets to do, to follow a circular orbit. But in Newtonian physics, objects which have no force on them move in straight lines, at constant speed (in other words, have constant velocity). So, to follow a circular orbit, some kind of force is required.
     To find the force, we estimate the acceleration (the rate at which velocity changes). This requires some geometry to do accurately, but can be approximated by considering the change that occurs during half of an orbit.
     At the start, the object is moving in some direction, at some velocity v. Half an orbit later, the object is on the other side of the orbit, moving in the opposite direction, at the same velocity. Hence the net change in velocity is 2 v (the difference between the velocity being one way or the other is twice the individual velocities).
     This change of velocity occurs in half an orbit, which takes half an orbital period, or 1/2 P. Therefore, the average rate of change of the velocity is
     (total velocity change) / (time required)
     or 2 v / (1/2 P) = 4 v / P.
     
     but velocity = distance / time
     circumference / period
     2 π (size of the orbit = semi-major axis) / period
     2 π ‘a’ / P
     
     acceleration = 4 v / P = 4 (2 π ‘a’ / P) / P
     acceleration = size / square of period
     But Kepler says square of period = cube of size
     Acceleration = size / cube of size = 1 / square of size
     
     Acceleration = (constant) / r-squared
     Force required to move something
     F = m times acceleration
     Force = mass / square of distance
     
     Action at a distance = force acting through empty space (????)
     
     Fluxions
     Everything is in a state of flux (change) in an orbit
     Distance is changing, speed is changing, and direction of motion is changing all the time
     Differential calculus
     
     Even for ellipses, force points at Sun (always), and is an inverse square force
     ??? is the force ???
     Apples are not pulled down by the Earth physically
     They just fall.
     
     Gravity is a strange thing.
     You can look at it many ways.
     
     Greek:
     Weight = desire to reach the ground
     Falls = allow it to accomplish its desire.
     
     Newton:
     Fall = Earth pulls on them with a force = gravity
     Weight = pull of the Earth on them
     Strange force, because all things fall the same way
     
     Force Law =>> F = m a (acceleration)
     Different masses acted on by a given force will have different accelerations.
     But gravity ‘knows’ that if something is harder to move (has more mass) it should be pulled on harder (has more weight)
     
     W = m g (mass times acceleration of gravity)
     
     Fictitious Forces
     ‘Fictitious’ force is a force that appears to exist, but does not ‘really’ exist, because you are in an accelerated reference frame.
     
     Accelerating car
     Car goes forward, and you and anything attached to the car are pushed forward (accelerated) at the SAME rate = acceleration of the car
     ANYTHING NOT NAILED DOWN will NOT move forward, and will therefore appear to move BACKWARDS at that same acceleration.
     If a ping pong ball and a bowling ball are on the dash, the bowling ball hits you much harder, because it moves with greater force, at the same acceleration.
     The backward force it exerts on you appears real. For that matter, there is an apparent backward force on you, throwing you backwards into the seat, as you move forward.
     REALLY, the seat pushes you forward, and you have a reaction force that pushes the seat backward. The REAL force is the forward one; but it feels as though the backward one is real.
     When the bowling ball hits you, it is not hitting you. You are running into it.
     
     As the Earth rotates, someone at the Pole sees things fall the ‘correct’ way.
     At the Equator, the horizon is ‘falling out’ from under the falling object, because you’re going around the Earth while it falls. So it LOOKS like it is falling slower. And if you weigh it, the ground is falling out from under the scale, so it weighs too little, as well. It appears that everything is falling and weighing 1/3 % less than at the Pole.
     
     Newton has used Kepler’s Laws, and his new math, fluxions, to find that the force making the planets move around the Sun points at the Sun, and is an inverse square force
     F = (constant) / r-squared
     
     He guesses that it is GRAVITY that does this.
     
     He has to justify and PROVE it, or nobody will believe him.
     To justify it, he looks at the Moon, to see if it obeys Kepler’s Laws.
     Things seemed to work just fine.
     
     Moon has an elliptical orbit. It obeys the Law of Areas,
     
     And to get the relative motions of the apple and the Moon requires an inverse-square reduction in the gravity
     Problem (PROVE)
     Inverse (backwards)
     Kepler’s Laws --> gravity
     Gravity --> Kepler’s Laws???
     
     YES!
     1. Orbits are conic sections, Sun at a focus.
     Conic sections = circles, ellipses, parabolas, hyperbolas.
     Only practical orbits for planets are circles and ellipses, and a circle is an ellipse of eccentricity zero, so -- Orbits are ellipses!
     2. Law of Areas -- exactly the same
     (‘equivalent’ to conservation of angular momentum)
     Start with physics --> Law of Areas
     (a) as you go out, you are pulled backwards, so you slow down; as you come in, you are pulled forwards, so you speed up. How much? Go through the math, and you get the Law of Areas.
     3. Mass times P-squared = size - cubed
     
     W = mg
     More mass requires more force
     IF the force in planetary motion is gravity, a pebble, a boulder, a moon, a planet should all ‘fall’ around the Sun the same.
     So, at the orbit of Jupiter, Io, Jupiter’s moon, should ‘fall’ (due to the Sun) the SAME as Jupiter, even though Jupiter is thousands of times heavier.
     At the orbit of the Earth, our moon should ‘fall’ around the Sun the same as the Earth, even though the Earth is heavier.
     
     F = constant / r-squared
     BUT since it’s gravity
     F = constant times (mass of object) / r-squared
     Action = reaction
     If we would pull the moon harder, if it were heavier, then the reaction pull of the moon would pull us harder, if we were heavier.
     F = constant times mass1 times mass2 / r-squared
     F = G m M / r-squared
     
     P-squared = a-cubed
     M times P-squared = a-cubed


Orbital Motion (Third Semester)
Kepler’s First Law Orbit of a planet is an ellipse, with the Sun at a focus.
      Force Law (Newton’s Second Law)
      F = m a
      Force = mass (times) acceleration
      Acceleration = change of motion or speed or both
      Mass = ‘inertia’ of object

      If you push ‘forward’, it will move faster (in the same direction)
      If you push ‘backward’, it will move slower (in the same direction)
      If you push ‘sideways’, it will move sideways (and maybe, faster or slower, depending on force)

      In the text, where the Greek concept of ‘uniform circular motion’ is discussed, ancient Greeks would have said, that such motion occurs because it is ‘natural’ for the objects involved (celestial bodies), and NO FORCE is required.
      However, in Newtonian (Galilean) physics, things go in straight lines, with constant speed, if there is no force on them (Law of Inertia, or Newton’s First Law)
      When there IS a force, the Force Law says that how hard or easy it is, to change the motion of something (to produce an acceleration) depends upon its inertia (which we call ‘mass’). If something has a small mass (inertia), it is easy to move; whereas if it has a big mass (or inertia), it is hard to move (you can still move it, if you ignore friction, and other forces, but it accelerates much more slowly).
      AS AN EXAMPLE:
      The Earth and Moon are going around each other. We normally think of the Moon as going around us, but we also ‘go around’ the Moon, or more accurately, around a point, in between us, called the center of mass of the Earth-Moon system (think of a teeter-totter, and where it would balance).
      Now, Newton has three laws of motion (first two are above). The Third Law, called the Law of Action and Reaction, says that the force acting on one body, due to another (‘action’), is equal and opposite to the force acting on the other body, due to the first one (‘reaction’). Opposite in the sense that they are in opposite directions, and act on the opposite object.
      So if you push on a wall it will push back on you in the opposite direction, with the same force. Or if you tug on a rope, someone on the other end, if they resist in any way, will automatically tug back on you with an equal and opposite force. And in the case of the Earth-Moon system, this means that whatever force acts on the Moon due to the Earth, the same force acts on the Earth due to the Moon.
      Force on Earth = mass of Earth (times) its acceleration
      Force on Moon = mass of Moon (times) its acceleration
(minor typographical and grammatical errors removed from the following, Mar 23, 2012)
      The Earth is 81.6 (approximately 80) times ‘heavier’ (more massive) than the Moon, so with THE SAME FORCE, the Earth moves 81.6 times less. The Moon goes around us in an orbit approximately 240,000 miles in ‘radius’ at a speed of 2,000 miles per hour (taking 27 1/3 days to go once around), and we go around in an ‘orbit’ 81.6 times smaller and slower, or only about 25 miles per hour (since we go around it in the same 27 1/3 days), covering an ‘orbital motion’ only about 3,000 miles in ‘radius’ (81.6 times smaller).
      We EACH go around the center of mass, but the Moon moves in a path 80 times larger, 80 times faster, because the same force acting on an object 80 times less massive moves it 80 times more.
      ALL FORCES WORK LIKE THIS. The amount of motion or acceleration that you get when you ‘act’ on an object depends both on the force, and the inertia or mass of the object. This means that under normal circumstances larger objects will move less when acted on by a force, and smaller ones will move more. However, that leads to the question: Why, if different masses should move at different rates when exposed to a given force, do all things fall in exactly the same way under the influence of gravity?
      Small thing, F = m (times) acceleration of gravity Big thing, F = M (times) acceleration of gravity
      The acceleration of gravity is always the same. SO, if something has a big or small mass, to make this work gravity has to exert more force on things that have more mass:
      FORCE LAW FOR GRAVITY
      W = m (times) g
      Weight = mass (times) acceleration of gravity
      Now, in Greek astronomy, everything goes around us, in some combination of uniform circular motions, because that’s the ‘natural’ thing to do. But in Newton/Galileo’s physics, straight-line motion is the ‘natural’ thing, and if something goes round in a curve there must be some kind of sideways force acting on it.
      For orbits that is the force of gravity.
      Prior to Newton it was presumed that gravity was a local phenomenon, on the Earth, and did not apply to things in space, which being up in the heavens were presumably made of different stuff from the Earth and therefore behaved differently.
      After Newton everything on the Earth and in space is presumed to follow the same natural laws, and the motions we see (at least for the planets) are due to gravitational forces between them and the Sun. And since the law of gravity on the Earth involves the mass of things (bigger masses requiring bigger forces), this means that the law of gravity in space must also do so:
      Newton’s Law of Universal Gravity
      F = m (time) M (divided by) r-squared
      The force (of gravity) is the product of the masses of the two objects, divided by the square of the distance between them (referred to as an inverse-square relationship)
      This discovery, that the force that makes the planets go around the Sun (and the moons of the planets go around them, and later on, when binary stars are discovered, stars go around each other, and still later on, when galaxies are discovered, stars go around galaxies) depends upon the masses involved changes Kepler’s Third Law to look like this:
      Kepler's version of his third law: Period-squared = size of orbit-cubed
      Newton's version of Kepler's third law: Mass of system (times) P-squared = a-cubed
      Now for all things in the Solar System the mass on the left is the mass of the Sun plus the mass of the planet; and planets are very small compared to the Sun, so to better than 1/10th of 1% accuracy this is always the same number and you can ignore it and get Kepler’s version of this formula.
      Example of how this works:
      Moon takes 27 1/3 days, to go around the Earth. We will call this one ‘unit’ of orbital period.
      Moon’s orbit is 240,000 miles in ‘radius’.
      We will call this one ‘unit’ of orbital size.
     
      So for our Moon, Kepler’s 3rd Law = 1-squared = 1-cubed (1 = 1)
      This works because we’re 'cheating', so to speak. We’re defining things to work this way.
      But now let’s do this for Io, a moon of Jupiter.
      Io’s orbit is about the same size as our Moon’s orbit, so it is 1 unit, approximately.
      Io’s orbital period is only about 1 1/2 days, or about 18 times less than for our Moon.
      1/18 squared = 1/324 ? = ? 1-cubed = 1 ?
      Of course not!
      UNLESS YOU PUT IN THE MASS.
      Earth + Moon = 1 unit
      Jupiter + Io = ? units
      ? (times) 1/324 = 1
      THEREFORE JUPITER MUST ‘WEIGH’ 324 mores than Earth. (using accurate values, the actual number is about 318)
      THIS IS HOW WE WEIGH THE UNIVERSE.
      For the Sun the numbers work out like this:
      The orbit of the Earth is 400 times the size of the orbit of the Moon. So on the right a-cubed is approximately 65,000,000.
      The orbital period of the Earth, however, is only about 13 ‘moonths’ -- about 13 times the orbital period of the Moon. So on the left, p-squared is only about 170.
      SUN-EARTH SYSTEM
      Mass (times) 170 = 65,000,000
      Therefore, Mass of Sun = 333,000 times the mass of Earth