This post will show a way to calculate the approximate distance between the earth and the moon. This value can be determined from knowing the time period of the Moon’s orbit around the Earth and assuming the Moon’s orbit is circular. The computation here uses the equation for the gravitational force developed by Isaac Newton and published in his book “Principia” in 1687.
The following need to be obtained from elsewhere:
M = Mass of the Earth = 5.96 × 10^{24} kg (from my previous science posts)
G = universal gravitational constant = 6.673×10^{−11} Nm^2/kg^2 (first lab measurement by Cavendish in 1798)
T = average period of the Moon’s orbit = 27 days = 2332800 seconds (full cycle of Moon phases)
The following will be calculated:
R = the Earth-Moon distance
Calculate the approximate distance between the Earth and the Moon:
1. For the Moon to be in the circular orbit, the gravitational force between the Earth and the Moon is equal to the centripetal force of the Moon about the Earth:
Let m be the mass of the Moon.
(G*M*m)/R^2 = (m*v^2)/R
(Note that m cancels out of both sides of the equation.)
2. Now solve for R:
R = G*M/v^2
3. If T is the period of the orbit of the Moon, then:
v = 2*π*R/T
4. Now insert the expression for v into the equation for R:
R = G*M/(2*π*R/T)^2
R = \sqrt[3]{( G*M*T^2 / (4*π^2) )}
R =\sqrt[3]{ (6.673×10^{−11} Nm^2/kg^2) * (5.96 × 10^{24} kg) * (2332800)^2 / (4 * 3.142^2)}
Approximate distance between the Earth and the Moon: R = 379800 km
Modern value for average distance from the Earth to the Moon: 384403 km
The percent difference is: 1.2%
Source of errors:
The sources of error in this approximation are the uncertainty in the values of the universal gravitational constant, the mass of the Earth, and the Moon’s orbital period. Note also that the Moon’s orbit is elliptical so that the actual Earth-Moon distance varies from 356,500 km at the perigee to 406,700 km at apogee. The average Earth-Moon distance is 384403 km.
Source of modern value for distance between the earth and the moon:
The modern value for the Earth-Moon distance is determined from laser ranging. This involves firing a laser at a mirror that astronauts had placed on the Moon and calculating the Earth-Moon distance from the round trip time it takes for the laser light to return to the Earth after reflecting off the mirror on the Moon. This is repeated several times during the Moon’s orbital period to get the full range of Earth-Moon distances that define the Moon’s elliptical orbit about the Earth.
The modern values are from the NASA Fact Sheets, located at:
https://nssdc.gsfc.nasa.gov/planetary/factsheet/
The notes on the NASA Fact Sheets state the following:
“Note that the values listed on the fact sheets are not “official” values, there is no single set of agreed upon values. They are based on ongoing research and as such are under study and subject to change at any time.
Every effort has been made to present the most up-to-date information, but care should be exercised when using these values.”