python - 地球卫星下方的天场可见区域

Question

我将如何计算地球卫星下方的面积，以便我可以绘制卫星经过时所覆盖的大片土地？

Skyfield有什么可以促进这一点的吗？

编辑：只是想我会澄清卫星下方区域的意思。鉴于地球是一个椭球体，我需要绘制卫星下方可能观察到的最大区域。我知道如何绘制卫星路径，但现在我需要绘制一些线来表示该卫星在地球上空飞行时可见的区域。

Answer 1

您的编辑清楚地表明了您想要什么。可以轻松计算卫星的可见区域（当地球被视为球体时）。可以在此处找到获得可见部分背景的良好来源。当地球被视为扁球体时，计算可见区域将更加困难（甚至可能是不可能的）。我认为最好修改这部分问题并将其发布在数学上。

如果要计算地球被视为球体时的可见区域，我们需要在 中进行一些调整Skyfield。使用 TLE api 加载卫星后，您可以轻松获得一个带有地球位置的子点。图书馆将其称为Geocentric位置，但实际上它是Geodetic位置（地球被视为扁球体的位置）。为了纠正这个问题，我们需要调整subpoint类Geocentric以使用Geocentric位置而不是Geodetic位置的计算。由于函数中的错误和缺失信息，reverse_terra我们还需要替换该函数。我们需要能够检索地球半径。这导致以下结果：

from skyfield import api
from skyfield.positionlib import ICRF, Geocentric
from skyfield.constants import (AU_M, ERAD, DEG2RAD,
                                IERS_2010_INVERSE_EARTH_FLATTENING, tau)
from skyfield.units import Angle

from numpy import einsum, sqrt, arctan2, pi, cos, sin

def reverse_terra(xyz_au, gast, iterations=3):
    """Convert a geocentric (x,y,z) at time `t` to latitude and longitude.
    Returns a tuple of latitude, longitude, and elevation whose units
    are radians and meters.  Based on Dr. T.S. Kelso's quite helpful
    article "Orbital Coordinate Systems, Part III":
    https://www.celestrak.com/columns/v02n03/
    """
    x, y, z = xyz_au
    R = sqrt(x*x + y*y)

    lon = (arctan2(y, x) - 15 * DEG2RAD * gast - pi) % tau - pi
    lat = arctan2(z, R)

    a = ERAD / AU_M
    f = 1.0 / IERS_2010_INVERSE_EARTH_FLATTENING
    e2 = 2.0*f - f*f
    i = 0
    C = 1.0
    while i < iterations:
        i += 1
        C = 1.0 / sqrt(1.0 - e2 * (sin(lat) ** 2.0))
        lat = arctan2(z + a * C * e2 * sin(lat), R)
    elevation_m = ((R / cos(lat)) - a * C) * AU_M
    earth_R = (a*C)*AU_M
    return lat, lon, elevation_m, earth_R

def subpoint(self, iterations):
    """Return the latitude an longitude directly beneath this position.

    Returns a :class:`~skyfield.toposlib.Topos` whose ``longitude``
    and ``latitude`` are those of the point on the Earth's surface
    directly beneath this position (according to the center of the
    earth), and whose ``elevation`` is the height of this position
    above the Earth's center.
    """
    if self.center != 399:  # TODO: should an __init__() check this?
        raise ValueError("you can only ask for the geographic subpoint"
                            " of a position measured from Earth's center")
    t = self.t
    xyz_au = einsum('ij...,j...->i...', t.M, self.position.au)
    lat, lon, elevation_m, self.earth_R = reverse_terra(xyz_au, t.gast, iterations)

    from skyfield.toposlib import Topos
    return Topos(latitude=Angle(radians=lat),
                    longitude=Angle(radians=lon),
                    elevation_m=elevation_m)

def earth_radius(self):
    return self.earth_R

def satellite_visiable_area(earth_radius, satellite_elevation):
    """Returns the visible area from a satellite in square meters.

    Formula is in the form is 2piR^2h/R+h where:
        R = earth radius
        h = satellite elevation from center of earth
    """
    return ((2 * pi * ( earth_radius ** 2 ) * 
            ( earth_radius + satellite_elevation)) /
            (earth_radius + earth_radius + satellite_elevation))


stations_url = 'http://celestrak.com/NORAD/elements/stations.txt'
satellites = api.load.tle(stations_url)
satellite = satellites['ISS (ZARYA)']
print(satellite)

ts = api.load.timescale()
t = ts.now()

geocentric = satellite.at(t)
geocentric.subpoint = subpoint.__get__(geocentric, Geocentric)
geocentric.earth_radius = earth_radius.__get__(geocentric, Geocentric)

geodetic_sub = geocentric.subpoint(3)

print('Geodetic latitude:', geodetic_sub.latitude)
print('Geodetic longitude:', geodetic_sub.longitude)
print('Geodetic elevation (m)', int(geodetic_sub.elevation.m))
print('Geodetic earth radius (m)', int(geocentric.earth_radius()))

geocentric_sub = geocentric.subpoint(0)
print('Geocentric latitude:', geocentric_sub.latitude)
print('Geocentric longitude:', geocentric_sub.longitude)
print('Geocentric elevation (m)', int(geocentric_sub.elevation.m))
print('Geocentric earth radius (m)', int(geocentric.earth_radius()))
print('Visible area (m^2)', satellite_visiable_area(geocentric.earth_radius(), 
                                                    geocentric_sub.elevation.m))