def cartesian2Geo(julian_date, x, y, z, precess_j2000=False):
""" Convert Cartesian ECI coordinates of a point (origin in Earth's centre) to geographical coordinates.
Arguments:
julian_date: [float] decimal julian date
X: [float] X coordinate of a point in space (meters)
Y: [float] Y coordinate of a point in space (meters)
Z: [float] Z coordinate of a point in space (meters)
Keyword arguments:
precess_j2000: [bool] The given coordinates are in J2000. False by default, which means that they
should be in the epoch of date.
Return:
(lon, lat, ele): [tuple of floats]
lat: longitude of the point in radians
lon: latitude of the point in radians
ele: elevation in meters
"""
# Precess the coordinates to epoch of date, if they are not already in it
if precess_j2000:
### Precess coordinates from J2000 to epoch of date ###
# Convert rectangular to spherical coordiantes
re, delta_e, alpha_e = cartesianToSpherical(x, y, z)
# Dynamical Julian date
jd_dyn = jd2DynamicalTimeJD(julian_date)
# Precess coordinates to J2000
alpha_ej, delta_ej = equatorialCoordPrecession(J2000_JD.days, jd_dyn, alpha_e, delta_e)
# Convert coordinates back to rectangular
x, y, z = sphericalToCartesian(re, delta_ej, alpha_ej)
###
# Calculate LLA
lat, r_LST, ele = ecef2LatLonAlt(x, y, z)
# Calculate proper longitude from the given JD
lon, _ = LST2LongitudeEast(julian_date, np.degrees(r_LST))
# Convert longitude to radians
lon = np.radians(lon)
# Convert the height from WGS84 to MSL
ele = wmpl.Utils.GeoidHeightEGM96.wgs84toMSLHeight(lat, lon, ele)
return lat, lon, ele
def calcOrbit(radiant_eci, v_init, v_avg, eci_ref, jd_ref, stations_fixed=False, reference_init=True, \
rotation_correction=False):
""" Calculate the meteor's orbit from the given meteor trajectory. The orbit of the meteoroid is defined
relative to the centre of the Sun (heliocentric).
Arguments:
radiant_eci: [3 element ndarray] Radiant vector in ECI coordinates (meters).
v_init: [float] Initial velocity (m/s).
v_avg: [float] Average velocity of a meteor (m/s).
eci_ref: [float] reference ECI coordinates in the epoch of date (meters, in the epoch of date) of the
meteor trajectory. They can be calculated with the geo2Cartesian function. Ceplecha (1987) assumes
this to the the average point on the trajectory, while Jennsikens et al. (2011) assume this to be
the first point on the trajectory as that point is not influenced by deceleration.
NOTE: If the stations are not fixed, the reference ECI coordinates should be the ones
of the initial point on the trajectory, NOT of the average point!
jd_ref: [float] reference Julian date of the meteor trajectory. Ceplecha (1987) takes this as the
average time of the trajectory, while Jenniskens et al. (2011) take this as the the first point
on the trajectory.
Keyword arguments:
stations_fixed: [bool] If True, the correction for Earth's rotation will be performed on the radiant,
but not the velocity. This should be True ONLY in two occasions:
- if the ECEF coordinate system was used for trajectory estimation
- if the ECI coordinate system was used for trajectory estimation, BUT the stations were not
moved in time, but were kept fixed at one point, regardless of the trajectory estimation
method.
It is necessary to perform this correction for the intersecting planes method, but not for
the lines of sight method ONLY when the stations are not fixed. Of course, if one is using the
lines of sight method with fixed stations, one should perform this correction!
reference_init: [bool] If True (default), the initial point on the trajectory is given as the reference
one, i.e. the reference ECI coordinates are the ECI coordinates of the initial point on the
trajectory, where the meteor has the velocity v_init. If False, then the reference point is the
average point on the trajectory, and the average velocity will be used to do the corrections.
rotation_correction: [bool] If True, the correction of the initial velocity for Earth's rotation will
be performed. False by default. This should ONLY be True if the coordiante system for trajectory
estimation was ECEF, i.e. did not rotate with the Earth. In all other cases it should be False,
even if fixed station coordinates were used in the ECI coordinate system!
Return:
orb: [Orbit object] Object containing the calculated orbit.
"""
### Correct the velocity vector for the Earth's rotation if the stations are fixed ###
##########################################################################################################
eci_x, eci_y, eci_z = eci_ref
# Calculate the geocentric latitude (latitude which considers the Earth as an elipsoid) of the reference
# trajectory point
lat_geocentric = np.arctan2(eci_z, np.sqrt(eci_x**2 + eci_y**2))
# Calculate the dynamical JD
jd_dyn = jd2DynamicalTimeJD(jd_ref)
# Calculate the geographical coordinates of the reference trajectory ECI position
lat_ref, lon_ref, ht_ref = cartesian2Geo(jd_ref, *eci_ref)
# Initialize a new orbit structure and assign calculated parameters
orb = Orbit()
# Calculate the velocity of the Earth rotation at the position of the reference trajectory point (m/s)
v_e = 2 * np.pi * vectMag(eci_ref) * np.cos(lat_geocentric) / 86164.09053
# Calculate the equatorial coordinates of east from the reference position on the trajectory
azimuth_east = np.pi / 2
altitude_east = 0
ra_east, dec_east = altAz2RADec(azimuth_east, altitude_east, jd_ref,
lat_ref, lon_ref)
# Compute velocity components of the state vector
if reference_init:
# If the initial velocity was the reference velocity, use it for the correction
v_ref_vect = v_init * radiant_eci
else:
# Calculate reference velocity vector using the average point on the trajectory and the average
# velocity
v_ref_vect = v_avg * radiant_eci
# Apply the Earth rotation correction if the station coordinates are fixed (a MUST for the
# intersecting planes method!)
if stations_fixed:
### Set fixed stations radiant info ###
# If the stations are fixed, then the input state vector is already fixed to the ground
orb.ra_norot, orb.dec_norot = eci2RaDec(radiant_eci)
# Apparent azimuth and altitude (no rotation)
orb.azimuth_apparent_norot, orb.elevation_apparent_norot = raDec2AltAz(orb.ra_norot, orb.dec_norot, \
jd_ref, lat_ref, lon_ref)
# Estimated average velocity (no rotation)
orb.v_avg_norot = v_avg
# Estimated initial velocity (no rotation)
orb.v_init_norot = v_init
### ###
v_ref_corr = np.zeros(3)
# Calculate the corrected reference velocity vector/radiant
v_ref_corr[0] = v_ref_vect[0] - v_e * np.cos(ra_east)
v_ref_corr[1] = v_ref_vect[1] - v_e * np.sin(ra_east)
v_ref_corr[2] = v_ref_vect[2]
else:
# MOVING STATIONS
# Velocity vector will remain unchanged if the stations were moving
if reference_init:
v_ref_corr = v_init * radiant_eci
else:
v_ref_corr = v_avg * radiant_eci
### ###
# If the rotation correction does not have to be applied, meaning that the rotation is already
# included, compute a version of the radiant and the velocity without Earth's rotation
# (REPORTING PURPOSES ONLY, THESE VALUES ARE NOT USED IN THE CALCULATION)
v_ref_nocorr = np.zeros(3)
# Calculate the derotated reference velocity vector/radiant
v_ref_nocorr[0] = v_ref_vect[0] + v_e * np.cos(ra_east)
v_ref_nocorr[1] = v_ref_vect[1] + v_e * np.sin(ra_east)
v_ref_nocorr[2] = v_ref_vect[2]
# Compute the radiant without Earth's rotation included
orb.ra_norot, orb.dec_norot = eci2RaDec(vectNorm(v_ref_nocorr))
orb.azimuth_apparent_norot, orb.elevation_apparent_norot = raDec2AltAz(orb.ra_norot, orb.dec_norot, \
jd_ref, lat_ref, lon_ref)
orb.v_init_norot = vectMag(v_ref_nocorr)
orb.v_avg_norot = orb.v_init_norot - v_init + v_avg
### ###
##########################################################################################################
### Correct velocity for Earth's gravity ###
##########################################################################################################
# If the reference velocity is the initial velocity
if reference_init:
# Use the corrected velocity for Earth's rotation (when ECEF coordinates are used)
if rotation_correction:
v_init_corr = vectMag(v_ref_corr)
else:
# IMPORTANT NOTE: The correction in this case is only done on the radiant (even if the stations
# were fixed, but NOT on the initial velocity!). Thus, correction from Ceplecha 1987,
# equation (35) is not needed. If the initial velocity is determined from time vs. length and in
# ECI coordinates, whose coordinates rotate with the Earth, the moving stations play no role in
# biasing the velocity.
v_init_corr = v_init
else:
if rotation_correction:
# Calculate the corrected initial velocity if the reference velocity is the average velocity
v_init_corr = vectMag(v_ref_corr) + v_init - v_avg
else:
v_init_corr = v_init
# Calculate apparent RA and Dec from radiant state vector
orb.ra, orb.dec = eci2RaDec(radiant_eci)
orb.v_init = v_init
orb.v_avg = v_avg
# Calculate the apparent azimuth and altitude (geodetic latitude, because ra/dec are calculated from ECI,
# which is calculated from WGS84 coordinates)
orb.azimuth_apparent, orb.elevation_apparent = raDec2AltAz(
orb.ra, orb.dec, jd_ref, lat_ref, lon_ref)
orb.jd_ref = jd_ref
orb.lon_ref = lon_ref
orb.lat_ref = lat_ref
orb.ht_ref = ht_ref
orb.lat_geocentric = lat_geocentric
# Assume that the velocity in infinity is the same as the initial velocity (after rotation correction, if
# it was needed)
orb.v_inf = v_init_corr
# Make sure the velocity of the meteor is larger than the escape velocity
if v_init_corr**2 > (2 * 6.67408 * 5.9722) * 1e13 / vectMag(eci_ref):
# Calculate the geocentric velocity (sqrt of squared inital velocity minus the square of the Earth escape
# velocity at the height of the trajectory), units are m/s.
# Square of the escape velocity is: 2GM/r, where G is the 2014 CODATA-recommended value of
# 6.67408e-11 m^3/(kg s^2), and the mass of the Earth is M = 5.9722e24 kg
v_g = np.sqrt(v_init_corr**2 -
(2 * 6.67408 * 5.9722) * 1e13 / vectMag(eci_ref))
# Calculate the radiant corrected for Earth's rotation (ONLY if the stations were fixed, otherwise it
# is the same as the apparent radiant)
ra_corr, dec_corr = eci2RaDec(vectNorm(v_ref_corr))
# Calculate the Local Sidreal Time of the reference trajectory position
lst_ref = np.radians(jd2LST(jd_ref, np.degrees(lon_ref))[0])
# Calculate the apparent zenith angle
zc = np.arccos(np.sin(dec_corr)*np.sin(lat_geocentric) \
+ np.cos(dec_corr)*np.cos(lat_geocentric)*np.cos(lst_ref - ra_corr))
# Calculate the zenith attraction correction
delta_zc = 2 * np.arctan2(
(v_init_corr - v_g) * np.tan(zc / 2), v_init_corr + v_g)
# Zenith distance of the geocentric radiant
zg = zc + np.abs(delta_zc)
##########################################################################################################
### Calculate the geocentric radiant ###
##########################################################################################################
# Get the azimuth from the corrected RA and Dec
azimuth_corr, _ = raDec2AltAz(ra_corr, dec_corr, jd_ref,
lat_geocentric, lon_ref)
# Calculate the geocentric radiant
ra_g, dec_g = altAz2RADec(azimuth_corr, np.pi / 2 - zg, jd_ref,
lat_geocentric, lon_ref)
### Precess ECI coordinates to J2000 ###
# Convert rectangular to spherical coordiantes
re, delta_e, alpha_e = cartesianToSpherical(*eci_ref)
# Precess coordinates to J2000
alpha_ej, delta_ej = equatorialCoordPrecession(jd_ref, J2000_JD.days,
alpha_e, delta_e)
# Convert coordinates back to rectangular
eci_ref = sphericalToCartesian(re, delta_ej, alpha_ej)
eci_ref = np.array(eci_ref)
######
# Precess the geocentric radiant to J2000
ra_g, dec_g = equatorialCoordPrecession(jd_ref, J2000_JD.days, ra_g,
dec_g)
# Calculate the ecliptic latitude and longitude of the geocentric radiant (J2000 epoch)
L_g, B_g = raDec2Ecliptic(J2000_JD.days, ra_g, dec_g)
# Load the JPL ephemerids data
jpl_ephem_data = SPK.open(config.jpl_ephem_file)
# Get the position of the Earth (km) and its velocity (km/s) at the given Julian date (J2000 epoch)
# The position is given in the ecliptic coordinates, origin of the coordinate system is in the centre
# of the Sun
earth_pos, earth_vel = calcEarthRectangularCoordJPL(
jd_dyn, jpl_ephem_data, sun_centre_origin=True)
# print('Earth position:')
# print(earth_pos)
# print('Earth velocity:')
# print(earth_vel)
# Convert the Earth's position to rectangular equatorial coordinates (FK5)
earth_pos_eq = rotateVector(earth_pos, np.array([1, 0, 0]),
J2000_OBLIQUITY)
# print('Earth position (FK5):')
# print(earth_pos_eq)
# print('Meteor ECI:')
# print(eci_ref)
# Add the position of the meteor's trajectory to the position of the Earth to calculate the
# equatorial coordinates of the meteor (in kilometers)
meteor_pos = earth_pos_eq + eci_ref / 1000
# print('Meteor position (FK5):')
# print(meteor_pos)
# Convert the position of the trajectory from FK5 to heliocentric ecliptic coordinates
meteor_pos = rotateVector(meteor_pos, np.array([1, 0, 0]),
-J2000_OBLIQUITY)
# print('Meteor position:')
# print(meteor_pos)
##########################################################################################################
# Calculate components of the heliocentric velocity of the meteor (km/s)
v_h = np.array(earth_vel) + np.array(
eclipticToRectangularVelocityVect(L_g, B_g, v_g / 1000))
# Calculate the heliocentric velocity in km/s
v_h_mag = vectMag(v_h)
# Calculate the corrected heliocentric ecliptic coordinates of the meteoroid using the method of
# Sato and Watanabe (2014).
L_h, B_h, met_v_h = correctedEclipticCoord(L_g, B_g, v_g / 1000,
earth_vel)
# Calculate the solar longitude
la_sun = jd2SolLonJPL(jd_dyn)
# Calculations below done using Dave Clark's Master thesis equations
# Specific orbital energy
epsilon = (vectMag(v_h)**2) / 2 - SUN_MU / vectMag(meteor_pos)
# Semi-major axis in AU
a = -SUN_MU / (2 * epsilon * AU)
# Calculate mean motion in rad/day
n = np.sqrt(G * SUN_MASS / ((np.abs(a) * AU * 1000.0)**3)) * 86400.0
# Calculate the orbital period in years
T = 2 * np.pi * np.sqrt(
((a * AU)**3) / SUN_MU) / (86400 * SIDEREAL_YEAR)
# Calculate the orbit angular momentum
h_vect = np.cross(meteor_pos, v_h)
# Calculate inclination
incl = np.arccos(h_vect[2] / vectMag(h_vect))
# Calculate eccentricity
e_vect = np.cross(v_h, h_vect) / SUN_MU - vectNorm(meteor_pos)
eccentricity = vectMag(e_vect)
# Calculate perihelion distance (source: Jenniskens et al., 2011, CAMS overview paper)
if eccentricity == 1:
q = (vectMag(meteor_pos) +
np.dot(e_vect, meteor_pos)) / (1 + vectMag(e_vect))
else:
q = a * (1.0 - eccentricity)
# Calculate the aphelion distance
Q = a * (1.0 + eccentricity)
# Normal vector to the XY reference frame
k_vect = np.array([0, 0, 1])
# Vector from the Sun pointing to the ascending node
n_vect = np.cross(k_vect, h_vect)
# Calculate node
if vectMag(n_vect) == 0:
node = 0
else:
node = np.arctan2(n_vect[1], n_vect[0])
node = node % (2 * np.pi)
# Calculate argument of perihelion
if vectMag(n_vect) != 0:
peri = np.arccos(
np.dot(n_vect, e_vect) / (vectMag(n_vect) * vectMag(e_vect)))
if e_vect[2] < 0:
peri = 2 * np.pi - peri
else:
peri = np.arccos(e_vect[0] / vectMag(e_vect))
peri = peri % (2 * np.pi)
# Calculate the longitude of perihelion
pi = (node + peri) % (2 * np.pi)
### Calculate true anomaly
true_anomaly = np.arccos(
np.dot(e_vect, meteor_pos) /
(vectMag(e_vect) * vectMag(meteor_pos)))
if np.dot(meteor_pos, v_h) < 0:
true_anomaly = 2 * np.pi - true_anomaly
true_anomaly = true_anomaly % (2 * np.pi)
###
# Calculate eccentric anomaly
eccentric_anomaly = np.arctan2(np.sqrt(1 - eccentricity**2)*np.sin(true_anomaly), eccentricity \
+ np.cos(true_anomaly))
# Calculate mean anomaly
mean_anomaly = eccentric_anomaly - eccentricity * np.sin(
eccentric_anomaly)
mean_anomaly = mean_anomaly % (2 * np.pi)
# Calculate the time in days since the last perihelion passage of the meteoroid
dt_perihelion = (mean_anomaly * a**(3.0 / 2)) / 0.01720209895
if not np.isnan(dt_perihelion):
# Calculate the date and time of the last perihelion passage
last_perihelion = jd2Date(jd_dyn - dt_perihelion, dt_obj=True)
else:
last_perihelion = None
# Calculate Tisserand's parameter with respect to Jupiter
Tj = 2 * np.sqrt(
(1 - eccentricity**2) * a / 5.204267) * np.cos(incl) + 5.204267 / a
# Assign calculated parameters
orb.lst_ref = lst_ref
orb.jd_dyn = jd_dyn
orb.v_g = v_g
orb.ra_g = ra_g
orb.dec_g = dec_g
orb.meteor_pos = meteor_pos
orb.L_g = L_g
orb.B_g = B_g
orb.v_h_x, orb.v_h_y, orb.v_h_z = met_v_h
orb.L_h = L_h
orb.B_h = B_h
orb.zc = zc
orb.zg = zg
orb.v_h = v_h_mag * 1000
orb.la_sun = la_sun
orb.a = a
orb.e = eccentricity
orb.i = incl
orb.peri = peri
orb.node = node
orb.pi = pi
orb.q = q
orb.Q = Q
orb.true_anomaly = true_anomaly
orb.eccentric_anomaly = eccentric_anomaly
orb.mean_anomaly = mean_anomaly
orb.last_perihelion = last_perihelion
orb.n = n
orb.T = T
orb.Tj = Tj
return orb
def geo2Cartesian(lat_rad, lon_rad, h, julian_date, precess_j2000=False):
""" Convert geographical Earth coordinates to Cartesian ECI coordinate system (Earth center as origin).
The Earth is considered as an elipsoid.
Arguments:
lat_rad: [float] Latitude of the observer in radians (+N), WGS84.
lon_rad: [float] Longitde of the observer in radians (+E), WGS84.
h: [int or float] Elevation of the observer in meters (EGS96 convention).
julian_date: [float] Julian date, epoch J2000.0.
Keyword arguments:
precess_j2000: [bool] Precess ECI coordinates to J2000. False by default.
Return:
(x, y, z): [tuple of floats] a tuple of X, Y, Z Cartesian ECI coordinates
"""
lon = np.degrees(lon_rad)
# Convert MSL height (i.e. height above sea level) to WGS84 height
h = wmpl.Utils.GeoidHeightEGM96.mslToWGS84Height(lat_rad, lon_rad, h)
# Calculate ECEF coordinates
ecef_x, ecef_y, ecef_z = latLonAlt2ECEF(lat_rad, lon_rad, h)
# Get Local Sidereal Time (apparent)
LST_rad = np.radians(jd2LST(julian_date, lon)[0])
# Calculate the Earth radius at given latitude
Rh = math.sqrt(ecef_x**2 + ecef_y**2 + ecef_z**2)
# Calculate the geocentric latitude (latitude which considers the Earth as an elipsoid)
lat_geocentric = math.atan2(ecef_z, math.sqrt(ecef_x**2 + ecef_y**2))
# Calculate Cartesian ECI coordinates (in meters), in the epoch of date
x = Rh * np.cos(lat_geocentric) * np.cos(LST_rad)
y = Rh * np.cos(lat_geocentric) * np.sin(LST_rad)
z = Rh * np.sin(lat_geocentric)
if precess_j2000:
### Precess coordinates to J2000 ###
# Convert rectangular to spherical coordiantes
re, delta_e, alpha_e = cartesianToSpherical(x, y, z)
# Dynamical Julian date
jd_dyn = jd2DynamicalTimeJD(julian_date)
# Precess coordinates to J2000
alpha_ej, delta_ej = equatorialCoordPrecession(jd_dyn, J2000_JD.days,
alpha_e, delta_e)
# Convert coordinates back to rectangular
x_ej, y_ej, z_ej = sphericalToCartesian(re, delta_ej, alpha_ej)
###
return x_ej, y_ej, z_ej
else:
# Leave the coordinates in the epoch of date
return x, y, z
