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Moon-Earth distance

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Aristarchus went a step further and computed the distance from earth, together with its size, obtaining a value of 20 earth radius for the distance (the real value is 60. The earth radius was known since Eratosthenes)

Symbol Meaning
s Radius of the Sun
S Distance to the Sun
l Radius of the Moon
L Distance to the Moon
t Radius of the Earth
D Distance to the vertex of Earth's shadow cone
n d/l, a directly observable quantity during a lunar eclipse
\phi Directly observed.
x S/L, which is derived from \phi

His values, then, are computed as:

Quantity Formula Value Actual value
x n/a ~19 390
n n/a 2 2.587
\theta n/a 1 0.259
s/t (1+x)/(1+n) 6.67 109
t/\ell x (1+n)/(1+x) 2.85 3.67
L/t (\ell/t)(180/\pi\theta) 20 60.32
S/t (L/t)(S/L) 380 23,500

The error in this calculation comes primarily from the poor values for x and \theta. The poor value for \theta is especially surprising, since Archimedes writes that Aristarchus was the first to determine that the Sun and Moon had an apparent diameter of half a degree. This would give a value of \theta=0.25, and a corresponding distance to the moon of 80 earth radii, a much better estimate.

A similar procedure was later used by Hipparchus, who estimated the mean distance to the moon as 67 earth radii, and Ptolemy, who took 59 earth radii for this value.


Lunar Laser Ranging experimentEdit

The ongoing Lunar Laser Ranging Experiment measures the distance between the Earth and the Moon using laser ranging. Lasers on Earth are aimed at retroreflectors previously planted on the Moon and the time delay for the reflected light to return is determined. Since the speed of light is known with very high accuracy, the distance to the moon can be calculated. This distance has been measured with increasing accuracy for more than 35 years.

The distance continually changes for a number of reasons, but averages about 384,467 kilometers (238,897 miles). The time delay in the reflected light is about 2½ seconds.

The experiment was first made possible by a retroreflector array installed on July 21, 1969, by the crew of Apollo 11. Two more retroreflector arrays left by the Apollo 14 and Apollo 15 missions have contributed to the experiment.

The unmanned Soviet Lunokhod 1 and Lunokhod 2 rovers carried smaller arrays. Reflected signals were initially received from Lunokhod 1, but no return signals have been detected since 1971, at least in part due to some uncertainty in its location on the Moon. Lunokhod 2's array continues to return signals to Earth.[1]


File:AS15-85-11468.jpg

The Apollo 15 array is three times the size of the arrays left by the two earlier Apollo missions. Its size made it the target of three-quarters of the sample measurements taken in the first 25 years of the experiment. Improvements in technology since then have resulted in greater use of the smaller arrays, by sites such as the Côte d'Azur Observatory in Grasse, France, and the Apache Point Observatory in New Mexico. The first measurements were made by the McDonald Observatory in Texas, although lunar laser ranging at this site stopped in 2009.[2]

At the Moon's surface, the beam is only about 6.5 kilometers (four miles) wide[3] and scientists liken the task of aiming the beam to using a rifle to hit a moving dime 3 kilometers (two miles) away. The reflected light is too weak to be seen with the human eye, but under good conditions, one photon will be received every few seconds (they can be identified as originating from the laser because the laser is highly monochromatic). This is one of the most precise distance measurements ever made, and is equivalent in accuracy to determining the distance between Los Angeles and New York to one hundredth of an inch.[4] As of 2002 work is progressing on increasing the accuracy of the Earth-Moon measurements to near millimeter accuracy.

File:AS14-67-9386.jpg

Some of the findings of this long-term experiment are:

  • The moon is spiralling away from Earth at a rate of 38 mm per year.[3]
  • The moon probably has a liquid core of about 20% of the Moon's radius.[1]
  • The universal force of gravity is very stable. The experiments have put an upper limit on the change in Newton's gravitational constant G of less than 1 part in 1011 since 1969.[1]
  • The likelihood of any "Nordtvedt effect" (a composition-dependent differential acceleration of the Moon and Earth towards the Sun) has been ruled out to high precision,[5][6] strongly supporting the validity of the Strong Equivalence Principle.
LunarPhotons

APOLLO Collaboration photon pulse return times

Additionally, the accuracy of these experiments has improved historic knowledge of the Moon's orbit enough to permit timing of solar eclipses up to 3,400 years ago.[1]

The presence of reflectors on the Moon has been used to rebut claims that the Apollo landings were faked.

For example, the APOLLO Collaboration photon pulse return graph, shown here, has a pattern consistent with a retroreflector array near a known landing site.

Planet Hypothesis of the MoonEdit

From Angular mometum conservation,

 
I_m \omega_m sin \theta_m = I_e  \omega_e sin \theta_e

If we assume equivalent density for moon and earth,

R_m/R_e =2.626.

From angular momentun conservation, tht size and distance are 9.549 times larger than those of conventional solar system.

ReferencesEdit

  1. 1.0 1.1 1.2 1.3 1.4 James G. Williams and Jean O. Dickey. "Lunar Geophysics, Geodesy, and Dynamics" (PDF). ilrs.gsfc.nasa.gov. Retrieved on 2008-05-04. 13th International Workshop on Laser Ranging, October 7-11, 2002, Washington, D. C.
  2. McKie, Robin (June 21, 2009), "After 40 years' reflection, laser moon mirror project is axed", The Guardian, http://www.guardian.co.uk/technology/2009/jun/21/mcdonald-observatory-space-laser-funding .
  3. 3.0 3.1 Fred Espenak (August 1994). "NASA - Accuracy of Eclipse Predictions". eclipse.gsfc.nasa.gov. Retrieved on 2008-05-04.
  4. "Apollo 11 Experiment Still Going Strong after 35 Years". www.jpl.nasa.gov (July 20, 2004). Retrieved on 2008-05-04.
  5. Adelberger, E.G., Heckel, B.R., Smith, G., Su, Y., and Swanson, H.E. (1990-Sep-20), "Eötvös experiments, lunar ranging and the strong equivalence principle", Nature 347: 261-263, http://www.nature.com/nature/journal/v347/n6290/abs/347261a0.html 
  6. Williams, J.G., Newhall, X.X., and Dickey, J.O. (1996), "Relativity parameters determined from lunar laser ranging", Phys. Rev. D 53: 6730-6739, http://prola.aps.org/abstract/PRD/v53/p6730_1 

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