Download Lesson 18 notes – Orbits - science

Lesson 18 notes – Orbits
Objectives
(i) analyse circular orbits in an inverse square law field by relating the
gravitational force to the centripetal acceleration it causes;
(j) define and use the period of an object describing a circle;
(k) derive the equation T2 = 4 π2 x r3
GM
from first principles;
2
(l) select and apply the equation T = 4 π2 x r3
GM
for planets and satellites
(natural and artificial);
(m) select and apply Kepler’s third law T 2 = k r 3 to solve problems;
(n) define geostationary orbit of a satellite and state the uses of such
satellites.
Outcomes
Be able to define the period of an object describing a circle.
Be able to look at data like period, radius, gravitational field strength, to relate
gravitational force to acceleration.
Be able to define a geostationary orbit of a satellite and state the uses of such
satellites.
Be able to select and apply Kepler’s third law T 2 = k r 3 to solve problems in
different situations.
Be able to select and apply the equation T2 = 4 π x r3
GM
correctly for planets
and satellites (natural and artificial);
Be able to derive the equation T2 = 4 π x r3
GM
from first principles;
Orbits
An orbit is the (usually elliptical) path described by one celestial body in its
revolution about another. The time it takes for one complete revolution is
called the period T. The closer a body is to the object it is orbiting the faster it
will go. This is because as you move closer to an object with mass its
gravitational field increases and so the centripetal force goes up making the
orbiting object accelerate.
History
The first true laws of planetary motion were proposed by Johannes Kepler
between 1609 and 1619 as a result of his work on the twenty years of
planetary observations made by Tycho Brahe, the astronomer royal to the
King of Denmark.
Kepler used Brahe's data to come up with three laws of planetary motion
1. The planets move in ellipses with the Sun at one focus
2. A line drawn from the planet to the Sun sweeps out equal areas in
equal times (see Figure 1)
Kepler’s 3rd Law
The ratio of the square of the period (T) of the planet about the Sun to
the cube of the mean orbit radius (r) is a constant or T2/r3 = constant.
T2 is directly proportional to r3
T2=kr3
Calculating k
The Earth is 150x109 m from the Sun and it takes one year (3.16x107 s) to
orbit the Sun.
Therefore Kepler's constant ( r3/T2) for the Solar System is
(150x109)3/(3.16x107)2 = 3.37x1018m3s-2
The geostationary satellite
A geostationary satellite has a period of exactly one day and so remains
constantly over one point on the Earth's surface. To be geostationary it must
have an orbit that lies in the
plane of the equator.
All Geosynchronous
satellites are in an orbit
42000 km from the Earth's
centre.
Extension
Deriving T2 = 4 π2
GM
x r3
The gravitational force F exerted by the Sun on a planet is given by:
F=GMm/r2
The centripetal force for an object in orbit is given by:
F = mv2/r
The instantaneous velocity of an orbiting object is given by:
v = 2πr/T
So we have 4π2/T2 = GM/r3
and so
r3 (4π2/GM) = T2
So our constant k = (4π2/GM)
Rearranging we get: r = (goR2T2/4π2)1/3
Using this we can check that the distance of our Geostationary Satellites are
at 42000km.