Module 4: The Wanderers

Activity 2:

The Universal Force of Gravity

Summary:

In this Activity, we will investigate

(a) elliptical orbits and Kepler’s Laws,

(b) Newton’s Law of Gravitation, and

(c) apparent weightlessness in orbit.

Some orbits in the Solar System cannot be approximated at all well by circles

(a) Elliptical Orbits and Kepler’s Laws

- for example, Pluto’s separation from the Sun varies by about 50% during its orbit!

According to Kepler’s First Law, closed orbits arounda central object under gravity are ellipses.

As a planet moves in an elliptical orbit, the Sun is at one focus (F or F’) of the ellipse.

F’ FC

r

v

The line that connects the planet’s point of closest approachto the Sun, the perihelion ...

F’ FC

perihelion

r

v

F’ FC

aphelion

perihelion

As a planet moves in an elliptical orbit, the Sun is at one focus (F or F’) of the ellipse

… and its point of greatest separation from the Sun, the aphelion

is called the major axis of the ellipse. r

v

F’ FC

r

v

The only other thing we need to know about ellipses is howto identify the length of the “semi-major axis”, because that determines the period of the orbit.

a a

“Semi” means half, and so the semi-major axis a is half thelength of the major axis:

v

m

M

r

For circular orbits around one particular mass - e.g. the Sun - we saw that the period of the orbit (the time for one completerevolution) depended only on the radius r

r

- that was Kepler’s 3rd Law:

the orbital period squared is proportional to the orbital radius cubed.

For objects orbiting a common central body (e.g. the Sun)in approximately circular orbits,

where G and 42 are constants and M is the mass of the Sun

radius

RPeriod

P

m

We can write Kepler’s third law:the orbital period squared is proportional

to the orbital radius cubed

in mathematical notation as:

Now here’s the mathematics….

This comes from equating the gravitational and centrifugal forces:

=

and noting that the circular velocity:

F’ FC

r

v

a a

For elliptical orbits,the period dependsnot on r, but on thesemi-major axisa instead.

a a

Let’s see what determines the period for an elliptical orbit:

For objects orbiting a common central body (e.g. the Sun)

It turns out that Kepler’s 3rd Law applies to all ellipticalorbits, not just circles, if we replace “orbital radius”by “semi-major axis”:

the orbital period squared is proportional to the orbital radius cubed.the orbital period squared is proportional to the semi-major axis cubed.

Each of these orbits has the same semi-major axis length a:

Note that a circleis a special case ofan ellipse, wherer = a.

So if each of these orbits is around the same massiveobject (e.g. the Sun),

then as they all have the same semi-major axis length a,

then, by Kepler’sThird Law, they have the sameorbital period.

Click here to see a simulation illustrating Kepler’s Third Law.

So, as you saw in the simulation, bodies orbiting at large distances have much longer orbital periods.

For the mathematicallyinclined, the square of the period P of the orbit increases in proportion to the cube of the semi-majoraxis a:

a semi-major axis

P period

distant planets havemuch large orbital periods

We haven’t yet met Kepler’s Second Law.

That’s because it’s not at all interesting for circular, oralmost circular orbits.

But if we look at a quite eccentric elliptical orbit, for example, that of Halley’s comet:

Comet Halley in 1910

Sun

Neptune

Comet Halley

Note that Comet Halley’s orbit is retrograde, which means that it orbits the Sun clockwise when viewed from the north pole. This is the the opposite sense to that of the planets.

An object in a highly elliptical orbit travels very slowlywhen it is far out in the Solar System,

… but speeds up as it passes the Sun.

According to Kepler’s Second Law,

… the line joining the object and the Sun ...

… sweeps out equal areas in equal intervals of time.

equal areas

That is, Kepler’s Second Law states that

the line joining a planet and the Sun sweeps outequal areas in equal intervals of time.

We call the force which keeps the Moon in its orbit around the Earth gravity.

Sir Isaac Newton’s conceptual leap in understandingof the effects of gravity largely involved his realisationthat the same force governs the motion of a falling objecton Earth - for example, an apple - and the motion of the Moon in its orbit around the Earth.

(b) Newton’s Law of Gravitation

Isaac Newton discovered that two bodies share a gravitational attraction, where the force of attraction depends on both their masses:

MSun MEarth

Both bodies feel the same force, but in opposite directions.

MSun MEarth

This is worth thinking about - for example, drop a pen to the floor. Newton’s laws say that the force with which the pen is attracting the Earth is equal and opposite to the force with which the Earth is attracting the pen, even though the pen is much lighter than the Earth!

Newton also worked out that if you keep the masses of the two bodies constant, the force of gravitational attraction depends on the distance between their centres:

mutual force of attraction

Note that the gravitational force is larger the closer the objects are together.

For any two particular masses, the gravitational force between them depends on their separation as:

distance between the masses, R

magnitude of the

gravitational force

between 2 fixed

masses, Fgrav

as the separation between the masses is increased, the gravitational force of attractionsbetween them decreases quickly.

Your pen dropping to the floor and a satellite in orbit around the Earth have something in common - they are both in freefall.

To see this, remember Newton’s thought experiment from the Activity Solar System Orbits:

On all these trajectories, the projectile is in free fall

under gravity.(If it were not, it would

travel in a straight line - that’s Newton’s

First Law of Motion.)

- that is, it falls to Earth eventually.

If the ball is not given enough “sideways” velocity, its trajectory intercepts the Earth

On all these trajectories, the projectile is in free fall.On the trajectories which make complete orbits, the projectile is travelling “sideways” fast enough ...

… that as it falls, the Earth curves away underneathit, and the projectile completes entire orbits without ever hitting the Earth.

This astronaut on a space walk is alsoin free fall.

The astronaut’s “sideways” velocityis sufficient to keephim or her in orbitaround the Earth.

(c) Apparent Weightlessness in Orbit

Why do astronauts in the Space Shuttle in Earth orbit feel weightless?

Let’s take a little time to answer the following question:

Some common misconceptions which become apparent in answers to this question are:

(a) there is no gravity in space,(b) there is no gravity outside the Earth’s atmosphere, or(c) at the Shuttle’s altitude, the force of gravity is very small.

Click on each alternative to see why we claim that it’s a misconception!

Then see if you agree with our explanation ...

On all these trajectories, the projectile is in free fall.

In a spacecraft (like the Shuttle) in Earth orbit, astronauts are in free fall, at the same rate as their spaceships.

That is why they experience weightlessness: just as a platform diver feels while diving down towards a pool, or a sky diver feels while in free fall.

On all these trajectories, the projectile is in free fall.

In the next Module we’ll look at one Solar Systemorbit in particular- that of the Moon around the Earth.

NASA: View of Australia

http://nssdc.gsfc.nasa.gov/image/planetary/earth/gal_australia.jpg

NASA: Halley in 1910

http://pds.jpl.nasa.gov/planets/gif/smb/hal1910.gif

NASA: Space Shuttlehttp://lisar.larc.nasa.gov/LISAR/IMAGES/SMALL/EL-1994-00718.jpeg

Image Credits

Now return to the Module home page, and read more about gravity in the Textbook Readings.

Hit the Esc key (escape) to return to the Module 4 Home Page

(a) There is no gravity in space?

At face value, this statement doesn’t bear too much examination, because Newton’s Law of Gravitation has been applied right from its inception to the motion of the Moon and planets - and they are in space.

When people make this assumption, perhaps what theyare really saying is that the sort of gravity which makesus feel heavy only exists on planetary surfaces- but Newton developed the Law in the first place by realizing that gravity as it acts on Earth (e.g. on an apple)is the same force as that which acts on the Moon and planets.

Back to the alternative answers

(b) There is no gravity outside the Earth’s atmosphere?

Like (a), at first glance this misconceptions seems naïve, because Newton’s Law of Gravitation has been applied right from its inception to the motion of the Moon and planets - and they are in space.

However what this statement might really be revealing is a link many people perceive between gravity and air: in other words, the mistaken idea that gravity does not exist in a vacuum - that air, in some way, makes things heavy.

In fact, air actually makes objects feel very, very slightly lighter - like buoyancy in a tank full of water, except that air is so much less dense than water that the effect is not noticeable.

The misconception that links gravity and air shows up in some science fiction movies too - watch for the one where a Concorde-type plane “mistakenly” ends up in Earth orbit. The passengersinside the plane can walk around, with a bit of care, butastronauts sent up to help them float weightlessly aboutoutside!

Back to the alternative answers

(c) At the Shuttle’s altitude, the force of gravity is very small?

This statement sounds reasonable - after all, the Shuttle is way out in space - until you check it with calculations.

In fact, compared to the radius of the Earth (6378 km), atypical Shuttle altitude above the Earth’s surface of 200 kmor so is pretty negligible.At that altitude, the force of gravity is only 5% less than on the Earth’s surface.

Back to the alternative answers