Transformation from Geocentric Coordinates into Equatorial Coordinates?

Question

I have the moon's position in equatorial coordinates (RA and Dec) and now I want to calculate same moon's position in terms of earth centered coordinates i.e. Geocentric (latitude & longitude). My main goal is to establish a c++-code that gives me this transformation from RA,Dec to Lat,Lon and vice versa.

EDIT: More precisely, I have moon’s position in equatorial geocentric coordinates with following details:

Origin: Center of Earth
Lat : Declination With fundamental Plane: Celestial Equator
Lon : R.A. With Primary direction: Vernal Equinox

And I am able to obtain the moon’s position corresponding to a specific Modified Julian Date (MJD). I am also able to calculate moon’s position in xyz-coordinates by following calculations

$X= EarthMoon*sin(PI/2.-dec_moon*deg2rad)*cos(ra_moon*deg2rad);$ $Y= EarthMoon*sin(PI/2.-dec_moon*deg2rad)*sin(ra_moon*deg2rad);$ $Z= EarthMoon*cos(PI/2.-dec_moon*deg2rad);$

where EarthMoon is the distance between Earth and Moon. Further, I am converting them into Horizon Coordinates (topocentric) w.r.t to an observer at a specific location by using my subroutine and this part done.

Further, I am initiating my simulated CR-particles from observer site. E.g. vCR[3]= {R,lat,lon}

vCR[3]={Rearth+2300, 31.30, 55.74} //Geocentric CS (earth-based)

And converting them into Geocentric Cartesian coordinates(x,y,z). Finally, I have

vM[3]={X,Y,Z}; in Equatorial geocentric coordinate system
vCR[3]={x,y,z}; in Geocentric Coordinate System i.e. Earth based

That is my particular question in which I want to make vM[moon position] and vCR[CR position] to lie in same coordinates to get my final result. i.e. transformation from Geocentric (R,lat,lon) into Equatorial Geocentric (h,R.A,Dec).
Note: For vCR[3]: zero-longitude is the Prime Meridian, Greenwich whereas for vM[3] it’s vernal equinox in celestial sphere there is the difference.

Answer 1

If I understand your question correctly than your problem is that the primary direction of vCR[3] is the intersection between the prime meridian and the Earth's equator, instead of the vernal equinox. The geographical latitude is the same as the declination, but to get the right ascension you will need to rotate your coordinates around the z-axis by an amount that is equal to the sidereal time at Greenwich ($\theta_G$). This is done, simply by replacing the the longitude (lon) with the sidereal time at your location $\theta = \theta_G-lon$.

x =  R* cos(lat) * cos(theta);
y=   R* cos(lat) * sin(theta);
z =  R* sin(lat);

vCR[3]={x,y,z};

where theta is the sidereal time $\theta$.

The sidereal time is not a time but actually an angle that gives the right ascension of objects that are at that moment crossing the local meridian (i.e. the line between South and the Celestial north pole, for observers on the northern hemisphere). It is in a sense the longitude of your position on Earth if the geographical coordinate system were to rotate along with the celestial sphere.

EDIT: To calculate the Sidereal time, you can use the method explained in Astronomical Algorithms by Jean Meeus:

$$T = \frac{JD-2451545.0}{3625}$$ $$\theta_0 = 100.46061837 + 36000.770053608 T + 0.000387933 T^2 - \frac{T^3}{38710000}$$

The sidereal time is expressed in degrees.

Then express the UTC time as an angle:

$$t = 15h+0.25m+\frac{0.25}{60}s$$

where $h$ is the number of hours, $m$, the number of minutes, and $s$ the number of seconds.

The sidereal time at Greenwich at the specified time is then:

$$\theta_G = \theta + 1.00273790935*t $$

and the local sidereal time is:

$$\theta = \theta_G - L$$

Where $L$ is the geographical longitude (positive west from Greenwich).

Answer 2

I think, you are mixing two different concepts. Equatorial coordinates are coordinates in the equatorial coordinate system (as opposed to the ecliptical, galactic, super galactic, or horizontal coordinate systems). The coordinates also refer to a specific origin. This origin might be heliocentric (centre of the Sun), baryocentric (centre of mass of the Solar System), geocentric (centre of the Earth), or topocentric (a specific location on the Earth's ellipsoid).

So you may have heliocentric equatorial coordinates, geocentric equatorial coordinates, or topocentric equatorial coordinates.

From your question I infer that you have geocentric coordinates of the Moon and you want to convert them to topocentric (not geocentric!) coordinates for an observer at a specific location (latitude and longitude) on the Earth.

The best way to go about this would be to:

• First, convert you geocentric equatorial coordinates to its rectangular equivalent (i.e. find the $X_M$, $Y_M$, and $Z_M$ equatorial coordinates).

• Second calculate the $X_o$, $Y_o$, and $Z_o$ coordinates of the observer on Earth in the equatorial system.

• Third, subtract the $X_o$, $Y_o$, and $Z_o$ coordinates of the observer from the $X_M$, $Y_M$, and $Z_M$ coordinates of the Moon.

• Finally, convert the coordinates back to spherical coordinates, i.e. the $X$, $Y$, and $Z$ coordinates back to right ascension, declination, and distance.

If you have the geocentric right ascension $\alpha_M$, declination $\delta_M$, and distance $d_M$, then you can get the rectangular coordinates from:

$$ X_M = d_M \cos(\alpha_M) \cos(\delta_M) $$ $$ Y_M = d_M \sin(\alpha_M) \cos(\delta_M) $$ $$ Z_M = d_M \sin(\delta_M) $$

The $X$-axis is pointed towards the vernal equinox, and the $Z$-axis is pointed towards the celestial north pole (i.e. parallel with the Earth's rotation axis).

For the rectangular equatorial coordinates of the observer, you need the geographical latitude ($\phi$) and longitude ($L$), the elevation ($H$) and the sidereal time ($\theta$).

To calculate the rectangular coordinates of the observer is a bit more difficult as the Earth's surface is (approximately) an ellipsoid. The following is taken from Jean Meeus book, Astronomical Algorithms:

$$b' = 0.99664719$$

which is a measure of the ellipsoid.

$$ u = \arctan(b' \tan(\phi)) $$ $$ \rho\sin(\phi) = R_E b' \sin(u) + H \sin(\phi)) $$ $$ \rho\cos(\phi) = R_E \cos(u) + H \cos(\phi) $$

where $R_E = 6378140.0 \textrm{m}$ is the Earth's radius. We can then calculate the rectangular geocentric equatorial coordinates of the observer:

$$ X_o = \rho \cos(\phi) \cos(\theta)$$ $$ Y_o = \rho \cos(\phi) \sin(\theta)$$ $$ Z_o = \rho \sin(\phi) $$

The topocentric rectangular equatorial coordinates of the Moon are then:

$$ X = X_M - X_o $$ $$ Y = Y_M - Y_o $$ $$ Z = Z_M - Z_o $$

Then convert back to spherical coordinates:

$$ d' = \sqrt{X^2 + Y^2 + Z^2} $$ $$ \alpha' = \arctan \left(\frac{Y}{X} \right) $$ $$ \delta' = \arcsin \left(\frac{Z}{d'} \right) $$

You then have the topocentric equatorial coordinates of the Moon: right ascension $\alpha'$, declination $\delta'$ and the distance $d'$.

You will have to make sure that $\alpha'$ is in the correct quadrant. Programmatically this can be easily achieved with the atan2 function, i.e. alpha = atan2(y,x).

I haven't tested these formulae so I hope that they are correct.