If it only passes by the sun and does not circle back, how can the position be calculated?
Instead of a "complete circle" cycle, use the closest distance and the time elapsed at that moment.
Planets and Small Solar System Objects
For a two-body orbit that is exactly parabolic, the celestial body does not have a finite period like an ellipse. We can still use the distance and days between perihelion to calculate its position before and after perihelion.
Guess first
Can a parabolic orbit specify a finite revolution period like an ellipse?
Show answer and explanation
cannot — An ideal two-body parabola is not a closed periodic orbit.
Then connect experience and science
The track just doesn't close anymore
The parabola with e=1 does not have a finite revolution period, nor can it copy the finite semi-major axis a and elliptical translational motion.
Use perihelion distance q and time difference
Set up the Barker equation
Calculate W from q and the number of days from perihelion, and solve the cubic equation. Symbols remain before and after perihelion.
s³+3s=W；W=3k(t−T)/(√2 q^(3/2))
Change solution back to position
s=tan(ν/2), distance q(1+s²). When reaching perihelion, s=0 and the radius is exactly q.
ν=2atan(s)；r=q(1+s²)
Compare before and after with numerical residuals
The distance between positive and negative time is the same and the direction is mirrored; in addition, the closed form solution of the hyperbolic function is used to check the iteration.
s=2sinh(asinh(W/2)/3)
AU is used for q, day is used for t−T, and k=0.01720209895. First obtain the dimensionless W, and then find the dimensionless s; the positive and negative of s distinguish before and after perihelion, and ν=2atan(s) corresponds to the diagram after being converted to gyration degrees.
give it a try
First read the Stonehouse comet calculation example in the original book, then change the time to the opposite number and compare the mirror position.
No need to memorize symbols
- q
- Perihelion distance AU
- W / s
- Dimensionless quantities of Barker's equation
- ν
- Signed true periapsis angle
Algorithm, accuracy and scope of application
Chapter 34: Barker iteration of equations 34.1–34.4, and the closed form of equivalent hyperbolic function is used for independent verification. Use the Gaussian constant k=.01720209895; in the book 0.03649116245 is the rounded value of 3k/√2. Example 34.a C/1998 H1 q=1.487469, t−T=112.5642 days, we should get ν≈66.78862°, r≈2.133911AU. The complete direction parameters or geocentric position not given in the original example are not made up.
my understanding check
What is r equal to at the moment of perihelion?
q — q is the closest distance.
What are the signs of s and ν before perihelion?
usually negative — The symbols before and after the retention time help determine direction.
Is the projection trajectory given here the star map seen on the earth?
no — It is a heliocentric space orbital projection that does not yet have Earth observers and light travel time processing.
It is possible to calculate the position without a finite period: starting from the perihelion distance and time difference, solve the Barker equation.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 34 chapter
Planets and Small Solar System Objects · All interactive courses