This post shows a way to calculate the approximate mass of the earth. The computation uses Isaac Newton’s second law and his equation for the gravitational force, both published in his book “Principia” in 1687.
The following need to be obtained from elsewhere:
R = average radius of Earth = 6370 km = 6370000 m (get from my previous science posts)
G = universal gravitational constant = 6.673×10^{11} Nm^2/kg^2 (first lab measurement was made by Cavendish in 1798)
g = approximate average gravitational acceleration at the Earth’s surface = 9.8 m/s^2 (can be easily measured in a lab at sea level. A reasonably accurate value of g was already calculated by Cavendish’s day.)
The following will be calculated:
M = mass of Earth
Calculate the approximate mass of the earth:
1. Let {m}_{o} be the mass of an arbitrary object at the Earth’s surface.
2. Set up Newton’s law for the gravitational force between M and mo.
gravitational force = (G*M*{m}_{o})/R^2 = {m}_{o}*g
(Note that {m}_{o} cancels out of both sides of the equation.)
3. Now solve for the mass of Earth, M:
M = (g*R^2) / G
M = (9.8 m/s^2)(6370000 m)^2 / (6.673×10^{−11} Nm^2/kg^2)
M = 5.95911 × 10^{24} kg
Approximate mass of Earth: 5.9591 × 10^{24} kg
Modern value for mass of Earth: 5.9722 × 10^{24} kg
Percent difference: .2 %
Sources of error:
The sources of error in this computation are due to the imprecise values of the universal gravity constant G, the approximate mean gravitational acceleration at sea level, g, and the approximate value for Earth’s average radius, R.
Currently, it is not possible to determine G or the mass of Earth with high precision, even with today’s advanced technology. On the other hand, a reasonably precise value for the product GM is not difficult to obtain. It is determined indirectly during the satellite orbit determination process. In this approach several parameters, including the GM product are varied in order to minimize the difference between the modeled laser ranging data and the measured laser ranging data. One of the parameters varied to reduce the differences is the product GM.
Unfortunately having a precise value for the product GM will not allow us to calculate a precise value for the Earth’s mass, M, since G, the gravitational constant, is not known with high precision. The best value that can be computed with today’s technology for the Earth’s mass, though not of high precision, is 5.9722 × 10^{24} kg.
Source of modern value for mass of 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.”