3D Geometry: Calculate the position of the north star

[ElijahRockers]

Homework Statement

Ok I'm able to track one star in the sky, over a period of one hour. We use three measurements to find the angle of the arc traced by the star. The three measurements also constitute two vectors. We can take the cross product of those vectors and it will give us a vector that is perpendicular to both of those, and hence points in the direction of the north star. Divide by its magnitude for a unit vector. Then, we can use the formula $z=\rho*cos\varphi_{c}$ Where rho is arbitrary (in this case, 1) and phi is the colatitude of the north star.

Homework Equations

Measurement 1: Longitude=0, Colatitude=45
Measurement 2: Longitude=-8.840 Colatitude=48.729
Measurement 3: Longitude=-15.561 Colatitude=51.95

$x=sin\varphi_{c}cos\theta$
$y=sin\varphi_{c}sin\theta$
$z=cos\varphi_{c}$

The Attempt at a Solution

Using the right hand rule, I deduce that M₂M₁ would be vector A and M₂M₃ should be vector B. AxB should point fron the origin (earth) in the direction of the north star.

AxB/||AxB|| = <-.24957, -.43297, .8662>

These are the euclidean coordinates for the north star, but the answer is needed in sphereical coords, $(\theta,\varphi)$

I get the correct latitude, 60°, but the longitude (which is supposed to be 240°) does not come out right. Instead I get 300°.

Any help would be much appreciated...

OK. When I use the formula $x=sin\varphi_{c}cos\theta$ I get 120 degrees for theta, and I notice that 360-120 is 240. What am I missing here? When I use $y=sin\varphi_{c}sin\theta$ I get like -60°. What's going on?

 

[kurt.physics]

OK, so I finally figured it out. The problem was that I was using the wrong formula for converting from Euclidean to spherical coordinates. The correct formula is x=cos\varphi_{c}cos\theta , y=sin\varphi_{c}sin\theta and z=cos\varphi_{c}. Once I used this formula I got the correct longitude of 240°.