Zur Hauptnavigation Zum Inhalt
The fundamental relationship for the PZX triangle is: sin(a) = sin(φ) sin(δ) + cos(φ) cos(δ) cos(H)
The Local Sidereal Time (LST) at any moment is directly related to the Right Ascension (RA) of a star crossing the local meridian. The fundamental relationship is:
0=sinϕsinδ+cosϕcosδcosH0 equals sine phi sine delta plus cosine phi cosine delta cosine cap H
Determining when and where a star rises or sets is another classic problem. The condition for a star to be on the horizon is a = 0° . Setting sin(a) = 0 in the cosine formula gives: 0 = sin(φ) sin(δ) + cos(φ) cos(δ) cos(H) Thus, the hour angle of rising or setting is given by: cos(H) = - tan(φ) tan(δ) This equation has two solutions: one for rising ( H negative) and one for setting ( H positive). The diurnal motion of the star across the meridian, as well as the concept of circumpolar stars (those that never set at certain latitudes), can be analyzed from this equation.
. Solving spherical astronomy problems requires three fundamental formulas applied to a spherical triangle with sides and opposing angles The Spherical Law of Cosines Used when dealing with three sides and one angle: spherical astronomy problems and solutions
For Dr. Elias Thorne, the dome was a sanctuary of geometry. While the rest of the world slept, Elias engaged in the ancient, silent war against the chaos of the night sky. His weapon was a slide rule, his battlefield was a sheaf of graph paper, and his enemy was a faint, erratic speck of light designated Asteroid 2045-KJ.
sin(0∘)=sinϕsinδ+cosϕcosδcosHsine open paren 0 raised to the composed with power close paren equals sine phi sine delta plus cosine phi cosine delta cosine cap H
Hs=arccos(-0.5466)≈123.13∘cap H sub s equals arc cosine negative 0.5466 is approximately equal to 123.13 raised to the composed with power
The Earth's axis wobbles (precession) and nods (nutation). As a result, Right Ascension and Declination change over time. An RA/Dec from 1950 is not valid today. The fundamental relationship for the PZX triangle is:
) must be treated as negative numbers in trigonometric functions. Functions like arcsinarc sine arccosarc cosine
cos(90∘−δ)=cos(90∘−ϕ)cos(90∘−h)+sin(90∘−ϕ)sin(90∘−h)cosZcosine open paren 90 raised to the composed with power minus delta 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 h 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 h close paren cosine cap Z Simplify using identity transformations:
Because the Earth rotates at a rate of 15∘ per hour15 raised to the composed with power per hour
cosine z equals cosine open paren 50 raised to the composed with power close paren cosine open paren 70 raised to the composed with power close paren plus sine open paren 50 raised to the composed with power close paren sine open paren 70 raised to the composed with power close paren cosine open paren 45 raised to the composed with power close paren Setting sin(a) = 0 in the cosine formula
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: $$ \cos H = - \tan \phi \tan \delta $$
ϕ≥90∘−31∘53′phi is greater than or equal to 90 raised to the composed with power minus 31 raised to the composed with power 53 prime
, angular distance north or south of the celestial equator).
Two stars are observed due east. One is rising, and the other is 30∘30 raised to the composed with power above the horizon. Which culminates first? Solution: Understand Position: A star that is already 30∘30 raised to the composed with power
The azimuth is found using the sine formula: sin(H) / sin(ZX) = sin(360° - A) / sin(PX) Rearranging: sin(360° - A) = sin(H) × sin(PX) / sin(ZX) Substituting: sin(360° - A) = sin(124°10′30″) × sin(47°39′) / sin(67°55′26″) sin(360° - A) = (0.8271 × 0.7396) / 0.9266 ≈ 0.6117 / 0.9266 = 0.6600 Now, 360° - A = arcsin(0.6600) , which has two possible values: 41°17′06″ or 138°42′54″ . To resolve the ambiguity, we use the cosine formula again, which is unambiguous for the angle: cos(PX) = cos(PZ) cos(ZX) + sin(PZ) sin(ZX) cos(360° - A) Solving for cos(360° - A) yields cos(360° - A) ≈ 0.7518 , which corresponds to 360° - A = 41°17′06″ . Thus, the azimuth A = 318°42′54″ (measured eastward from north).