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How could I determine the distance from the Earth to the moon by using the method of simultaneous parallax? I know the right ascension and declination as seen from two different points on earth. I also know the right ascension and declination of a distant star.

I have location 1 in Buenos Aires and location 2 in Shanghai, with both observing the moon at the same time. The moon has a different right ascension and declination in each location.

Knowing the right ascension and declination of the star Sirius, how might I calculate the distance to the moon? None of the equations I have tried gives me the correct answer.

  • Moon Right Ascension from Buenos Aires: 0.0363099165337 Radians
  • Moon Declination from Buenos Aires: 0.220376340654 Radians
  • Moon Right ascension from Shanghai: 0.0366537040004 Radians
  • Moon Declination from Shanghai: 0.197477937143 Radians
  • Sirius Right Ascension (same from the entire globe): 1.76780036611 Radians
  • Sirius Declination (same from the entire globe): -0.291750983093 Radians
James K
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Luis Sarmiento
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    You need the precise time of the observation also. The location of sirius isn't needed. The basic idea is: compute the geocentric xyz coordinates of the observers; rotate those coordinates based on the sidereal time of the observations; represent the RA/Dec from each site as lines, and compute the intersection of the two lines. – Greg Miller Feb 02 '23 at 02:35
  • related: Using parallax to find distance to the moon and Determine distance to Moon after diurnal parallax? and How to solve numerical problems based on diurnal parallax method (formulas, examples and calculations)? – uhoh Feb 03 '23 at 21:48
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    Welcome to Astronomy SE! "None of the equations I have tried gives me the correct answer." Please show or at least link to all of the equations you have tried. It is possible you have chosen the wrong ones, or the right ones but used them incorrectly. The more information about what you have tried so far, the better an answer can address your situation. Thanks! – uhoh Feb 03 '23 at 21:52
  • BTW, The location of the Moon makes little sense. on RA close to zero (in this case if I convert to degrees is less than 2 deg); we would expect the declination to be somewhere between -3 and +7 deg. Hence Moon declination of ~0.2Rad is way way too high. – d_e Feb 03 '23 at 22:59
  • @d_e, I agree that the location of the Moon seems off. I wrote a little Python script to try and answer this question and it returned the wrong value for the distance of the Moon, despite working for the current RA/dec values for Shanghai and Buenos Aires. – Roy Smart Feb 04 '23 at 00:12
  • What time were those observations made? – PM 2Ring Feb 04 '23 at 22:55
  • d_e thank you for your response. It is possible I got the wrong values for the RA and DEC. I got these values from the moon being observed around a month ago, so it is not the current position of the Moon. Nevertheless, where would you recommend I get information? I sadly do not own a telescope so an online database would be truly helpful. Thank you. – Luis Sarmiento Feb 04 '23 at 23:53
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    i calculated the angular separation using that from https://www.skythisweek.info/angsep.pdf – Luis Sarmiento Feb 04 '23 at 23:55
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    You can get data for Solar System bodies (including some spacecraft) from JPL Horizons – PM 2Ring Feb 04 '23 at 23:59
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    (Edit) I calculated the angular separation from the moon to the star from both locations using the equation from https://www.skythisweek.info/angsep.pdf I then subtracted one from the other to get the parallax angle. As shown in https://www.mccarthyobservatory.org/pdfs/pm020102.pdf Page 4. I then used equation

    Distance to moon = (Distance between two points/ (2*sin(parallax angle/2))

     – Luis Sarmiento Feb 05 '23 at 00:04

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