Knowing the orbit of a comet, how do you calculate where it will be on the star map tonight?
Follow the orbit from perihelion time to the target date, and then change the observation starting point to the earth.
Planets and Small Solar System Objects
A heliocentric point in the orbit cannot be directly used to find comets on the ground. You need to convert it to the same standard equatorial coordinates as the sun's vector, and remember that the telescope is receiving light that started earlier.
Guess first
Can the heliocentric position given by the orbit be directly regarded as the right ascension and declination seen from the earth?
Show answer and explanation
No, we need to change the starting point of observation — The geocentric vector and optical path processing of the same reference system of the sun are also required.
Then connect experience and science
First find the heliocentric position along the orbit
Use perihelion time T and a, e to find M, E, ν, r, and then turn to the J2000 equatorial axis according to the orbital direction.
Equation 33.6–33.9
Change the starting point of observation to the earth
Add the geocentric sun vector at the same receiving time to get the geocentric target vector. The three coordinate axes must be consistent.
ξ=X+x，η=Y+y，ζ=Z+z
Iterate the optical path to obtain the astrometric direction
Only the position of the target at an earlier time is recalculated, and the sun's receiving time vector remains unchanged; no nutation or optical aberration is added here.
Astrometric position: including optical path, excluding annual optical aberration/nutation
The brightness can also be estimated, but the model must be stated
Comet uses g+5logΔ+klogr, and asteroid uses H–G phase function. Empirical brightness is not an actual measurement, nor can the two objects be confused with one.
H–G is only used within the phase angle range given in this book
Add the geocentric vector of the sun to the heliocentric vector of the small celestial body to obtain ξ, eta, and ζ, and then iterate the optical path to find the measurement direction of the celestial body. The H–G brightness model also relies on the heliocentric/geocentric distance and phase angle, and position and magnitude cannot be mixed and added together.
give it a try
First check the direction with the date 33.b in the original book of Comet Encke, and then switch the optical path. The brightness parameters are clearly fictitious teaching inputs and do not represent Enke's real photometry.
No need to memorize symbols
- T / t
- Perihelion time/query time
- ξ,η,ζ
- Geocentric J2000 equatorial vector
- H,G / g,k
- Asteroid phase parameters / comet empirical brightness parameters
Algorithm, accuracy and scope of application
Complete the second route 33.6-33.12 of Chapter 33: Encke root number in the original book → heliocentric equatorial vector → geocentric solar vector in the same reference system → light path iteration → J2000 astronomical measurement of right ascension and declination, distance from the earth, solar distance angle and phase angle. Sun uses Chapter 26 of this project to truncate the VSOP87 J2000 vector in real time; the original book example uses complete VSOP87, so the truncation/rounding difference is noted in the verification. In addition, the empirical brightness of 33.13 comets and 33.14 asteroids H–G are achieved. The latter’s phase angle is limited to 0–120°, and returns null if it crosses the boundary. Instead of perturbation integrals, orbital parameters are not extended to annual forecasts.
my understanding check
When iterating the light path, at what time does the sun vector take?
Time of receiving light — The second route in the original book retains the sun vector at the reception time.
Are astrometric positions and apparent positions the same thing?
no — Only the optical path is counted here, and annual optical aberration and nutation are not added.
If H and G of H–G are set to the teaching values casually, can we get the true brightness of the comet?
No — It is not only an empirical model of different objects, but also requires real photometric parameters.
The orbit description is seen from the sun; to find stars, you need to look at the earth and then deal with the time it takes for the light to travel.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 33 chapter
Planets and Small Solar System Objects · All interactive courses