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    Re: Upcoming Planets 2023 close encounters
    From: Paul Hirose
    Date: 2023 Feb 27, 16:43 -0800

    On 1/12/2023 11:00 PM, Antoine Couëtte wrote:
    > Lady Venus back in the evening sky.
    > She will "fly by" Saturn at a minimum geocentric distance close from
    > 20.7 ' (slighly above a Moon apparent Semi-Diameter) on Jan 22nd, 2023
    > by 22h14m UT.
    I tried to solve this as relative motion problem, similar to "closest
    point of approach" on a maneuvering board.
    geocentric apparent ICRS positions at 1-day interval:
    21h47m55.520s -14°36'33.2" 2023-01-21 00:00:00 Saturn
    21h39m52.073s -15°39'28.3" 2023-01-21 00:00:00 Venus
    21h48m22.253s -14°34'16.8" 2023-01-22 00:00:00 Saturn
    21h44m45.273s -15°14'50.1" 2023-01-22 00:00:00 Venus
    21h48m49.100s -14°31'59.7" 2023-01-23 00:00:00 Saturn
    21h49m37.200s -14°49'47.2" 2023-01-23 00:00:00 Venus
    Distance and direction (position angle) from Saturn to Venus:
    2°12'33.1" 241°24'11.1" 2023-01-21 00:00
    1°06'16.6" 232°09'23.0" 2023-01-22 00:00
       21'15.4" 146°50'55.8" 2023-01-23 00:00
    Re-orient the spherical coordinate frame so the first and third points
    are on the equator and the second is on the prime meridian. In this new
    frame, Saturn's coordinates are constant but nonzero. Latitudes and
    +0°00'00.0"  -1°07'58.6" 2023-01-21 00:00:00 Venus
    -0°00'02.9"  +0°00'00.0" 2023-01-22 00:00:00 Venus
    +0°20'40.1"  +1°02'57.3" Saturn
    +0°00'00.0"  +1°07'55.5" 2023-01-23 00:00:00 Venus
    Note the first and third Venus positions have nearly symmetrical
    longitudes about the prime meridian, and the middle Venus position is
    only about 3" off the equator. That shows the relative motion of Venus
    with respect to Saturn is closely approximated by constant velocity on a
    great circle.
    Time at closest approach can be deduced from longitude. An improved
    estimate is Jan 22.0, plus 1°03' / 1°08' of a day, or about Jan 22 22:15
    TT. Two more iterations, and reduction of the time interval to 60
    minutes, gives minimum separation at 22:15:12:
    +0°00'00.0"   -0°02'49.8" 2023-01-22 21:15:12 Venus
    +0°20'40.5"   -0°00'00.0" Saturn
    -0°00'00.0"   -0°00'00.0" 2023-01-22 22:15:12 Venus
    -0°00'00.0"   +0°02'49.8" 2023-01-22 23:15:12 Venus
    Saturn's zero longitude indicates a perfect solution at this precision.
    In this case the relative motion of Venus is exactly on a great circle
    (middle latitude = 0), so Saturn's latitude is also the distance of
    closest approach. But that's not true at higher precision, or if
    relative motion departs from a great circle.
    A re-computation with the old DE406 ephemeris gives an identical time.
    This encounter could have been accurately computed 25 years ago.
    > She will "fly by" Jupiter at a minimum geocentric distance close from
    > 29.3' (about a Moon apparent angular Diameter) on March 02nd, 2023 by
    > 05h05m UT.
    I get 29'21.0" at 05:06:28 TT. On the HASTRO-L mailing list someone got
    0.48918° (29' 21.05") at 05:06:27 from the SOLEX program.
    Jean Meeus (Astronomical Algorithms, 1991) gives the example of Mercury
    and Saturn at 1978 Sept 13. My solution with the DE431 ephemeris quickly
    converges to 3'44.8" at 15:06:35 TT, compared to 3'44" at 15:06 UT by Meeus.
    However, results are not so good with the Venus / Mars encounter this
    year on July 1. My solution with the above method does not converge. I
    had to use trial and error to get 3°33'54.342" separation at 07:10:59
    TT. My difficulties are probably due to a combination of slow relative
    motion which does not fit a great circle well, plus large separation.
    Paul Hirose

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