Module 4: The Wanderers

Activity 1:

Solar System Orbits

Summary

In this Activity, we will investigate

(a) motion under gravity, and

(b) circular orbits in the Solar System and

Kepler’s First and Third Laws.

(a) Motion under Gravity

In the last Activity we saw that the length of planetary years increases and the orbital speed decreases as one moves out from the neighbourhood of the Sun.

To understand these trends we need to know a little about how objects move under gravity.

When Newton was studyinghow objects move under

gravity, he found it helpfulto imagine throwing a

ball from the top of a gigantic mythical mountain on Earth.

As the ball falls towards Earth, aphysicist would say that it gains

energy of motion (kinetic energy) at the expense of its

potential energy (which depends on how far it is from

the Earth’s centre).

This is a formal way of saying that the lower it gets, the faster it falls! (... until it enters the atmosphere and air resistance sets in.)

Depending on how much total energy (kinetic plus potential) the projectile has, it might ...

take one of a number of possible orbital paths:

The projectile never manages to escape

the Earth alongthese paths

-these are bound

elliptical orbits..

Remember that circles are special cases of ellipses.

In particular, Kepler’s First Law states that all orbits of planets in our Solar System are ellipses with the Sun at one focus.

But let’s get back to our ball thrown off a mythical mountain:

If it were thrown just hard enough, the ball could conceivably keep going until it escapes the

Earth’s gravity entirely!

(The path it takesthis time is

called a parabola.)

The minimum launch speed from the Earth’s surface for a projectile to escape the Earth entirely is 11.2 km/s.

This is called the escape velocity from Earth - the velocity an object needs to be moving at to escape the Earth’s gravitational attraction.

The more massive planets (e.g. Jupiter, Saturn, Uranus) have higher escape velocities.

mass

escape velocity

A graph of escape velocity versus mass would look like this:

more massive planetshave higher escape velocities

The escape velocity, ve, is proportional to the square-root of the planet’s mass, M:

Escape velocity doesn’t just depend on the planet’s mass - it also depends on the distance between the planet and the escaping object.

The velocity ve of an object is inversely proportional to its distance d from the planet:

distance from planet

escape velocity

A graph of escape velocity versus distance from a planet looks like:

The further away theplanet is, the smalleris the velocity needed toescape it

… this will become important when we talk about the escape velocity from very small, extremely dense objects like white dwarfs and neutron stars (and even black holes) in the Stars and the Milky Way unit.

rp

ve

The full equation for the escape velocity ve from the surface of a planet depends on the planet’s mass M and its radius rp and is given by:

where G = Gravitational Constant = 6.67 x 10-11 N m2/kg2

As the object moves away from the surface of the planet, we can replace the planet radius rp with the distance d from the (centre of the) planet.

Planet Escape Velocity (km/sec)

We can compare escape velocities from each of the planets:

(where the escape velocities for the giant gas planets, Jupiter, Saturn & Uranus, are calculated at cloud tops,as these planets have no distinct solid surfaces).

Mercury 4.3

Venus 10.3

Earth 11.2

Mars 5.0

Jupiter 61

Saturn 35.6

Uranus 22

Neptune 25

Pluto 1.2

(b) Circular orbits in the Solar System

Nearly all Solar System orbits are good approximations to circles.

v

m

M

r

When objects do travel in circles, the time they take to do a complete orbit - the period - depends on the radius of the orbit (r) and the mass they are orbiting (M), but not on the object’s mass (m).

v

m

M

r

(This isn’t always strictly true,but it works well when the object - e.g. the Earth

- is much less massive than whatit’s orbiting - in this case, the Sun!)

So for planets orbiting the same object - the SunThe period of an orbit increases with its radiuslike this:

orbital radius

orbital period

Distant planets have much longer “years” than do planets near the Sun

… which explains the increase we saw in the last Activity in planetary orbital period for the planets as we move out from the Sun:

Planet (Sidereal) Year

*(measured in multiples of Earth years)

*planetary

orbital period

Orbital periods of the planets:

Mercury 0.241

Venus 0.615

Earth 1.00

Mars 1.88

Jupiter 11.9

Saturn 29.5

Uranus 84.0

Neptune 165

Pluto 249

The relationship between orbital period and orbital radius is worth writing down:

This is Kepler’s Third Law, applied to circular orbits.

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

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

(… yes, we have skipped the Second Law - back to it later!)

It essentially means that, as you look at larger and larger orbits, the orbital period (the “year” for each planet)increases even faster than does the orbital radius.

For example, planets with very large orbital radii such as Neptune and Pluto have such long periods that we haven’t observed them go through an entire year yet.

In this Activity we looked mainly at circular orbits.

In the next Activity we will focus on the more generalcase of elliptical orbits.

NASA: Pluto & Charon http://pds.jpl.nasa.gov/planets/welcome/thumb/plutoch.gif

NASA: View of Australia

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

Image Credits

Now return to the Module home page, and read more about Solar System orbits in the Textbook

Readings.

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