When taking continuous photos of planets, why are they sometimes far apart and sometimes crowded together?
Take a picture every time the same time passes, and the gaps between the pictures will tell you whether it is going fast or not.
Planets and Small Solar System Objects
Take a picture of a passing bicycle. If you take a picture every second, when it is fast, the positions of the two pictures will be far apart. Now look at a planet the same way: in the same amount of time, the distance it travels along its orbit is not always the same.
Guess first
Take a photo at the same time, the farther apart the photos are, explain?
Show answer and explanation
Go faster — The distance covered in the same time is longer, so the movement is faster.
Then connect experience and science
First draw the orbit and the sun
The sun is at one focus of the ellipse, not at the center. The larger e is, the flatter the orbit is.
e = focal length / semi-major axis
Use a uniform angle to represent time
M grows at a constant rate, indicating the proportion of periods that have elapsed since perihelion, but not the true angular position of the planet.
M = 360° × elapsed time/cycle
Solve for auxiliary angle E
Use correction after correction to find E that satisfies the equation. The residual table below shows how the error shrinks.
M = E − e sin E (calculate in radians)
Return to true orbital position
Get the point on the ellipse from E, and then look at it from the sun to get the true angle ν.
x/a = cos E − e；y/a = √(1−e²) sin E
Expand the calculation
M弧Spend = MSpend × π/180 f(E) = E − e sin E − M E新 = E − f(E)/(1 − e cos E) x/a = cos E − e y/a = √(1−e²) sin E ν = atan2(y/a, x/a)
M, E, and trigonometric functions in iterations are in radians. Finally, convert E and ν back to degrees; you cannot directly subtract E in degrees from e sin E.
give it a try
First set e to 0, then change it to 0.8. Keep M = 60° and see how much the true angle changes.
No need to memorize symbols
- e
- Eccentricity, in this lesson is 0 ≤ e ≤ 0.9
- M
- The average near point angle increases uniformly with time
- E
- Near point angle, angle on the auxiliary circle
- ν
- True periapsis, the direction angle of a planet as viewed from the sun
- r/a
- The ratio of the distance from the sun to the planet and the semi-major axis
Algorithm, accuracy and scope of application
Chapter 30 of the original book gives a variety of solutions; Newton iteration is used here, and different solution routes of AA+ and residual verification of the substitution equation are used. The graph is normalized by the semi-major axis, which does not correspond to the real orbit of a planet; it only supports elliptical orbits and does not handle parabolas or hyperbolas.
my understanding check
What does the sparseness of isochronous snapshots near the Sun mean for planets?
faster — The snapshot intervals are the same, and a wider distance between positions means going faster.
If the orbit is changed to a perfect circle, what will happen to the snapshot spacing?
tend to be the same — The angular velocity and velocity in a circular orbit are both constant.
Does the analogy of bicycle continuous shooting explain that planets have engines?
cannot — The analogy only helps to understand the speed meaning of "distance/time"; the motion of the planet is determined by physical laws such as gravity.
The more spread out the positions taken at the same time, the faster the planet moves there.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 30 chapter
Planets and Small Solar System Objects · All interactive courses