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    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
    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.
    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
    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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