EarthSatellite 下方的 Skyfield 可见区域

Question

我将如何计算 EarthSatellite 下方的面积，以便我可以绘制卫星经过时覆盖的土地面积？

Skyfield 有什么东西可以促进这一点吗？

编辑：只是想我会澄清我所说的卫星下方区域的意思。鉴于地球是一个球体，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

你的编辑清楚地表明了你想要什么。卫星的可见区域可以很容易地计算出来（当地球被视为一个球体时）。可以在此处找到获得有关可见部分背景的良好来源。当地球被视为扁球体时，计算可见面积会困难得多（甚至可能是不可能的）。我认为最好将问题的那部分进行改革并将其发布在数学上。

如果要计算将地球视为球体时的可见区域，我们需要在Skyfield. 通过使用 TLE api 加载卫星，您可以轻松获得地球上的位置的子点。图书馆称之为Geocentric位置，但实际上它是Geodetic位置（地球被视为扁球体的位置）。要纠正这一点，我们需要调整subpoint的的Geocentric类来使用计算的Geocentric位置，而不是Geodetic位置。由于reverse_terra函数中的错误和信息缺失，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))