Among those little dots next to Saturn, which one is Titan?
First, calculate the positions of the eight satellites one by one according to Dourneau's theory, and then look at their arrangement next to Saturn.
Planets and Small Solar System Objects
Saturn has more than one moon surrounding it, and the eight main moons vary greatly in period, inclination, and distance. We calculate them according to Dourneau's theory, and then use the same field of view to simultaneously look at the inner small orbit and Iapetus in the distance.
Guess first
If the satellite looks close to the planet's disk in the photo, does it mean it must be in front of the planet?
Show answer and explanation
Not necessarily, maybe later — Two-dimensional projection does not provide complete depth and must be judged in conjunction with the Z direction.
Then connect experience and science
First, consider the center of the planet as the zero point
The transverse X is positive toward the west, and the longitudinal Y is along the north pole of the planet; Z is the depth, and negative values are closer to the earth. Not the celestial right ascension and declination.
Unit: planetary equatorial radius
First, calculate how long it has taken light to reach the earth.
What we see is the earlier light. The planetary distance and light travel time are first iterated, and then the satellite phase is calculated at the corresponding moment.
Luminous time = TD − planetary light travel time
Each of the eight satellites has its own periodic term
Orbital theory includes periodic perturbations, eccentricities and tilts, followed by coordinate rotation to the Earth's line of sight and corrections for differential light travel and perspective. Dourneau's theory adds resonance terms to the inner four satellites, and complete periodic parameters for each of the outer four satellites.
Periodic term → 3D coordinate → apparent projection
The front and back and lighting determine whether it is blocked or not.
Looking from the direction of the Earth, the center enters and exits the disk to distinguish between eclipses and occultations; looking again from the direction of the Sun, distinguishes eclipses from occultations. The precise contact time is not sought here.
Separate judgments from the earth's perspective and the sun's perspective
The orbit is first calculated according to the unique periodic terms of each Dourneau satellite, and then rotated and projected to the observation direction. The coordinate unit is the equatorial radius of Saturn; visible position and whether it enters the shadow require different viewing directions.
give it a try
First look at all the satellites, then click "6 hours later" to see the arrangement changes; "Look at Iapetus" will jump to the outermost one. Hollow points indicate that the model determines that it is occluded or in the shadow, and the position is still retained for learning.
No need to memorize symbols
- X / Y
- West/Planetary north projection, in units of planetary equatorial radius
- Z
- Depth; negative values are closer to the observer than the center of the planet
- When light travels
- Propagation time from planet to observation point
- Ling / cover / food / shadow Ling
- Different states of center projection in the direction of the earth or the sun
Algorithm, accuracy and scope of application
Chapter 46 Dourneau's eight-satellite theory: Mimas/Enceladus/Tethys/Dione resonance terms and eccentricity corrections, complete period parameters of Rhea/Titan/Hyperion/Iapetus, unified fifth-order central difference subroutine for the outer four stars, geocentric direction rotation to B1950 according to 21.5/21.7, three-dimensional multiple rotations, differential light travel time, perspective correction. The Rhea eccentricity auxiliary item adopts the errata 0.001 clearly pointed out by AA+, and does not use the 0.01 typographical error in the original book; therefore, the Rhea coordinates in Book Example 46.a are different from the printed table, and the rounding accuracy of other satellites is consistent with the example question. Chapter 32 of Saturn/Earth position multiplexing truncates VSOP87, and the satellite time adopts iterative light travel time; the sun perspective independently obtains the shadow state. The convergence limit of the light travel time of 0.17 seconds is consistent with AA+. Provides true calculated positions and classification of central overhangs and eclipses, and does not calculate finite-size contact moments, mutual events, ring occlusions, or center-center observation corrections.
my understanding check
With the same abscissa on the map, can we directly determine whether the satellite is in front or behind the planet?
No, depth Z is required — The front and rear positions determine whether it is a forward or a cover, and looking at X/Y alone is not enough.
What are the units of X and Y in the picture?
planet equatorial radius — They are projected coordinates relative to the center of the planet's disk.
Is the central state here equivalent to precise contact time forecast?
not equal to — This lesson only classifies the relationship between satellite centers and planetary projections, and does not cover all events such as limited satellite disc contact, mutual eclipse and mutual occultation.
From Mimas to Iapetus, the period and distance vary greatly; the same set of projection geometry determines who is visible.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 46 chapter
Planets and Small Solar System Objects · All interactive courses