python - 地球卫星下方的天空可见区域

Question

如何计算地球卫星下方的面积，以便可以绘制卫星经过时覆盖的土地带？

Skyfield 有什么东西可以促进这一点吗？

编辑:只是想澄清一下卫星下方区域的含义。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卫星飞越地球时的可见区域。

Answer 1

您的编辑清楚地表明了您想要什么。卫星的可见面积可以很容易地计算出来(本地球被视为一个球体时)。可以找到获取可见部分背景的好来源 here 。本地球被视为扁球体时，计算可见面积将会困难得多(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到数学上。

如果你想计算本地球被视为球体时的可见面积，我们需要在 Skyfield 中进行一些调整。通过使用 TLE api 加载卫星，您可以轻松获取地球上位置的子点。图书馆将此称为 Geocentric位置，但实际上是 Geodetic位置(地球被视为扁球体)。为了纠正这个问题，我们需要调整 subpoint Geocentric的类来使用 Geocentric 的计算位置而不是 Geodetic位置。由于reverse_terra中的错误和信息缺失函数我们还需要替换该函数。我们需要能够检索地球半径。结果如下:

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))