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It is a well-known fact that launching a satellite is a captial-intensive and fuel-exhaustive process. Moreover, maintaining high accuracy and precision in any satellite orbit analysis is paramount to be able to comprehend helpful information from them.
Detecing some peculiar phenomena associated with satellites, which we call "events", could provide beneficial insights about their orbit dynamics for further treatment. While some could provide critical scientific information and help us formulate efficient space strategies and policies, the potentially disastrous ones, like satellite collisions, could help us take further steps to prevent such contingencies.
This notebook gives a glimpse of hapsira
's event detection capabilities. The procedure to track
an event during an orbit's propagation is fairly simple:
- Instantiate the desired event class/classes.
- Pass the
Event
object(s) as an argument toCowellPropagator
. - Detect events! Optionally, the
terminal
anddirection
attributes can be set as required.
# Imports
import numpy as np
from numpy.linalg import norm
import matplotlib.pyplot as plt
import astropy
import astropy.units as u
from astropy.time import Time
from astropy.coordinates import (
CartesianRepresentation,
get_body_barycentric_posvel,
)
from hapsira.bodies import Earth, Sun
from hapsira.twobody.events import (
AltitudeCrossEvent,
LatitudeCrossEvent,
NodeCrossEvent,
PenumbraEvent,
UmbraEvent,
)
from hapsira.twobody.orbit import Orbit
from hapsira.twobody.propagation import CowellPropagator
from hapsira.twobody.sampling import EpochsArray
from hapsira.util import time_range
Let's define some natural perturbation conditions for our orbit so that its altitude decreases with time.
from hapsira.constants import H0_earth, rho0_earth
from hapsira.core.perturbations import atmospheric_drag_exponential
from hapsira.core.propagation import func_twobody
R = Earth.R.to_value(u.km)
# Parameters of the body
C_D = 2.2 # Dimensionless (any value would do)
A_over_m = ((np.pi / 4.0) * (u.m**2) / (100 * u.kg)).to_value(
u.km**2 / u.kg
) # km^2/kg
# Parameters of the atmosphere
rho0 = rho0_earth.to_value(u.kg / u.km**3) # kg/km^3
H0 = H0_earth.to_value(u.km) # km
def f(t0, u_, k):
du_kep = func_twobody(t0, u_, k)
ax, ay, az = atmospheric_drag_exponential(
t0, u_, k, R=R, C_D=C_D, A_over_m=A_over_m, H0=H0, rho0=rho0
)
du_ad = np.array([0, 0, 0, ax, ay, az])
return du_kep + du_ad
We shall use the CowellPropagator
with the above perturbating conditions and pass the events we want to keep track of, in this case only the AltitudeCrossEvent
.
tofs = np.arange(0, 2400, 100) << u.s
orbit = Orbit.circular(Earth, 150 * u.km)
# Define a threshold altitude for crossing.
thresh_alt = 50 # in km
altitude_cross_event = AltitudeCrossEvent(thresh_alt, R) # Set up the event.
events = [altitude_cross_event]
method = CowellPropagator(events=events, f=f)
rr, _ = orbit.to_ephem(
EpochsArray(orbit.epoch + tofs, method=method),
).rv()
print(
f"The threshold altitude was crossed {altitude_cross_event.last_t} after the orbit's epoch."
)
Let's see how did the orbit's altitude vary with time:
altitudes = np.apply_along_axis(
norm, 1, (rr << u.km).value
) - Earth.R.to_value(u.km)
plt.plot(tofs[: len(rr)].to_value(u.s), altitudes)
plt.title("Altitude variation")
plt.ylabel("Altitude (in km)")
plt.xlabel("Time (in s)")
Refer to the API documentation of the events to check the default values for terminal
and direction
and change it as required.
Similar to the AltitudeCrossEvent
, just pass the threshold latitude while instantiating the event.
orbit = Orbit.from_classical(
Earth,
6900 << u.km,
0.75 << u.one,
45 << u.deg,
0 << u.deg,
0 << u.deg,
130 << u.deg,
)
thresh_lat = 35 << u.deg
latitude_cross_event = LatitudeCrossEvent(orbit, thresh_lat, terminal=True)
events = [latitude_cross_event]
tofs = np.arange(0, 20 * orbit.period.to_value(u.s), 150) << u.s
method = CowellPropagator(events=events)
rr, _ = orbit.to_ephem(EpochsArray(orbit.epoch + tofs, method=method)).rv()
print(
f"The threshold latitude was crossed {latitude_cross_event.last_t} s after the orbit's epoch"
)
Let's plot the latitude varying with time:
from hapsira.core.spheroid_location import cartesian_to_ellipsoidal
latitudes = []
for r in rr:
position_on_body = (r / norm(r)) * Earth.R
_, lat, _ = cartesian_to_ellipsoidal(
Earth.R, Earth.R_polar, *position_on_body
)
latitudes.append(np.rad2deg(lat))
plt.plot(tofs[: len(rr)].to_value(u.s), latitudes)
plt.title("Latitude variation")
plt.ylabel("Latitude (in degrees)")
plt.xlabel("Time (in days)")
The orbit's latitude would not change after the event was detected since we had set terminal=True
.
Since the attractor is Earth
, we could use GroundtrackPlotter
for showing the groundtrack of the
orbit on Earth.
from hapsira.earth import EarthSatellite
from hapsira.earth.plotting import GroundtrackPlotter
from hapsira.plotting import OrbitPlotter
es = EarthSatellite(orbit, None)
# Show the groundtrack plot from
t_span = time_range(orbit.epoch, end=orbit.epoch + latitude_cross_event.last_t)
# Generate ground track plotting instance.
gp = GroundtrackPlotter()
gp.update_layout(title="Latitude Crossing")
# Plot the above-defined earth satellite.
gp.plot(
es,
t_span,
label="Orbit",
color="red",
marker={
"size": 10,
"symbol": "triangle-right",
"line": {"width": 1, "color": "black"},
},
)
Viewing it in the orthographic
projection mode,
gp.update_geos(projection_type="orthographic")
gp.fig.show()
and voila! The groundtrack terminates almost at the 35 degree latitude mark.
Users can detect umbra/penumbra crossings using the UmbraEvent
and PenumbraEvent
event classes,
respectively. As seen from the above examples, the procedure doesn't change much.
from hapsira.core.events import eclipse_function
attractor = Earth
tof = 2 * u.d
# Classical orbital elements
coe = (
7000.137 * u.km,
0.009 * u.one,
87.0 * u.deg,
20.0 * u.deg,
10.0 * u.deg,
0 * u.deg,
)
orbit = Orbit.from_classical(attractor, *coe)
Let's search for umbra crossings.
umbra_event = UmbraEvent(orbit, terminal=True)
events = [umbra_event]
tofs = np.arange(0, 600, 30) << u.s
method = CowellPropagator(events=events)
rr, vv = orbit.to_ephem(EpochsArray(orbit.epoch + tofs, method=method)).rv()
print(
f"The umbral shadow entry time was {umbra_event.last_t} after the orbit's epoch"
)
Note: Even though the eclipse events UmbraEvent
and PenumbraEvent
take the Orbit
as input, they are not used in propagation but used only to access some helpful attributes of the orbit.
Let us plot the eclipse functions' variation with time.
k = Earth.k.to_value(u.km**3 / u.s**2)
R_sec = Sun.R.to_value(u.km)
R_pri = Earth.R.to_value(u.km)
# Position vector of Sun wrt Solar System Barycenter
r_sec_ssb = get_body_barycentric_posvel("Sun", orbit.epoch)[0]
r_pri_ssb = get_body_barycentric_posvel("Earth", orbit.epoch)[0]
r_sec = ((r_sec_ssb - r_pri_ssb).xyz << u.km).value
rr = (rr << u.km).value
vv = (vv << u.km / u.s).value
eclipses = [] # List to store values of eclipse_function.
for i in range(len(rr)):
r = rr[i]
v = vv[i]
eclipse = eclipse_function(k, np.hstack((r, v)), r_sec, R_sec, R_pri)
eclipses.append(eclipse)
plt.xlabel("Time (s)")
plt.ylabel("Eclipse function")
plt.title("Eclipse function vs time")
plt.plot(tofs[: len(rr)].to_value(u.s), eclipses)
We could get some geometrical insights by plotting the orbit:
# Plot `Earth` at the instant of event occurence.
Earth.plot(
orbit.epoch.tdb + umbra_event.last_t,
label="Earth position at event instance",
)
plotter = OrbitPlotter()
plotter.plot(orbit)
plotter.set_orbit_frame(orbit)
# Convert satellite coordinates to a `CartesianRepresentation` object.
coords = CartesianRepresentation(
rr[:, 0] << u.km, rr[:, 1] << u.km, rr[:, 2] << u.km
)
plotter.plot_trajectory(coords, color="orange")
It seems our satellite is exiting the umbra region, as is evident from the orange colored trajectory!
This event detector aims to check for ascending and descending node crossings. Note that it could yield inaccurate results if the orbit is near-equatorial.
r = [-3182930.668, 94242.56, -85767.257] << u.km
v = [505.848, 942.781, 7435.922] << u.km / u.s
orbit = Orbit.from_vectors(Earth, r, v)
As a sanity check, let's check the orbit's inclination to ensure it is not near-zero:
print(orbit.inc)
Indeed, it isn't!
node_event = NodeCrossEvent(terminal=True)
events = [node_event]
tofs = [0.01, 0.1, 0.5, 0.8, 1, 3, 5, 6, 10, 11, 12, 13, 14, 15] << u.s
method = CowellPropagator(events=events)
rr, vv = orbit.to_ephem(EpochsArray(orbit.epoch + tofs, method=method)).rv()
print(f"The nodal cross time was {node_event.last_t} after the orbit's epoch")
The plot below shows us the variation of the z coordinate of the orbit's position vector with time:
z_coords = [r[-1].to_value(u.km) for r in rr]
plt.xlabel("Time (s)")
plt.ylabel("Z coordinate of the position vector")
plt.title("Node Crossing")
plt.plot(tofs[: len(rr)].to_value(u.s), z_coords)
We could do the same plotting done in LatitudeCrossEvent
to check for equatorial crossings:
es = EarthSatellite(orbit, None)
# Show the groundtrack plot from
t_span = time_range(
orbit.epoch - 1.5 * u.h, end=orbit.epoch + node_event.last_t
)
# Generate ground track plotting instance.
gp = GroundtrackPlotter()
gp.update_layout(title="Node Crossing")
# Plot the above-defined earth satellite.
gp.plot(
es,
t_span,
label="Orbit",
color="red",
marker={
"size": 10,
"symbol": "triangle-right",
"line": {"width": 1, "color": "black"},
},
)
gp.update_geos(projection_type="orthographic")
gp.fig.show()
Indeed, we can observe that it's an ascending node crossing! If we want to only detect
either of the two crossings, the direction
attribute is at our disposal!
If we would like to track multiple events while propagating an orbit, we just need to add the concerned events inside events
.
Below, we show the case where NodeCrossEvent
and LatitudeCrossEvent
events are to be detected.
# NodeCrossEvent is detected earlier than the LatitudeCrossEvent.
r = [-6142438.668, 3492467.56, -25767.257] << u.km
v = [505.848, 942.781, 7435.922] << u.km / u.s
orbit = Orbit.from_vectors(Earth, r, v)
# Node Cross event
node_cross_event = NodeCrossEvent(terminal=True)
# Latitude event
thresh_lat = 60 * u.deg
latitude_cross_event = LatitudeCrossEvent(orbit, thresh_lat, terminal=True)
events = [node_cross_event, latitude_cross_event]
tofs = [1, 2, 4, 6, 8, 10, 12] << u.s
method = CowellPropagator(events=events)
rr, vv = orbit.to_ephem(EpochsArray(orbit.epoch + tofs, method=method)).rv()
print(f"Node cross event termination time: {node_cross_event.last_t} s")
print(
f"Latitude cross event termination time: {latitude_cross_event.last_t} s"
)
When detecting multiple events, the propagation stops as soon as any event, with the terminal
property set to True
, is detected.