If you know which day a comet is closest to the sun, can you calculate where it is before and after?
Let time pass through perihelion and observe the position, distance and speed in the elliptical orbit.
Planets and Small Solar System Objects
An orbit is not a circle of constant velocity. Given the ellipse size, oblateness and perihelion time, we first replace the number of days apart with the average perihelion angle, and then find the true position step by step.
Guess first
In the same amount of time, does it always take the same long distance on an elliptical orbit?
Show answer and explanation
No, it's usually faster closer to the sun — Time and true angular position need to be related through Kepler's equations.
Then connect experience and science
First agree on the day with the closest sun
t−T is the number of days from perihelion. Negative numbers mean closeness, positive numbers mean distance; the ellipse will come back after a full circle.
n=k/a^(3/2)；M=n(t−T)
Find the true position
The average periapsis angle increases uniformly with time; only by solving Puller's equation can we obtain the real periapsis angle and distance at a non-uniform speed.
E−e sinE=M；r=a(1−e cosE)
Putting planar orbits into three-dimensional space
After adding i, Ω, and ω, the same plane position is rotated to the specified reference ecliptic. The origin is still at the sun.
x,y,z are rotated orthogonally by the orbital plane vector
It is faster to check perihelion with speed
The speed is related to the distance from the current position; the energy relationship check algorithm can be used instead of just guessing based on the speed of the animation.
v=k√(2/r−1/a) AU/day
Expand the calculation
k = 0.01720209895 n = k/a^(3/2) M = n(t−T) E − e sin E = M r = a(1−e cos E) x轨道 = a(cos E−e)；y轨道 = a√(1−e²)sin E v = k√(2/r−1/a)
a and r are in AU, t−T is in days, n is rad/day, and the rate v is AU/day. The frame rate is converted to km/s; the three-dimensional orientation only rotates the position and does not change the distance.
give it a try
Compare the same number of days before and after perihelion: the distance is the same, but the true perihelion angle direction is opposite. Changing i, Ω, and ω only changes the spatial placement.
No need to memorize symbols
- T / t
- Perihelion time/query time
- n
- Horizontal motion, rad/day
- E / ν
- Partial approach angle / true approach angle
- r
- Heliocentric distance AU
Algorithm, accuracy and scope of application
Chapter 33 Elliptical motion of small celestial bodies: M, Kepler solution, true periapsis angle, radius vector, three-dimensional orbit placement and instantaneous velocity. The parameters are teaching input and do not pretend to be the orbit of a real comet. In addition, the planet-light-time course covers the geocentric positions and light travel times of major planets. This class does not output fictitious right ascension and declination or observation positions that are not supported by Earth's ephemeris.
my understanding check
t−T is negative, what does it mean?
before perihelion — The negative one is the time offset, not the physical distance.
Will rotating the orbit change the heliocentric distance r?
Won't — Orthogonal rotation changes the direction component but not the vector length.
Should the ellipse e satisfy?
0≤e<1 — e=1yes抛物线，e>1yeshyperbola。
The perihelion moment gives the starting point of time, Kepler's equation turns time into position, and the orbital orientation turns the plane position into a space vector.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 33 chapter
Planets and Small Solar System Objects · All interactive courses