Data from Hipparcos-2 via ESASky (Science Mode), epoch J1991.25.

Below, "inclination" is spherical polar coordinate from the celestial north pole (in contrast to declination which is measured from the equator; these terms really ought to be swapped). "cartesian" is Cartesian coordinates after projecting onto a unit sphere.

Altair: RA (h,m,s) = 19.0 50.0 46.68 ; Declination (degree, arcminute, arcsecond) = 8.0 52.0 2.6 ; Decimal = 19.8463 8.86738888888889 ; azimuth = 297.6945 degree ; inclination = 81.13261111111112 degree ; (inclination, azimuth) in radians = 1.4160311946290238 5.19574919007826 ; cartesian = 0.4592021652056551 -0.8748552750817722 0.15414804290692427

Vega: RA (h,m,s) = 18.0 36.0 56.19 ; Declination (degree, arcminute, arcsecond) = 38.0 46.0 58.8 ; Decimal = 18.615608333333334 38.783 ; azimuth = 279.234125 degree ; inclination = 51.217 degree ; (inclination,azimuth) in radians = 0.8939052829939358 4.873554865175412 ; cartesian = 0.12508948352782784 -0.7694218935484297 0.626372549557442

Deneb: RA (h,m,s) = 20.0 41.0 25.91 ; Declination (degree, arcminute, arcsecond) = 45.0 16.0 49.2 ; Decimal = 20.690530555555554 45.28033333333333 ; azimuth = 310.35795833333333 degree ; inclination = 44.71966666666667 degree ; (inclination,azimuth) in radians = 0.7805054237276908 5.416768232684039 ; cartesian = 0.45564889763867855 -0.536182264248891 0.7105579931192031

We calculated central angle from the dot product of Cartesian coordinates. This method unfortunately loses precision, especially in tests with small angles. Future: consider other methods. Some stackexchange discussion, referencing two papers by Kahan: Miscalculating Area and Angles of a Needle-like Triangle, and Computing Cross-Products and Rotations in 2- and 3-Dimensional Euclidean Spaces.

Central angle between Vega and Deneb = acos 0.9146212800796072 = 0.41622610373088237 radian = 23.847999066953967 degree

Central angle between Deneb and Altair = acos 0.7878479666638791 = 0.66348948085889 radian = 38.01514700454042 degree

Central angle between Altair and Vega = acos 0.8271282666595854 = 0.596817749577609 radian = 34.19513818929266 degree

The central angle in spherical triangles is analogous to the length of a side in planar geometry.

We applied the spherical law of cosines to find the angles of the triangle. We also verified the result with the law of haversines, which is recommended for small triangles.

Altair vertex angle: 0.707867723241909 radian = 40.557832995296 degree

Vega vertex angle: 1.4324375118272625 radian = 82.0726238439231 degree

Deneb vertex angle: 1.1286406097530866 radian = 64.66634352592364 degree

The angle at Vega, 82 degrees, makes it nearly a right triangle, confirmed visually.

sum = 187.29680036514276 degree, which is greater than 180, as expected for a spherical triangle.

area = 0.12735319123246525 steradian

percentage of the celestial sphere = 1.013%, which is surprisingly small given how large the triangle can look when near the horizon.

The calculations above have the stars when and where Hipparcos observed them at 1991.25 (which presumably denotes some sort of decimal time). (Have their locations been extrapolated backward in time, regressing their proper motions, to the same time point, because Hipparcos didn't simultaneously observe them all on 1991.25?) Future task, use the proper motion values (also measured by Hipparcos, given below) to calculate where the stars are now and redo the angle calculations. Both Altair and Vega (relatively nearby stars) have moved on the order of 10 arc-second in the 28 years since. Calculating the new locations will require some more spherical geometry. It will also require verifying whether the proper motion in right ascension has already been converted by (cos declination). It looks like Hipparcos proper motion values in ESAsky are motion along great circles, so do not need additional multiplication by (cos declination).

Proper motion in mas/yr for (ra, dec) ; visual magnitude:

Altair : Hipparcos catalog number # 97649 ; proper motion 536.23 385.29 ; mag 0.8273

Vega : Hipparcos catalog number # 91262 ; proper motion 200.94 286.23 ; mag 0.0868

Deneb : Hipparcos catalog number # 102098 ; proper motion 2.01 1.85 ; mag 1.2966

To calculate new location corrected for proper motion, calculate the motion in 3D and project back to the celestial sphere. Get radial velocity from VizieR.

radial velocity (km/s) and parallax (mas):

Altair: -26.10 194.95

Vega: -13.80 130.23

Deneb: -4.50 2.31

Which angle in the Summer Triangle is changing the fastest? Perhaps a live, constantly updating display.

It might be best to assume Deneb is not moving, at least, not in the sky.

Verify calculations by seeing if one can reproduce this this plot of the motion of Barnard's Star.

Barnard's Star: HIP 87937 ; RA (deg) 269.454 ; Dec 4.668 ; Proper motion ( -797.84 , 10326.93 ) ; Radial -109.70 ; Parallax 548.31

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