Why the Moon is Receding
This article proposes a model of the earth moon system not advocated by mainstream scientists. Do not copy it as you may get an 'F' on your term paper as well as be ridiculed by friends.
It is generally believed that the recession of the moon is due to the friction caused by the tidal bulge of the earth being carried forward of the moon by the earth's rotation. The bulge causes the earth's rotation to slow down and the loss energy is transferred to the moon, causing it to speed up and therefore recede.
In this article, the angular momentum transfer of an earth moon closed system is calculated using data from the Time Service Dept., U.S. Naval Observatory and NASA’s Moon Fact Sheet. The study concludes that tidal friction alone cannot explain the current recession rate. However, with an added recession described in the Hubble’s expansion model, the sum more closely matches the observed expansion of 3.8 cm/year.
We first focus on the recession of the moon due to loss in angular momentum of the earth. The amount of angular momentum gained by the orbital moon will have to be offset by an equal amount lost by the Earth.
Let ω be the angular velocity of the earth. According to the Time Service Dept., U.S. Naval Observatory, in the 188 years from 1820 to 2008, the angular rotation of the Earth decreased from ω = 2 π radians per 86,400 second to ω = 2 π radians per 86,400.002 seconds. Let Te be the time for the earth to complete one revolution, we have
dωe = (2 π /Te2)dTe
Substituting data from the U.S. Naval Observatory, we have
dωe ~ -1.683 x 10-12 radians/sec per 188 years or -8.95 x 10-15 radians/sec per year
Earth’s moment of inertia can be derived directly from NASA’s Moon Fact Sheet: I/MR2 = 0.3308, Me = 5.9736 x 1024 kg and Re = 6371 km
Ie = 0.3308 x 5.9736 x 1024 x (6371000) 2 = 8.021 x 10 37 Kg.m2
The change in the earth’s angular momentum (dLe) is the product of moment of inertia (Ie) and the change in angular velocity (dωe).
dLe = Ie dωe
Substituting above values yields
dLe = Ie dωe = 8.021 x 1037 dωe
dLe ~ -7.18 x 1023 Kg.m2/sec per year
As the earth slows down, this change in angular momentum is transferred to the orbital angular momentum (LT) of the moon
dLT = -dLe = 8.021 x 1037 dωe = 7.18 x 1023 Kg.m2/sec per year
Let Mm be the mass of the moon, Vo be the mean orbital velocity and Ro be the distance from the moon to the earth, we have
LT = MmVo Ro
and its derivative is
dLT = MmVo dRo + MmRo dVo
Since orbital velocity can be expressed in terms of gravitational constant (G), orbital radius (Ro) and earth mass (Me)
Vo = (G Me)0.5 Ro-0.5
Its rate of change is
dVo = -0.5 (G Me)0.5 Ro-1.5 dRo
dLT can be rewritten as
dLT = MmVo dRo - 0.5 (G Me)0.5 Ro-1.5 dRo
The equation can be rearranged as
dRo = dLT / ( MmVo – 0.5 (G Me)0.5 Ro-1.5 )
We can get the moon mass, moon's orbital velocity and the earth mass from NASA’s Moon Fact Sheet
Mm = 0.07349 x 1024 kg; Vo = 1023 m/s; Me = 5.9736 x 1024 kg
From NASA's "Measuring the Moon's Distance", the center to center distance of the moon to the earth is Ro = 385,000,000 m
Gravitational constant G = 6.673 × 10-11 m3 kg-1 s-2
Previously we derived dLT = -dLe = 8.021 x 1037 dωe ~ 7.18 x 1023 Kg m-2/sec per year
Substitutions of values to dRo = dLT / ( MmVo – 0.5 (G Me)0.5 Ro-1.5 ) yield
dRm = 2.12 x 1012 dωe
The recession due to tide and angular momentum transfer is approximately
ΔRo = 2.12 x 1012 x 8.95 x 10-15 = 0.019 m /year
According to NASA’s Moon Fact Sheet, the recessional rate of the moon from the earth is 0.038 m/year. As the angular momentum transfer only account for 50% of the recessional rate. There are at least 2 reasons for this under-estimate:
1) I made a mistake and the result is off by a factor of 2. If you see my err, kindly email me at CreationWord (at) live (dot) com.
2) Some other force contributes to the recession. Let's go down this route.
Taking the Hubble constant H = 2.3×10-18 s-1 and the distance from the moon to the earth as 385,000 km, the cosmic recessional velocity dRc as calculated by the expansion model of the cosmos is
dRc = H Ro
And the recessional distance is
ΔRc ~ H Ro Δt
The recession per year (31,556,930 seconds) is
ΔRc ~ 385,000,000 x 2.3×10-18 x 31,556,930 = 0.028 m /year
The sum of recessions due to cosmic expansion and angular momentum transfer is
ΣΔR = ΔRc + ΔRo ~ 0.028 + 0.019 = 0.047 m / year
This value is 24% higher than the observed recessional rate of 0.038 m per year and is closer than that from discounting the expansion effect. Assuming the above measurements and calculations are precise, accurate and correct, a model in which the recession of the moon from the earth is due to both the cosmic expansion and the transfer of the earth’s angular momentum does yield a better fit than one in which only one factor is considered.
Reference to the following sources does not imply their endorsement or agreement with the views and opinions expressed in this article.1 LEAP SECONDS. Time Service Dept., U.S. Naval Observatory, Washington, DC, http://tycho.usno.navy.mil/leapsec.html, 31 December 2008.
2 Dr. David R. Williams. Moon Fact Sheet. NASA Goddard Space Flight Center, http://nssdc.gsfc.nasa.gov/planetary/factsheet/moonfact.html, 02 February 2010.
3 Measuring the Moon's Distance. NASA Goddard Space Flight Center, http://eclipse.gsfc.nasa.gov/SEhelp/ApolloLaser.html, 2005 July 11.
I am not a career astrophysicist. This and other articles on the universe stem from my attempts to reach the Creator via examining the creation. I hope you find something useful as you read them judiciously. Thanks. D. N. Pham.
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