Transcript m04a01

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 studying
how objects move under
gravity, he found it helpful
to imagine throwing a
ball from the top of
a gigantic mythical
mountain on Earth.
As the ball falls towards Earth, a
physicist 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 along
these 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 takes
this 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.
The escape velocity, ve, is proportional to the square-root
of the planet’s mass, M:
A graph of escape velocity versus mass would look like this:
escape
velocity
more massive planets
have higher escape
velocities
mass
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:
A graph of escape velocity versus distance from a planet
looks like:
escape
velocity
The further away the
planet is, the smaller
is the velocity needed to
escape it
distance from planet
… 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.
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:
ve
rp
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.
We can compare escape velocities from each of the planets:
Planet
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
Escape Velocity (km/sec)
4.3
10.3
11.2
5.0
61
35.6
22
25
1.2
(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).
(b) Circular orbits in the Solar System
Nearly all Solar System orbits are
good approximations to circles.
M
r
m
v
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).
(This isn’t always strictly true,
but it works well when the object
- e.g. the Earth
- is much less massive than what
m
it’s orbiting
- in this case, the Sun!)
v
M
r
The period of an orbit increases with its radius
So for planets orbiting the same object - the Sun
like this:
orbital
period
Distant planets have
much longer “years” than
do planets near the Sun
orbital radius
… which explains the increase we saw in the last
Activity in planetary orbital period for the planets as
we move out from the Sun:
Orbital periods of the planets:
Planet
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
(Sidereal) Year *
0.241
0.615
1.00
1.88
11.9
29.5
84.0
165
249
*(measured in multiples of Earth years)
planetary
orbital period
The relationship between orbital period and orbital
radius is worth writing down:
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.
This is Kepler’s Third Law, applied to circular orbits.
(… 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 general
case of elliptical orbits.
Image Credits
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
Now return to the Module home page, and read
more about Solar System orbits in the Textbook
Readings.
Hit the Esc key (escape)
to return to the Module 4 Home Page