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I Inconsistencies re Astronomical constants

  1. Feb 22, 2016 #1
    I have been working on some calculations about the history of our Moon's orbit, and I have found an inconsistency with respect published value for the following four constants:

    G = Universal Gravitational Constant
    M = Earth's Mass
    R = Moon's orbit semi-major axis
    P = Moon's orbit sidereal period

    From https://en.wikipedia.org/wiki/Standard_gravitational_parameter
    and https://en.wikipedia.org/wiki/Elliptic_orbit
    There are two different ways to calculate a value for μ.
    μ = GM = 4 π2 R3 / T2

    From http://arxiv.org/pdf/1507.07956v1.pdf
    and http://physics.nist.gov/cgi-bin/cuu/Value?bg|search_for=gravity
    G = 6.67408 × 10-11 m3 / s2 kg​
    From http://asa.usno.navy.mil/static/files/2016/Astronomical_Constants_2016.pdf [Broken]
    M = 5.9722 × 1024 kg​
    From http://nssdc.gsfc.nasa.gov/planetary/factsheet/moonfact.html
    R = 384,400,000 m​
    P = 655.728 hrs = 2,360,620.8 s​

    The calculations:
    GM = 3.985894 × 1024 m3 / s2
    4 π2 R3 / T2 = 4.023995 × 1024 m3 / s2

    The difference between these two calculations shows an approximate error percentage of 1%. Does anyone have any explanation about why this error percentage is so large given the precision of the published values?

    Can someone suggestion about how I can find relatively recent published values which will result in a better match between the two different calculations of μ?

    I just had the thought that I should adjust the value for R to be the average of the distances of the moon at apogee and perigee from the center of mass of the Earth and Moon.
    Last edited by a moderator: May 7, 2017
  2. jcsd
  3. Feb 22, 2016 #2


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  4. Feb 22, 2016 #3
    Hi Russ:

    Thanks for the post.

    I assume you mean the first diagram, which I confess I have difficulty interpreting.

    I notice that the figure gives a value of a 384,405 km as some unclear kind of distance, and below gives 344,400 km as the Inverse sine parallax which matches the value I found on the NASA fact sheet for the Moon. Note [5] says
    This often quoted value for the mean distance is actually the inverse of the mean of the inverse of the distance, which is not the same as the mean distance itself.​
    So I am guessing that this is not the value I want to use. The figure also gives 384,748 km as the semi-major axis, so i will try that and see what happens.


    I plugged in the 384,748 km value and the percent error got a little worse. So I don't think that is the right approach either.
  5. Feb 22, 2016 #4


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    But have you calculated with a reduction of radius - barycenter = (6378 - 4641) km = 1737 km on the orbit R? I thought that are roughly ##(10^4*10^3)^3##, i.e. 21 digits which roughly matches the error margin?
  6. Feb 22, 2016 #5


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    You need to use (M(Earth) + M(Moon)) for the mass of the system. The Moon's mass is not negligible compared to the Earth. If you do that, using your numbers, I get

    G(MEarth + MMoon) =

    4π^2r^3/T^2 =

    This is better than 0.01%.
  7. Feb 22, 2016 #6


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    A couple of other facts to consider. We don't know neither G nor the mass of the Earth and Moon as accurately as we know GM for either

    For example, by using your values for G and the mass of the Earth, you got GM = 3.985894 × 10^14 m^3 / s^2, but the more accurate value for GM(Earth) is 3.986004418×10^14 m^3/s^2

    For the Moon, GM(Moon)=4.9048695×10^12 m^/s^2
    The gravitational parameter (GM) for a body is sometimes designated as [itex]\mu[/itex]

    The orbital period would be found by:

    [tex]T= 2 \pi \sqrt{\frac{a^3}{\mu_{E}+\mu_{M}}}[/tex]

    using the semi-major axis of 384,748 km from above, this an answer of 27.3215953252 days, vs. the listed value of 27.321661 days listed as the sidereal period for the Moon. This is an error of only ~0.00024%

    In addition, the sidereal period is in reference to the stars and the calculated period assumes that the semi-major axis of the orbit does not move relative to the stars. (in other words that after one sidereal period, the Moon returns to the exact same point in its orbit relative to the perigee of the orbit.) But for the Moon, this is not the case, there is a precession of the perigee. When the Moon returns to the same point relative to the stars, it will not have returned to the same point relative to the perigee and this is going to cause a small difference between the calculated and actual sidereal period.
    Last edited: Feb 23, 2016
  8. Feb 23, 2016 #7


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  9. Feb 28, 2016 #8


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    The authors omitted some measurements of G, and used the time of publication instead of the time of measurement for their data. There is a response discussing those issues. While a variation still leads to a slightly better fit, the result is not as significant as it looked like.

    A change in G of the size discussed in those papers would have a massive influence on the orbit of Moon, lunar laser ranging would have detected it long ago. Even the planetary orbits would be affected in a notable way. Unless you make the model more complicated to limit the effect to local measurements of G, which makes the whole thing even more obscure.
  10. Feb 28, 2016 #9
    Hi @SteamKing and @mfb:

    I confess I am confused by the Schlamminger, Gundlach, and Newman paper
    I get that they are criticizing the Phys Org fit of measured values of G to a sinusoid with a 5.9 year period,
    but I could find no preferred value of G in their paper.

    Wikipedia gives the value G = 6.674×10−11,
    and Phys Org gives G = 6.673889 × 10−11.

    I would like to use the most generally accepted precise value available for some calculations I am making. What would you advise?

  11. Feb 28, 2016 #10


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    The NIST has published a brief article on their web site which includes the "official" values of G as recommended by CODATA:

  12. Feb 28, 2016 #11

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    To say the same thing in a different way, G is known to 100 ppm. GM, on the other hand, where M is the mass of the earth, is known 500x better: to 0.2 ppm. This is 1000x smaller than the effect Anderson reports, so it would require that the Earth gain and lose a few quadrillion tons every year to keep GM constant. If you instead consider GM for the sun, that is known even better - 400 parts per trillion.
  13. Feb 28, 2016 #12
    Hi SteamKing:

    Thanks for your advice. I will use the CODATA value G = 6.67408 x 10-11.

  14. Mar 2, 2016 #13


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    Have you accounted for the margin of error in each of the measurements?
  15. Mar 2, 2016 #14
    Hi ohwilleke:

    Thanks for the reminder. I will be careful to calculate the error range for all the derived values I am working on while using different methods so I can be sure what I choose as my final answers will be have the smallest error ranges I can get.

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