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## The Orbital Velocities of the Planets

People are fascinated with high-speed travel. A Boeing 747 flies at about 550 miles per hour. The Concorde Supersonic Transport flew over twice as fast at about 1350 miles per hour. That is impressively fast, but consider the speeds that we passengers on planet Earth always travel. The circumference of the Earth at the equator is 24900 miles and the residents of the tropic areas travel that distance every 24 hours. They are traveling over a thousand miles an hour. At 38° latitude we are traveling about 820 miles per hour. That is quite impressive but it pales in comparison with the orbital speed of the Earth, The average distance of the Earth from sun is 93.5 million miles. This means that in 365.25 days the Earth travels 587.5 million miles. This works out to about 67 thousand miles per hour. This is the speed we travel 24 hours a day, 365.25 days a year.

The orbital velocity is 2πR/T where R is the average radius of the orbit and T is the length of the year. The orbital velocity of a planet relative to that of Earth's is then the relative radius divided by the relative length of the year.

The relative distances, lengths of the years and orbital velocities of the various planets are as follows:

Planet
Radius of Orbit
Relative to
that of Earth's
Length of Year
Relative to
Earth's Year
Orbital Velocity
Relative to
That of Earth's
Mercury0.3870.24091.607
Venus0.7230.6161.174
Earth1.01.01.000
Mars1.5241.90.802
Jupiter5.20312.00.434
Saturn9.53929.50.323
Uranus19.18840.228
Neptune30.061650.182
Pluto39.522480.159

Thus Mercury's orbital speed is 1.607(67,000)=107.7 thousand miles per hour, as befits a planet named for the god of speed. Mars is a bit of a laggard. Its speed is only 0.802(67,000)=53.7 thousand miles per hour. Pluto is veritably creeping around its orbit at only 10.7 thousand miles per hour.

There is an interesting computation which can be performed using the above figures. Let us look at the products of the square roots of the relative radii and the orbital velocities. The computations are given below:

PlanetR1/2 VV×R1/2
Mercury0.6221.6071.000
Venus0.8501.1740.998
Earth1.01.01.000
Mars1.2350.8020.991
Jupiter2.2810.4340.990
Saturn3.0890.3230.998
Uranus4.3800.2280.999
Neptune5.4830.1820.998
Pluto6.2870.1591.000

The results indicate that the product of the relative orbital velocity and the square root of the relatative orbit radius is close to unity. If all of the numbers were precisely correct the product would all have been exactly 1.0. Thus the relative orbital speed is given by:

#### V = 1/R½

This is just Kepler's Law in a different form. Kepler's Law is that for each planet the square of the length of its year is equal to the cube of the radius of its orbit. From Kepler's Law if you know how far a planet is from the sun you can tell how long it would take for that planet to go around the sun.

Thus in relative terms

#### T2 = R3so T = R3/2and hence V = R/T = R/R3/2 = 1.0/R1/2

Thus for a planet twice as far from the sun as Earth the orbital velocity relative to that of Earth's is 1/√2 = 0.707.

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