## Double Star Rho and Theta Measurements David Haworth and Tom Frey

Astronmetry is used to measure the RA and Dec of each the double stars in the image.

Next the distance and position angle is calculated from each star's position which is described by RA and Dec.

Below are formulas for calculating the the double star separation rho and position angle theta from RA and Dec. Formula from J. Smolinski and W. Osborn

Formula for calculating the double star separation rho and position angle theta from RA and Dec.
Reference: 2006RMxAC..25...65S

• RA1 = Primary star RA in radians
• RA2 = Secondary star RA in radians
• Dec1 = Primary star Dec in radians
• Dec2 = Secondary star Dec in radians
• rho = arccos[cos((RA2-RA1)cos(Dec1)cos(Dec2-Dec1)]
• theta = 90 degrees - arctan[(sin(Dec2-Dec1)/(cos(Dec2-Dec1)*sin((RA2-RA1)*cos(Dec1)))] Formula from Robert K. Buchheim

Formula for calculating the double star separation rho and position angle theta from RA and Dec.
Reference: CCD Double-Star Measurements at Altimira Observatory in 2007 Journal of Double Star Observations, Vol. 4 No. 1 Winter 2008 by Robert K. Buchheim

• Small angle approximation for |RA2-RA1|<<1 and |Dec2-Dec1|<<1
• RA1 = Primary star RA in radians
• RA2 = Secondary star RA in radians
• Dec1 = Primary star Dec in radians
• Dec2 = Secondary star Dec in radians
• rho = square root( ((RA2-RA1)cos(Dec1))^2 + (Dec2-Dec1)^2 )
• theta = (pi/2) - arctan((Dec2-Dec1)/((RA2-RA1)cos(Dec1))
• theta = arctan((Dec2-Dec1)cos(Dec1)/(Dec2-Dec1))
• Theta must be adjusted for the correct quadrant for the position angle starting at north and going to the east. Formula from Michael Greaney

Formula for calculating the double star separation rho and position angle theta from RA and Dec.
Reference:
Michael Greaney, Chapter 25, Observing and Measuring Visual Double Stars (The Patrick Moore Practical Astronomy Series) 2nd Edition R. W. Argyle

• RA1 = Primary star RA in radians
• RA2 = Secondary star RA in radians
• Dec1 = Primary star Dec in radians
• Dec2 = Secondary star Dec in radians
• cos(rho) = sin(Dec1)sin(Dec2)+cos(Dec1)cos(Dec2)cos(RA2-RA1)
• tan(theta) = (cos(Dec1)cos(Dec2)sin(RA2-RA1))/(sin(Dec2)-cos(pho)sin(Dec1)) Links  References  Thomas G. Frey Thomas G. Frey is Professor Emeritus of Chemistry at California Polytechnic State University, San Luis Obispo, CA. He has been an active member of the Central Coast Astronomical Society for over 25 years. He was a team leader at the Pine Mountain Observatory (PMO) Summer Science Research Work- shop, Bend, OR, 2009 and co-director of the PMO workshop, summer of 2011.  David Haworth enjoys astronomy imaging and processing those images to bring out details that cannot be seen easily by visual observing with the same size optics. David Haworth started astroimaging with a Cookbook CCD camera he built in 1996 and since then has used many types of cameras to image the sky. David wrote Chapter 2: "Afocal Photography with Digital Cameras" in the second edition of "The Art and Science of CCD Astronomy" which was published in December 2005. David's images have appeared in magazine front covers, articles, books, catalogs, videos, music CD covers, T-shirts, other web sites, etc. 