The calendar says perihelion, is the lowest point of the distance curve really there?
First use approximate dates to locate the same round of events, and then find the zero point of the derivative on the actual calculated distance curve.
Planets and Small Solar System Objects
The original book's perihelion and aphelion polynomials are fast, but the error for the outer planets can be weeks. Now that we have a periodic theory for calculating planetary distances by date, we can check when distances change from decreasing to increasing, or from increasing to decreasing, around this cycle.
Guess first
Can finding a minimum distance near an approximate date guarantee that it is the only "main perihelion"?
Show answer and explanation
No, there may be multiple local minima — The original book specifically discusses the major and minor minima of Neptune; the search scope and candidates must be explained.
Then connect experience and science
The approximate date is only responsible for positioning this round
First locate the event according to the polynomial in the original book. The upper limit of the search is fixed at 1/20 of one cycle of the planet, and the search does not become larger.
Prediction time ± P/20
Really calculate the slope of the distance
Use Appendix III to truncate VSOP87 to calculate R by date, and use the central difference to find the derivative; first find the small interval where the derivative changes sign.
dR/dt from negative to positive: extremely small
Find roots only in the enclosed local interval
Each sign-changing interval is divided into two halves separately, and the curvature type and distance between both sides are checked. You can't just pick a pixel that looks low.
Derivative zeros + extreme value type checking
Multiple local extrema, do not insist on being unique
Neptune may have major and minor minima. If there are multiple candidates in the same window, they are displayed one by one and the ambiguity is retained. Events cannot be changed silently.
Explicitly list when there are multiple candidates
First establish a time window near the prediction, look for the sign change of dR/dt, then find the zero point of the derivative and check the extreme value type. From negative to exactly minimum, from positive to negative is maximum; there is no guarantee that there will be no other extreme values outside the time window.
give it a try
First look at the vicinity of Venus and Mars, and then look at the vicinity of Neptune 2047. If there are multiple extreme values in the window, the page will list them clearly, instead of quietly treating another event as the answer.
No need to memorize symbols
- R(t)
- Planetary heliocentric distance, AU
- P
- Mean period of perihelion and aphelion in the original book
- dR/dt
- Derivative of distance with respect to date, AU/day
Algorithm, accuracy and scope of application
Complete the calculation route for finding the true local extremum from the distance function in Chapter 38. Using R(t) in Appendix III of the orbital-motion package, the fourth-order central difference derivative h=0.1 day; fix the 200-segment scan derivative change sign within the original polynomial predicted value ±P/20, and divide each segment into 45 rounds, and check the second-order sign. The fixed range is much smaller than the next main event period and does not expand the window infinitely; if there is no root or there are multiple similar roots, it will be displayed truthfully. Multiple extrema are not numerical failures, and Neptune in particular may have subminima. The fixed search window must also fall within the course verification interval of 1900–2100, otherwise the range prompt will be returned and no refinement will be performed. The results are limited by the truncated VSOP87 model error; there is no guarantee that the times listed in the original book with the complete theory will match second by second.
my understanding check
How should the sign of the distance derivative of perihelion change?
negative to positive — The distance first decreases and then increases.
The program does not find the root of the target type. Can the window be expanded indefinitely until it finds one?
This should not be done quietly — That might be replaced by another round of events that should report that the current range was not found.
Can it be called a modern precision numerical almanac by refining it with truncated VSOP87?
cannot — No matter how accurate the numerical root finding is, it cannot exceed the input ephemeris model.
Put the "date given by the formula" back into the distance curve check; if multiple extreme values are found, keep this fact.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 38 chapter
Planets and Small Solar System Objects · All interactive courses