def eci2RaDec(eci):
""" Convert Earth-centered intertial vector to right ascension and declination.
Arguments:
eci: [3 element ndarray] Earth-centered inertial coordinats
Return:
(ra, dec): [tuple of floats] right ascension and declinaton
"""
# Normalize the ECI coordinates
eci = vectNorm(eci)
# Calculate declination
dec = np.arcsin(eci[2])
# Calculate right ascension
ra = np.arctan2(eci[1], eci[0]) % (2 * np.pi)
return ra, dec
def calcSpatialResidual(jd, state_vect, radiant_eci, stat, meas):
""" Calculate horizontal and vertical residuals from the radiant line, for the given observed point.
Arguments:
jd: [float] Julian date
state_vect: [3 element ndarray] ECI position of the state vector
radiant_eci: [3 element ndarray] radiant direction vector in ECI
stat: [3 element ndarray] position of the station in ECI
meas: [3 element ndarray] line of sight from the station, in ECI
Return:
(hres, vres): [tuple of floats] residuals in horitontal and vertical direction from the radiant line
"""
meas = vectNorm(meas)
# Calculate closest points of approach (observed line of sight to radiant line) from the state vector
obs_cpa, rad_cpa, d = findClosestPoints(stat, meas, state_vect,
radiant_eci)
# Vector pointing from the point on the trajectory to the point on the line of sight
p = obs_cpa - rad_cpa
# Calculate geographical coordinates of the state vector
lat, lon, elev = cartesian2Geo(jd, *state_vect)
# Calculate ENU (East, North, Up) vector at the position of the state vector, and direction of the radiant
nn = np.array(ecef2ENU(lat, lon, *radiant_eci))
# Convert the vector to polar coordinates
theta = np.arctan2(nn[1], nn[0])
phi = np.arccos(nn[2] / vectMag(nn))
# Local reference frame unit vectors
hx = np.array([-np.cos(theta), np.sin(theta), 0.0])
vz = np.array([
-np.cos(phi) * np.sin(theta), -np.cos(phi) * np.cos(theta),
np.sin(phi)
])
hy = np.array([
np.sin(phi) * np.sin(theta),
np.sin(phi) * np.cos(theta),
np.cos(phi)
])
# Calculate local reference frame unit vectors in ECEF coordinates
ehorzx = enu2ECEF(lat, lon, *hx)
ehorzy = enu2ECEF(lat, lon, *hy)
evert = enu2ECEF(lat, lon, *vz)
ehx = np.dot(p, ehorzx)
ehy = np.dot(p, ehorzy)
# Calculate vertical residuals
vres = np.sign(ehx) * np.hypot(ehx, ehy)
# Calculate horizontal residuals
hres = np.dot(p, evert)
return hres, vres
stat = np.array(geo2Cartesian(lat, lon, h + 10, jd))
meas = np.array([0.0, 0.0, 1.0])
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
print(calcSpatialResidual(jd, state_vect, radiant_eci, stat, meas))
# Plot the origin
#ax.scatter(0, 0, 0)
# Plot the first point
ax.scatter(*state_vect)
# Plot the line from the origin
rad_x, rad_y, rad_z = -vectNorm(state_vect)
rst_x, rst_y, rst_z = state_vect
meteor_len = 1000000
ax.quiver(rst_x,
rst_y,
rst_z,
rad_x,
rad_y,
rad_z,
length=meteor_len,
normalize=True,
color='b',
arrow_length_ratio=0.1)
# Plot the radiant direction line
rad_x, rad_y, rad_z = -radiant_eci
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 checkFOVOverlap(self, rp, tp):
""" Check if two stations have overlapping fields of view between heights of 50 to 115 km.
Arguments:
rp: [Platepar] Reference platepar.
tp: [Platepar] Test platepar.
Return:
[bool] True if FOVs overlap, False otherwise.
"""
# Compute the FOV diagonals of both stations
reference_fov = np.radians(np.sqrt(rp.fov_v**2 + rp.fov_h**2))
test_fov = np.radians(np.sqrt(tp.fov_v**2 + tp.fov_h**2))
lat1, lon1, elev1 = np.radians(rp.lat), np.radians(rp.lon), rp.elev
lat2, lon2, elev2 = np.radians(tp.lat), np.radians(tp.lon), tp.elev
# Compute alt/az of the FOV centre
azim1, alt1 = raDec2AltAz(np.radians(rp.RA_d), np.radians(rp.dec_d),
rp.JD, lat1, lon1)
azim2, alt2 = raDec2AltAz(np.radians(tp.RA_d), np.radians(tp.dec_d),
tp.JD, lat2, lon2)
# Use now as a reference time for FOV overlap check
ref_jd = datetime2JD(datetime.datetime.utcnow())
# Compute ECI coordinates of both stations
reference_stat_eci = np.array(
geo2Cartesian(lat1, lon1, rp.elev, ref_jd))
test_stat_eci = np.array(geo2Cartesian(lat2, lon2, tp.elev, ref_jd))
# Compute ECI vectors of the FOV centre
ra1, dec1 = altAz2RADec(azim1, alt1, ref_jd, lat1, lon1)
reference_fov_eci = vectNorm(np.array(raDec2ECI(ra1, dec1)))
ra2, dec2 = altAz2RADec(azim2, alt2, ref_jd, lat2, lon2)
test_fov_eci = vectNorm(np.array(raDec2ECI(ra2, dec2)))
# Compute ECI coordinates at different heights along the FOV line and check for FOV overlap
# The checked heights are 50, 70, 95, and 115 km (ordered by overlap probability for faster
# execution)
for height_above_ground in [95000, 70000, 115000, 50000]:
# Compute points in the middle of FOVs of both stations at given heights
reference_fov_point = reference_stat_eci + reference_fov_eci*(height_above_ground \
- elev1)/np.sin(alt1)
test_fov_point = test_stat_eci + test_fov_eci * (
height_above_ground - elev2) / np.sin(alt2)
# Check if the middles of the FOV are in the other camera's FOV
if (angleBetweenVectors(reference_fov_eci, test_fov_point - reference_stat_eci) <= reference_fov/2) \
or (angleBetweenVectors(test_fov_eci, reference_fov_point - test_stat_eci) <= test_fov/2):
return True
# Compute vectors pointing from one station's point on the FOV line to the other
reference_to_test = vectNorm(test_fov_point - reference_fov_point)
test_to_reference = -reference_to_test
# Compute vectors from the ground to those points
reference_fov_gnd = reference_fov_point - reference_stat_eci
test_fov_gnd = test_fov_point - test_stat_eci
# Compute vectors moved towards the other station by half the FOV diameter
reference_moved = reference_stat_eci + vectorFromPointDirectionAndAngle(reference_fov_gnd, \
reference_to_test, reference_fov/2)
test_moved = test_stat_eci + vectorFromPointDirectionAndAngle(test_fov_gnd, test_to_reference, \
test_fov/2)
# Compute the vector pointing from one station to the moved point of the other station
reference_to_test_moved = vectNorm(test_moved - reference_stat_eci)
test_to_reference_moved = vectNorm(reference_moved - test_stat_eci)
# Check if the FOVs overlap
if (angleBetweenVectors(reference_fov_eci, reference_to_test_moved) <= reference_fov/2) \
or (angleBetweenVectors(test_fov_eci, test_to_reference_moved) <= test_fov/2):
return True
return False
def sampleTrajectory(dir_path, file_name, beg_ht, end_ht, sample_step):
""" Given the trajectory, beginning, end and step in km, this function will interpolate the
fireball height vs. distance and return the coordinates of sampled positions and compute the azimuth
and elevation for every point.
Arguments:
Return:
"""
# Load the trajectory file
traj = loadPickle(dir_path, file_name)
# Set begin and end heights, if not given
if beg_ht < 0:
beg_ht = traj.rbeg_ele / 1000
if end_ht < 0:
end_ht = traj.rend_ele / 1000
# Convert heights to meters
beg_ht *= 1000
end_ht *= 1000
sample_step *= 1000
# Generate heights for sampling
height_array = np.flipud(
np.arange(end_ht, beg_ht + sample_step, sample_step))
### Fit time vs. height
time_data = []
height_data = []
for obs in traj.observations:
time_data += obs.time_data.tolist()
height_data += obs.model_ht.tolist()
# Plot the station data
plt.scatter(obs.time_data,
obs.model_ht / 1000,
label=obs.station_id,
marker='x',
zorder=3)
height_data = np.array(height_data)
time_data = np.array(time_data)
# Sort the arrays by decreasing time
arr_sort_indices = np.argsort(time_data)[::-1]
height_data = height_data[arr_sort_indices]
time_data = time_data[arr_sort_indices]
# Plot the non-smoothed time vs. height
#plt.scatter(time_data, height_data/1000, label='Data')
# Apply Savitzky-Golay to smooth out the height change
height_data = scipy.signal.savgol_filter(height_data, 21, 5)
plt.scatter(time_data,
height_data / 1000,
label='Savitzky-Golay filtered',
marker='+',
zorder=3)
# Sort the arrays by increasing heights (needed for interpolation)
arr_sort_indices = np.argsort(height_data)
height_data = height_data[arr_sort_indices]
time_data = time_data[arr_sort_indices]
# Interpolate height vs. time
ht_vs_time_interp = scipy.interpolate.PchipInterpolator(
height_data, time_data)
# Plot the interpolation
ht_arr = np.linspace(np.min(height_data), np.max(height_data), 1000)
time_arr = ht_vs_time_interp(ht_arr)
plt.plot(time_arr, ht_arr / 1000, label='Interpolation', zorder=3)
plt.legend()
plt.xlabel('Time (s)')
plt.ylabel('Height (km)')
plt.grid()
plt.show()
###
# Take the ground above the state vector as the reference distance from the surface of the Earth
ref_radius = vectMag(traj.state_vect_mini) - np.max(height_data)
# Compute distance from the centre of the Earth to each height
radius_array = ref_radius + height_array
print('Beginning coordinates (observed):')
print(' Lat: {:.6f}'.format(np.degrees(traj.rbeg_lat)))
print(' Lon: {:.6f}'.format(np.degrees(traj.rbeg_lon)))
print(' Elev: {:.1f}'.format(traj.rbeg_ele))
print()
print("Ground-fixed azimuth and altitude:")
print(
' Time(s), Sample ht (m), Lat (deg), Lon (deg), Height (m), Azim (deg), Elev (deg)'
)
# Go through every distance from the Earth centre and compute the geo coordinates at the given distance,
# as well as the point-to-point azimuth and elevation
prev_eci = None
for ht, radius in zip(height_array, radius_array):
# If the height is lower than the eng height, use a fixed velocity of 3 km/s
if ht < traj.rend_ele:
t_est = ht_vs_time_interp(
traj.rend_ele) + abs(ht - traj.rend_ele) / 3000
time_marker = "*"
else:
# Estimate the fireball time at the given height using interpolated values
t_est = ht_vs_time_interp(ht)
time_marker = " "
# Compute the intersection between the trajectory line and the sphere of radius at the given height
intersections = lineAndSphereIntersections(np.array([0, 0, 0]), radius,
traj.state_vect_mini,
traj.radiant_eci_mini)
# Choose the intersection that is closer to the state vector
inter_min_dist_indx = np.argmin(
[vectMag(inter - traj.state_vect_mini) for inter in intersections])
height_eci = intersections[inter_min_dist_indx]
# Compute the Julian date at the given height
jd = traj.jdt_ref + t_est / 86400.0
# Compute geographical coordinates
lat, lon, ele_geo = cartesian2Geo(jd, *height_eci)
# Compute azimuth and elevation
if prev_eci is not None:
# Compute the vector pointing from the previous point to the current point
direction_vect = vectNorm(prev_eci - height_eci)
### Compute the ground-fixed alt/az
eci_x, eci_y, eci_z = height_eci
# 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 velocity of the Earth rotation at the position of the reference trajectory point (m/s)
v_e = 2 * np.pi * vectMag(height_eci) * 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,
lat, lon)
# The reference velocity vector has the average velocity and the given direction
# Note that ideally this would be the instantaneous velocity
v_ref_vect = traj.orbit.v_avg_norot * direction_vect
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
ra_norot, dec_norot = eci2RaDec(vectNorm(v_ref_nocorr))
azim_norot, elev_norot = raDec2AltAz(ra_norot, dec_norot, jd, lat,
lon)
###
else:
azim_norot = -np.inf
elev_norot = -np.inf
prev_eci = np.copy(height_eci)
print(
"{:s}{:7.3f}, {:13.1f}, {:10.6f}, {:11.6f}, {:10.1f}, {:10.6f}, {:10.6f}"
.format(time_marker, t_est, ht, np.degrees(lat), np.degrees(lon),
ele_geo, np.degrees(azim_norot), np.degrees(elev_norot)))
print(
'The star * denotes heights extrapolated after the end of the fireball, with the fixed velocity of 3 km/s.'
)
print('End coordinates (observed):')
print(' Lat: {:.6f}'.format(np.degrees(traj.rend_lat)))
print(' Lon: {:.6f}'.format(np.degrees(traj.rend_lon)))
print(' Elev: {:.1f}'.format(traj.rend_ele))