TLEが与えられた2つの衛星間の軌道距離の計算方法

I have 2 LEO satellites with the same orbital plane with same
inclination & longitude of the ascending node. I have the TLEs for
each satellite. How do I calculate the orbital distance between the
satellites in terms of time(sec) & space(km)?

I think I can use the time to periapsis for each satellite. I
referred to
this question about how to calculate time to periapsis
but the
equation mentioned in the answer requires eccentric anomaly. From
the TLE, I only have mean anomaly & according to wikipedia post about eccentric
anomaly
, there is no straightforward way to calculate eccentric
anomaly if mean anomaly is known.

So, how do I know the orbital distance (in secs & km) between 2
satellites in the same orbital plane given TLE? Assume reference
time as when TLE is generated. Assume that both TLE’s are generated
at same time.

ベストアンサー

I think I’ve found the answer to this question. Please
correct me if I am wrong as I’m new to this field

この質問で述べたように、周術期までの時間が鍵になるかもしれません。 Mean AnomalyのWikipediaの説明から、この式が見つかりました。
$$ M = n。(t- tau)$$
$ M $はMean Anomaly、$ n $はMean angular motion、$ tau
$は身体が周術期にある時間です。だから、$(t- tau)$は周術期までの時間です。

Mean anomaly(degrees) & Mean angular motion(revs/day) is given
in the TLE. I just need to convert them into same units.
$$ (t-tau) = frac{M(deg).(pi/180)}{n(rev/day).(2pi/86400)}
$$
This will give me time to periapsis in seconds for each satellite.
So the distance between them, in terms of time, can be easily found
by subtracting the two time to periapsis values.

To find the distance between two satellites in terms of space, I
need to find the
arc length of an ellipse
. The equation provided by the answer
is
$$ int_0^{theta_1} sqrt{a^2.sin^2(theta) + b^2.cos^2(theta)}
dtheta $$
$ a $ is semi-major axis, $ b $ is semi-minor axis & $ theta_1 $
is the angle of the arc, which can be easily found using Mean
angular motion.

I used WolfRam Alpha to compute the above
equation & found the answer to the distance between 2 satellites in
terms of space(km). I cross verified it using the general
distance-time formula
$$ frac{Orbit Circumference}{Orbit Period} = frac{Arc
Length}{Time to traverse Arc length}$$
Orbit Circumference can be found from semi-major, semi-minor axis &
eccentricity. I used Google calculator. Orbit period can be found using
Mean angular motion, Time to traverse Arc length is difference
between time to periapsis for 2 Satellites, so the Distance between
2 Satellites in terms of space(km) is the only unknown.

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