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Re: Jupiter-Venus angular separation
From: Antoine Couëtte
Date: 2022 May 6, 02:51 -0700

(1) - Geocentric approximation:

1.1 - 09:00 Geocentric Center to Center distance 1°31.1'

1.2 - 11:00 Geocentric Center to Center distance 1°35.2'

from which we derive Geometric Center to Center distance at 08:43 to be quite close from 1°30.5' (which is our Topocentric Target value).

Hence let's use 08:43 as a first - and probably rough - approximation for UT estimate, and see what actual sextant topocentric distance we get between Venus and Jupiter.

(2) - 1st UT Estimate with UT = 08:43 and topocentric approximation

From Atria and Venus published heights (15°07' and 27°52') considered to be measured both at 08:43, get S00°26.0' / W053°00.2' from which we compute (reverse engineering):

Venus observed Height 27°52.0' / Azimut 091.351°  and Jupiter observed Height 29°14.7' / Azimut 091.673°, i.e a sextant distance at 1°24.4' (includes everything : refraction, parallax, ...).

Too low since our target sextant distance is 1°35.0'

(3) - 2nd UT Estimate and topocentric approximation

Assuming that the topocentric rate of change between Venus and Jupiter Distances is equal to its geocentric rate of change, let's try everything again through assuming that UT = 11:30

From Atria and Venus published heights (15°07' and 27°52') considered to be measured both at 11:30, get S00°28.0' / W094°44.7' from which we compute (reverse engineering):

Venus observed Height 27°52.0' / Azimut 091.279°  and Jupiter observed Height 29°20.8' / Azimut 091.645°, i.e a sextant distance at 1°30.8', (includes everything : refraction, parallax, ...).

Slightly too high since our target sextant distance is 1°35.0'

(4) - 3rd UT Estimate at 11:22 and topocentric approximation

From (2) and (3) above, we perform a third trial with UT = 11:22

From Atria and Venus published heights (15°07' and 27°52')  considered to be measured both at 11:22, get S00°27.9' / W092°44.8' from which we compute (reverse engineering):

Venus observed Height 27°52.0' / Azimut 091.283°  and Jupiter observed Height 29°20.5' / Azimut 091.646°, i.e a sextant distance at 1°30.5' (includes everything : refraction, parallax, ...).

Here we are !

Computed Solution is close from :

UT = 11h22m at S00°27.9' / W092°44.8'

(5.1) - This result assumes a non moving observer.

(5.2) - Computed Position could be slightly refined through assuming that you observe Atria at 11h20, you then observe Venus at 11h21 and you carefully measure your Sextant distance of 1°30.5' at 11h22m . I leave this to other number crunchers. Should not change the result by that much.

(5.3) - HOW ACCURATE ? Between 08h43 and 11h30 the geocentric observed Sextant distance changes from 1°24.4' to 1°30.8' , i.e 6.4' in 2.8 hours. If we consider that our sextant distance built-in error cannot be less than 0.1', this translates into an uncertainty of slightly under 30 minutes on the UT determination, i.e. an uncertaintly of 7° on Longitude. If we assumed sextant error of .2' which would be already a nice and solid result - and a more realistic one - then our longitude guaranteed accuracy is just under 15°.

Longitude again, Longitude always ...

Antoine M. "Kermit" Couëtte

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