In this article, we will discuss some common problems in spherical astronomy and provide their solutions. We will cover topics such as celestial coordinates, time and date, astronomical triangles, and planetary motion.
1 hour=15∘,1 minute of time=15′ (arcminutes),1 second of time=15′′ (arcseconds)1 hour equals 15 raised to the composed with power comma space 1 minute of time equals 15 prime (arcminutes) comma space 1 second of time equals 15 double prime (arcseconds)
Z=arccos(-0.1365)≈97.8∘cap Z equals arc cosine negative 0.1365 is approximately equal to 97.8 raised to the composed with power Because the hour angle (
In spherical astronomy, we don't work with straight lines. We work with on a sphere of infinite radius (the celestial sphere). The Cosine Rule:
One misty evening, a frantic young captain named Marco burst into her observatory. His ship’s chronometer had broken, and his sextant’s vernier scale was jammed. He was supposed to sail to the island of Cypress Peak at dawn, but the fog would hide the horizon. “Without instruments, I’m lost,” he said. spherical astronomy problems and solutions
From the cosine formula, setting $h=0$: $$ 0 = \sin \phi \sin \delta + \cos \phi \cos \delta \cos H $$ $$ \cos H = - \frac\sin \phi \sin \delta\cos \phi \cos \delta $$ Or simplified: $$ \cos H = - \tan \phi \tan \delta $$
At the exact moment of theoretical sunrise or sunset, the center of the sun sits exactly on the horizon line, meaning its altitude is From Problem 1, we know:
sinAsina=sinBsinb=sinCsincthe fraction with numerator sine cap A and denominator sine a end-fraction equals the fraction with numerator sine cap B and denominator sine b end-fraction equals the fraction with numerator sine cap C and denominator sine c end-fraction 2. Coordinate Transformation (Horizontal to Equatorial) To convert between the Horizontal System (Altitude or Zenith Distance ) and the Equatorial System (Hour Angle , Declination ) at a given observer latitude , we use the (Pole-Zenith-Object).
cos(90∘−a)=cos(90∘−ϕ)cos(90∘−δ)+sin(90∘−ϕ)sin(90∘−δ)cosHcosine open paren 90 raised to the composed with power minus a close paren equals cosine open paren 90 raised to the composed with power minus phi close paren cosine open paren 90 raised to the composed with power minus delta close paren plus sine open paren 90 raised to the composed with power minus phi close paren sine open paren 90 raised to the composed with power minus delta close paren cosine cap H Using trigonometric identities ( In this article, we will discuss some common
Substitute the observatory's latitude:
This public link is valid for 7 days and shares a thread, including any personal information you added. This link or copies made by others cannot be deleted. If you share with third parties, their policies apply. Can’t copy the link right now. Try again later.
Thus (A \approx 119^\circ) (measured from north through east). (a \approx 57.5^\circ), (A \approx 119^\circ).
When he returned, he brought Elara a gift—a brass armillary sphere. “For teaching me,” he said, “that the sky is not a mystery. It’s a sphere — and every problem has a solution if you know which triangle to solve.” We work with on a sphere of infinite
cosθ=(-0.0941×0.4695)+(0.9956×0.8829×0.8434)cosine theta equals open paren negative 0.0941 cross 0.4695 close paren plus open paren 0.9956 cross 0.8829 cross 0.8434 close paren
Numerator: (0.9397 \times 0.5 = 0.46985) Divide: (0.46985 / 0.5373 \approx 0.8746) [ A \approx \arcsin(0.8746) \approx 61.0^\circ \ \textor \ 119.0^\circ ] Check (\cos A): (\cos A = (\sin\delta - \sin\phi\sin a)/(\cos\phi\cos a)) Numerator: (0.3420 - (0.6428\times0.8431) = 0.3420 - 0.5419 = -0.1999) Denominator: (0.7660 \times 0.5373 = 0.4116) (\cos A = -0.1999 / 0.4116 \approx -0.4857) → (A > 90^\circ).
The principal astronomical triangle (also called the or PZS triangle ) has vertices: