Ground longitude/latitude under a satellite (cartesian coordinates) at a specfic epoch

Question

The script I'm wanting to develop uses the cartesian coordinates (XYZ) from a satellite, and in conjunction with the range, elevation and azimuth from a location, I then take a satellite’s orbital information and get the ground longitude/latitude under that satellite at a given time.

One step further from this: imagne the signal from a satellite piercing the atmosphere at exactly 300km above sea level. At this particular point when altitude is 300km, I need to calculate the ground longitude/latitude.

In the pyemph module there appears to be already a method (ephem.readtle) that can achieve this, but for TLE (two line element) data only. I'd like to use a satellite's cartesian coordinates to develop this. Is there such a method already out there? Or perhaps somebody with experience in this domain can point me in the right direction.

A similar question already exists referring to ECEF from Azimuth, Elevation, Range and Observer Lat,Lon,Alt, but it's not the same problem.

Here's what I have developed already: - satellite cartesian coordinates, XYZ - azimuth, elevation and range of satellite from ground station - ground station coordinates in lat, long, height above sea level

Here's what I need: - ground longitude/latitude under a satellite at a specific epoch, and in particular where the piercing point in the atmosphere (the point which the signal from the satellite pierces the atmosphere) is 300km altitude.

Answer 1

I found what I was looking for via this:

def ionospheric_pierce_point(self, dphi, dlambda, ele, azi):
    Re = 6378136.3 # Earth ellipsoid in meters
    h = cs.SHELL_HEIGHT * 10**3 # Height of pierce point meters, and where maximum electron density is assumed
    coeff = Re / (Re + h)
    lat_rx = dphi
    long_rx = dlambda

# Degrees to radians conversions
ele_rad = np.deg2rad(ele)
azi_rad = np.deg2rad(azi)
lat_rx_rad = np.deg2rad(lat_rx)
long_rx_rad = np.deg2rad(long_rx)

psi_pp = (np.pi / 2) - ele_rad - np.arcsin(coeff * np.cos(ele_rad)) # Earth central angle between user and the Eart projection of the pierce point, in radians
psi_pp_deg = np.rad2deg(psi_pp)
lat_pp = np.arcsin(np.sin(lat_rx_rad)*np.cos(psi_pp) +
np.cos(lat_rx_rad)*np.sin(psi_pp)*np.cos(azi_rad)) # in radians

if (lat_rx > 70 and ((np.tan(psi_pp)*np.cos(azi_rad)) > np.tan((np.pi/2) - lat_rx_rad))) or (lat_rx < -70 and ((np.tan(psi_pp)*np.cos(azi_rad + np.pi)) > np.tan((np.pi/2) + lat_rx_rad))):
    long_pp = long_rx_rad + np.pi - np.arcsin((np.sin(psi_pp)*np.sin(azi_rad)) / np.cos(lat_pp))
else:
    long_pp = long_rx_rad + np.arcsin((np.sin(psi_pp)*np.sin(azi_rad)) / np.cos(lat_pp))

lat_pp_deg = np.rad2deg(lat_pp)
long_pp_deg = np.rad2deg(long_pp)

return lat_pp_deg, long_pp_deg