We have been discussing Newton's Laws of Motion and Newton's Law of Gravitation. To study orbits - objects in orbit around the Earth or Sun, and how to get them there - we need to put these laws together. We need to consider how objects accelerate when the force of gravity is applied. Then we will consider how to get objects into orbit and what their orbits look like once they are up there.

Newton's Laws - once again

First, we need to consider the trajectory of an object - that is, the path it takes - when a force is applied. The force could come from your muscles (as when you throw something), a rocket engine or gravity.

By now you have gotten the idea that gravity acts downwards (towards the center of the Earth) and that all things fall at the same rate (if we ignore air resistance). What happens when we add a horizontal force? If the force is purely horizontal (not adding or subtracting from gravity) then we can consider the effects of the forces separately: gravity determines the motion in the vertical direction and the horizontal force determines the motion in the horizontal direction.

 

(2) At exactly the same time that you drop a valuable crystal glass, your housemate throws a bottle horizontally. The bottle lands at the feet of your noisy neighbor--which smashes first--the glass or the bottle? (Hint: think of the horizontal and vertial motions separately). (3) If you drop an object it lands on your toes. If you throw it sideways it goes a way before hitting the ground. If you throw it harder sideways it goes farther. If you throw it up at an angle does it go farther? Always? (Remember: if you throw it directly up in the air it lands on your head)

Launching Up - and out into space

Rockets are generally launched directly upwards at first.

This is to get them out of the lower atmosphere, where there is lots of friction, quickly. Soon into the launch, the rocket is directed horizontally. It is the horizontal motion that puts the rocket into orbit.

Circular Orbits

We are not going to explain WHY this is the case (because it requires excessive amounts of geometry or calculus to do so). BUT - it is very useful to have a formula for a circular orbit - remember, a circle is an ellipse with no eccentricity. The speed of a circular orbit of radius r is described by

Vcirc = ( GM/r)1/2

For the Earth orbiting the Earth this speed is 30 km/s. For an object orbiting the Earth the orbital speed is 8 km/s.

Look at the formula and

(5) The Hubble Space Telescope (mass=2000 kg) is in orbit at about 580km above the Earth. Also orbiting the Earth at 580 km is one of the astronauts gloves (mass=0.5 kg). Which orbits the Earth faster, HST or the glove?

All Kinds of Orbits

Look again at the figure above (which also has better drawings of these types of orbits).

(6) Above are diagrams showing orbits when launched with increasing speed. Label them appropriately with the terms: A=Circular, B=elliptical, C=Escape, D=Sub-orbital

Newton's Version of Kepler's Third Law

This is probably THE MOST IMPORTANT FORMULA in all of astrophysics. It is used throughout the solar system, the Galaxy and the most distant universe. So we are going to take some time to work our way up to it.

 

(7) In a circular orbit 200 m above the surface of the Earth (that is 6400+200=6600 km from the center of the Earth)--there is a spy satellite looking down. We can work out the orbital period of this spy satellite (and, obviously, so could the people being spied on)--as follows: (a) What is the total circumference of the orbit? (b) Remember SPEED = DISTANCE / TIME, so TIME = (c) If the DISTANCE is in km, and the SPEED is in km/s, what are the units of TIME? (d) Now plug in the numbers and convert the orbital period of this spy satellite to hours.

Let's think about what we can now conclude about orbits:

• to know the orbital height of a satellite all you need to know is it's orbital period--and vice versa.
• but you CANNOT measure its mass from watching its orbit.
• The formula for orbital period also works for orbits around ALL planets, ALL stars, ALL galaxies, ALL black holes--EVERYTHING being orbited by ANYTHING.

This formula for orbital period is extremely useful - Especially if we do a little algebra.....

Let us write the general expression for the orbital period. And let's use the letter a for orbital distance (Why not? Besides, we might later want to think about orbits as ellipses rather than just circles--and remember our friend the semi-major axis? Remind yourself if necessary)

Wait a minute! This formula looks remarkably like Kepler's Third Law, does it not? With some extra bits. It is in fact Newton's Version of Kepler's Third Law (or NVKTL for short).

   4 2
P2  =  ------   a3
   G M

And it applies to ANYTHING being orbited by SOMETHING. Note that it does not involve the mass of the orbitER--just the orbitEE.

Let us start with planets. They orbit the Sun. So M=M_Sun. Comparing Jupiter's orbital period around the Sun compared with Earth's we have

That is EXACTLY Kepler's Third Law.

 

(8) Check that it still works for Jupiter where aJ = 5.3 A.U.--what is PJ (in years)?

NVKTL (Newton's Version of Kepler's Third Law) is supposed to work for ANYTHING orbited by SOMETHING. Not just planets.

G is Universal (originally measured in a lab by a guy called Cavendish in Cambridge, England).

• G = 6.67 x 10^-11 in SI units.
• = 3.14......
• 4=4=4=4=4=4=4=..........

We have here an equation with 3 unknown quantities. That means that if we measure 2 of them we can re-arrange the equation to determine the third. THIS IS VERY VALUABLE.

 

(10) Write out the equation for Newton's Version of Kepler's Third Law, re-arranging the terms with each of the 3 variables on the left hand side of the equation - as if you wanted to solve for (a) P = (b) a = (c) M = (11) Say we measure the orbital period, P, and the orbital distance, a, of one of the moons, say Europa, orbiting Jupiter. What can we determine from this equation? (a) the mass of Europa (b) the mass of Jupiter (c) the mass of the Earth (12) Stars often come in twos and threes rather than on their own. Say we detect a little star orbiting a big star. The little star has an orbital period around the big star of about one year. The distance between stars is harder to measure, but say we estimate it to be about 1 A.U. What is the mass of the big star compared to our Sun? (13) We can "weigh" a galaxy by measuring the speed of stars at its edge. Describe how you think this might be done using Newton's Version of Kepler's Third Law.

We are going to come back to Newton's version of Kepler's 3rd Law time and time again - to measure the mass of planets, of asteroids, distant stars,...black holes at the center of the Galaxy and whole galaxies.