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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

   

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