Re: [Phys-L] Earths tilt

[David Bowman <David_Bowman@georgetowncollege.edu>]

Regarding:

> If the earth wasn’t tilted, would the sun rise due east everywhere or would it change due to latitude?

Suppose a celestial object has a declination of δ (positive is north & negative is south of the celestial equator), and suppose an observer's latitude is λ (again, positive is north & negative is south).  Then (neglecting atmospheric refraction effects and assuming a spherical planet for the observer) the rising point/angle on the horizon *relative to due east* of the object is

arcsin(sin(δ)/cos(λ)), where a positive value is an angle north of east and a negative one is an angle south of east.

And the initial trajectory angle it rises from there relative to vertical is

arcsin(sin(λ)/cos(δ)), where a positive value is an angle that goes rightward, i.e. southward, relative to vertical, and a negative value is an angle that goes leftward, i.e. northward, relative to vertical.

This assumes that |λ ± δ| ≤ π/2.  But if |λ + δ| > π/2 then the object doesn't rise at all, but just stays above the horizon on a circumpolar path.  OTOH, if |λ - δ| > π/2 then the object always stays below the horizon and also never rises.

So if the earth's spin axis relative to the ecliptic normal was exactly zero then the Sun would always have a declination δ = 0 .  Knowing this it is fairly easy to see what would happen at various latitudes.  In particular, the Sun would always rise due east (except at the poles where things become singular and indeterminant).  But the direction angle of its rising from due east is just the observer's latitude.

Dave Bowman