P113 Gravitation: Lecture 3
• Escape speed from orbit
• Planets and satellites: Keplers Laws
• Orbital energy
2006: Assoc. Prof. R. J. Reeves
Gravitation 3.1
Escape from Earth’s Gravity
• We know from everyday experience by throwing stones that they
go higher up if we throw them with faster velocity. Can we
derive an equation that will give the height as a function of
velocity?
• A mass m starts at the surface of the earth with vertical velocity v.
It reaches an altitude h where it turns around and falls down.
Conservation of energy gives:
• Exercise: Complete the algebra to derive the following equation
for h.
2006: Assoc. Prof. R. J. Reeves
Gravitation 3.2
Escape from Earth’s Gravity - 2
• Solving for h gives
v (m/s)
2006: Assoc. Prof. R. J. Reeves
Altitude h (m)
1
0.06
10
5.6
100
560
1,000
56,000
10,000
3.4 x 107
100,000
– 6.7 x 106
Gravitation 3.3
Escape from Earth’s Gravity - 3
• The negative altitude reached when the velocity was 100,000 m/s
indicates the mass becomes unbound from the earth’s
gravitational field.
• The escape velocity from the earth’s surface is given by
Body
Mass (kg)
Radius (m)
Escape speed
(km/s)
Earth
5.98 x 1024
6.37 x 106
11.2
Earth’s moon
7.36 x 1022
1.74 x 106
2.38
Jupiter
1.90 x 1027
7.15 x 107
59.5
Sun
1.99 x 1030
6.96 x 108
618
2006: Assoc. Prof. R. J. Reeves
Gravitation 3.4
Escape from Earth’s Gravity - 4
• Escape from earth requires 11.2 km/s = 40,000 km/hr! How do
rockets manage this incredible speed?
• 11.2 km/s is the required speed at the earth’s surface without any
additional thrust being applied.
• We could escape from earth at 40 km/hr providing we applied
thrust greater than our weight all the way to the moon!
• Rockets achieve a balance:
– They start out less than 11.2 km/s but continue to apply thrust.
– The necessary escape speed decreases as the altitude
increases.
– Eventually the rocket is travelling faster than the escape speed
for its present altitude. The rocket motors can be turned off
and then it’s glide to the moon!
2006: Assoc. Prof. R. J. Reeves
Gravitation 3.5
Kepler’s Laws applied to the Planets - 1
1. THE LAW OF ORBITS: All
planets move in elliptical
orbits with the Sun at one
focus.
•
•
•
The orbit is characterised by
its semimajor axis a and its
eccentricity e.
Rp is the perihelion distance.
Ra is the aphelion distance.
2006: Assoc. Prof. R. J. Reeves
Gravitation 3.6
Kepler’s Laws applied to the Planets - 2
2. THE LAW OF AREAS: A line that connects the planet to the Sun
sweeps out equal areas in then plane of the planets orbit in equal
time intervals.
• Kepler’s second law is a statement of conservation of angular
momentum.
2006: Assoc. Prof. R. J. Reeves
Gravitation 3.7
Kepler’s Laws applied to the Planets - 3
3. THE LAW OF PERIODS: The square of the period of any planet
is proportional to the cube of the semimajor axis of its orbit.
a (1010 m)
e
T (yrs)
T2/a3
Mercury
5.79
0.2056
0.241
2.99
Venus
10.8
0.0068
0.615
3.00
Earth
15.0
0.0167
1.00
2.96
Mars
22.8
0.0934
1.88
2.98
Jupiter
77.8
0.0483
11.9
3.01
Saturn
143
0.0560
29.6
2.98
Uranus
287
0.0461
84.0
2.98
Neptune
450
0.0097
165
2.99
Pluto
590
0.2482
248
2.99
Planet
2006: Assoc. Prof. R. J. Reeves
Gravitation 3.8
Solar system simulation
• http://janus.astro.umd.edu/javadir/orbits/ssv.html
2006: Assoc. Prof. R. J. Reeves
Gravitation 3.9
Orbital Energy - 1
• Kepler’s second law stipulates that the speed of a planet in orbit is
faster when closer to the sun. Correspondingly, the gravitational
potential energy is smaller (more negative) when closer.
• Assume an exact circular orbit. Newton’s second law gives
• The kinetic energy is then
• The total energy of the orbit is
2006: Assoc. Prof. R. J. Reeves
Gravitation 3.10
Orbital Energy - 2
• If the orbit is elliptical, the total energy is
• The negative value for the total energy is a general indication of a
bound system.
• Question: Can you think of another bound system with a negative
energy?
2006: Assoc. Prof. R. J. Reeves
Gravitation 3.11
Problem
2006: Assoc. Prof. R. J. Reeves
Gravitation 3.12