Spherical Astronomy — Problems And Solutions

θ=arccos(0.8270)≈34.21∘theta equals arc cosine 0.8270 is approximately equal to 34.21 raised to the composed with power

δcircumpolar≥90∘−30.67∘delta sub circumpolar end-sub is greater than or equal to 90 raised to the composed with power minus 30.67 raised to the composed with power

sin(H)sin(45∘)=sin(120∘)sin(90∘−10.58∘)the fraction with numerator sine open paren cap H close paren and denominator sine open paren 45 raised to the composed with power close paren end-fraction equals the fraction with numerator sine open paren 120 raised to the composed with power close paren and denominator sine open paren 90 raised to the composed with power minus 10.58 raised to the composed with power close paren end-fraction spherical astronomy problems and solutions

cosA=sinδ−sinϕsinacosϕcosacosine cap A equals the fraction with numerator sine delta minus sine phi sine a and denominator cosine phi cosine a end-fraction

) : Angular distance measured westward along the equator from the observer's local meridian. : θ=arccos(0

At the moment of rising or setting, the altitude of the object is , which means the zenith distance Use the simplified horizontal cosine formula where

cosA=0.4226−(0.6428⋅0.7626)0.7660⋅0.6468=0.4226−0.49020.4954=-0.06760.4954≈-0.1365cosine cap A equals the fraction with numerator 0.4226 minus open paren 0.6428 center dot 0.7626 close paren and denominator 0.7660 center dot 0.6468 end-fraction equals the fraction with numerator 0.4226 minus 0.4902 and denominator 0.4954 end-fraction equals negative 0.0676 over 0.4954 end-fraction is approximately equal to negative 0.1365 the altitude of the object is

sina=(sin40∘⋅sin25∘)+(cos40∘⋅cos25∘⋅cos45∘)sine a equals open paren sine 40 raised to the composed with power center dot sine 25 raised to the composed with power close paren plus open paren cosine 40 raised to the composed with power center dot cosine 25 raised to the composed with power center dot cosine 45 raised to the composed with power close paren

Rearrange the formula to solve for semi-major axis (a): a = (r_a + r_p) / 2

, the object is either circumpolar (never sets) or never rises at that latitude. 🛰️ Problem 4: Correcting for Atmospheric Refraction

Astronomers apply optical refraction models based on the observed altitude.