2006: Assoc. Prof. R. J. Reeves Gravitation 3.1 P113 Gravitation: Lecture 3 Escape speed from orbit Planets and satellites: Keplers Laws Orbital energy

2006: Assoc. Prof. R. J. Reeves Gravitation 3.1

P113 Gravitation: Lecture 3• Escape speed from orbit• Planets and satellites: Keplers Laws• Orbital energy


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.


Escape from Earth’s Gravity - 2• Solving for h gives

v (m/s) 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


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


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!


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.



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.



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.

Planet 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


Solar system simulation• http://janus.astro.umd.edu/javadir/orbits/ssv.html


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


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?


Problem

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2006: Assoc. Prof. R. J. Reeves Gravitation 3.1 P113 Gravitation: Lecture 3 Escape speed from orbit Planets and satellites: Keplers Laws Orbital energy