# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: Deneb, Altair and Saturn**

**From:**Paul Hirose

**Date:**2019 May 16, 13:03 -0700

On 2019-05-14 22:46, Antoine Couëtte wrote: > Deneb, Altair and Saturn look approximately lined up and equidistant to northern observers in the morning skys nowadays. > It would be interesting to determine from a geocentric perspective the date and time when they are [almost] exactly lined up and their angular separations then. Geocentric apparent unit vectors (magnitude = 1) in the ICRS at 2019-05-15 0 h UTC: 0.4593056 -0.8747981 0.1541643 Altair 0.4557055 -0.5362104 0.7105004 Deneb (Star catalog data from the Hipparcos re-reduction, van Leeuwen, 2007.) The vector cross product Altair × Deneb = -.5388799 -.2560833 .1523659. This is a vector in the ICRS to the north pole of the great circle that contains the stars, such that the angle east from Altair to Deneb is less than 180°. (If you interchange Altair and Deneb in the cross product, and also in the preceding sentence, it's still true.) The vector magnitude (about .62) is equal to the sine of the separation angle between the stars. However, what's really needed is a unit vector, so divide the cross product by its magnitude to obtain -.8751169 -.4158679 .2474354, an ICRS unit vector to the north pole of the new coordinate system. In rectangular coordinates this is a vector on the +Z axis. Let Altair be the zero point on the pseudo equator. In rectangular coordinates the vector to Altair is on the +X axis, and the cross product (pole × Altair) is a unit vector on the +Y axis: .1523440 .2485602 .9565611. Again, these are the ICRS directions of the axes of the new coordinate system. Now construct a rotation matrix to convert coordinates from the ICRS to the new system. The rows of a rotation matrix are the basis vectors (unit vectors on the X, Y, and Z axes) of the new coordinate system in terms of the old. Those vectors have already been computed: [.4593056 -.8747981 .1541643] X axis [.1523440 .2485602 .9565611] Y axis [-.8751169 -.4158679 .2474354] Z axis Call this rotation matrix R. Lunar 4.4 says the unit vector to Saturn at 2019-05-15 0 h UTC is 0.3432012 -0.8643920 -0.3674770. (JPL DE431 ephemeris) Call this vector S. To convert S to the new coordinate system, multiply by rotation matrix R: R * S = .8571509 -.5140830 -.0317951 The negative Z coordinate means Saturn is a little "south" of the "equator". However, the search for the time when Z is zero requires repetition of the entire computation, since the "north pole" moves a quarter second of arc in one day. (This is almost entirely due to constantly varying effect of aberration on the apparent places of the two stars.) So, I wrote a small custom program which performs the above steps for any specified time. I get 2019 June 24 02:43:25 UTC. That's accurate to about a tenth second of arc. CHECK WITH USNO MICA 2.2.2 Add 37 s to convert UTC to TAI, and 32.184 s to convert TAI to TT. Result is 2019 June 24 02:44:34 TT. At that time, MICA gives RA and declination: 19.8623061 h +08.920707° Altair 20.7018785 h +45.347788° Deneb 19.3212096 h -21.843745° Saturn Convert those coordinates to vectors in rectangular form. Create a rotation matrix as before and transform the vectors to a new system in which the stars are on the equator and Altair is the longitude origin. Convert rectangular to spherical. All three bodies are on the "equator": +00.00000° +00.00000° Altair +38.00945° +00.00000° Deneb -31.77830° +00.00001° Saturn Geocentric apparent distances from Altair, per Lunar 4.4, agree with the "longitude" angles above. 38.00946° Deneb 31.77829° Saturn