Download Kepler`s Laws

AK/NATS 1745.06
Fall 2006
Exercise #3 : Kepler's Third Law
The large rockets and extensive technology associated with putting a satellite into orbit introduces a bruteforce element, leaving the impression that satellites and their movements can be arranged entirely to suit
our convenience. This is not so. After the rocket power has been shut off, the motion of a satellite is
determined by the effects of the earth's gravitational field, and this places severe restrictions on what is
possible. The outcome is exactly the same as the effect of the sun's gravitational field on the orbits of the
planets, and in this way the large number of satellites now orbiting the earth is quite similar to a small solar
system. This assignment deals with the similarity between the two systems, and the inherent natural laws
which they must obey.
Johannes Kepler derived, early in the 17th century, the geometry of the planetary orbits. Of particular
interest here is the Harmonic Law, which may be written in algebraic form as :
P2  k  a 3
where P is the period of revolution of the planet, a is the semi--major axis of the planet's orbit about the
sun, and k is a constant for all the planets set by the sun's gravity. In the case of the solar system,
k  1 when the period of the sun orbiting satellite is stated in years and the semi-major axis is measured in
astronomical units. The same algebraic form applies to the orbits of satellites around the earth, but in that
case the constant k will have a different value because the source of gravity is the earth rather than the
sun. Simply substitute ‘satellite' for ‘planet' and ‘centre of the earth' for ‘sun'. (The actual value of the
constant k is irrelevant for this lab exercise but for those interested, it can be determined simply by
calculating the slope of the line in Graph 1.)
The relation between satellite period and distance from the centre of the earth is shown graphically in
Graph 1. Recall that the average radius of the earth is 6371km.
The line on Graph 1 illustrates the behavior of the system of satellites around the earth. Each point on that
line represents a possible satellite either natural (like the moon) or artificial (like a space shuttle). Use the
graph to answer the following questions. When explaining any of your answers, be short and concise. Three
or four sentences should suffice for an answer.
In the case of an object orbiting one that is much larger, we can also use
M = a3/ P2
To give us the mass (M) of the larger object.
AK/NATS 1745.06
Fall 2006
Graph 1 : Kepler's Harmonic Law for Earth Satellites
Orbital Period (hours)
1000
100
10
1
1
10
100
1000
Distance from the Earth's Centre (x1000km)
Questions
1.
A satellite was placed in an orbit somewhat above the earth's atmosphere, at a distance of 15,000
km from the centre of the earth. Determine the orbital period (in hours) of the satellite.
2.
If the shuttle orbits about 300 km from the surface of the Earth and Earth’s radius is 6500 km,
how long would it take for the shuttle to go around Earth? (remember the orbital distance is
measured from the centre of Earth)
3.
The moon orbits the earth every 27.32 days (sidereal period). Based upon this information and
using Graph 1, what is the distance to the moon? If the distance to the moon was only half of the
distance you have just determined, what would be the corresponding period? (Obviously it is NOT
a linear relation between distance and period - not surprising given Kepler's third law.)
4.
The ideal setup for a communications satellite is to have it in a geo-stationary orbit, that is, to
have its orbital motion synchronized with the rotation of the Earth, so that it remains above the
same location on the Earth continuously. What would be the orbital period of such a satellite?
What would be its orbital radius (semi-major axis)?.
5.
Using Kepler's third law, determine the distance from the sun of an object that has a sidereal
period of 12 years.
6.
If the distance of this object from the sun had been 19 AU, what is it’s period?
AK/NATS 1745.06
7.
Fall 2006
For the moons of Jupiter, we can use their orbital periods and distances from Jupiter to calculate
the mass of Jupiter in terms of Earth’s mass. The periods of the moons must be in years and the
distance to Jupiter in A.U. (1 A.U. = 150,000,000 km) Fill in the numbers you convert on the
table on the answer page. Do the calculation for each moon, but you only have to show one
example of what you did. Is the mass from one of the moons not the same as the others?
AK/NATS 1745.06
Name:
Answer Sheet - Assignment # 2
1. Period in hours of Satellite?
2. Period in hours of shuttle?
3. Distance to moon and change in period?
4. Stationary satellite?
5. Distance of object?
6. Period of object ?
Fall 2006
Student Number:
.
AK/NATS 1745.06
Fall 2006
Name:
Answer Sheet - Assignment # 2
7.
Student Number:
.
Jupiter mass table
Moon
Io
Europa
Ganymede
Callisto
Period
(in days)
1.8
3.6
7.3
14.5
Distance
(in km)
421,600
670,900
1,070,000
1,883,000
Period
(in Years)
Distance
(in A.U. )
Jupiter mass (in Earth masses)
(show sample calculations for each of the 3 columns you have to fill in)