The ephemeris only records a few times, so how to find the "most recent" and "just past zero" moments?
In addition to filling in an intermediate value, interpolation can also find local minima or zero crossings; provided that the data is really nearby.
General Algorithms and Numerical Computing
In the table, the distance from the Sun to Mars first becomes smaller and then larger, and the minimum value may be hidden between two lines. Another table shows Mercury's latitude as one negative and one positive, and it also crosses zero between records. We use three or five points from the book to identify these moments.
Guess first
If you see a curve falling, can you extrapolate it very far to determine the lowest point?
Show answer and explanation
No, the lowest point should be supported by nearby data — The local interpolating polynomial may be completely different from the real function at a distance.
Then connect experience and science
First select the correct unit and center
The angles in the table have been converted into arcseconds, and the time offset divided by the table interval is n. The center should be close to the event being queried.
n=(query time−center time)/table interval
The information used at three points and five points is different.
Three points describe secondary bending; when the third difference cannot be ignored, five points retain higher-order changes. It's not just an average of five points.
Three-point quadratic / five-point fourth difference polynomial
Zero crossing and extreme value are two different conditions
When the latitude of Mercury passes zero, find y=0; when the distance to Mars is minimum, find the derivative y′=0. After finding the moment, the function value of that moment must be calculated.
Zero point y(n)=0; extreme value y′(n)=0
Only believe in the vicinity, do not regard extrapolation as prediction
If the calculated n is far from the center, the formula in the book may give absurd results. This lesson rejects this event output and prompts to change center or data.
Check the range and residuals after local roots
Expand the calculation
a = y₂−y₁；b = y₃−y₂；c = b−a y(n) = y₂ + n(a+b+nc)/2 y′(n) = (a+b)/2 + nc 零点：y(n)=0；极值：y′(n)=0
The three-point branch is expanded above, and the five-point branch adopts a fourth-order polynomial. The derivative is for dimensionless n; after finding the root, the range needs to be checked. When c=0, do not directly divide by c to find the extreme value.
give it a try
First look at the difference between three points and five points on the lunar surface, and then cut to the zero crossing of Mercury and the closest distance to Mars. Extreme values cannot be reliably extrapolated from a small, distant segment of data.
No need to memorize symbols
- n
- Offset from center time, divided by sampling interval
- a,b,c / A…K
- step-by-step difference
- y(n)
- Three-point quadratic or five-point quartic interpolation polynomial
- y′(n)=0 / y(n)=0
- Extreme value conditions/zero-crossing conditions
Algorithm, accuracy and scope of application
Complete Chapter 3 3.3–3.11. The three-point and five-point differential expansions are the same polynomial; the zero points are equivalent to the Newton correction of the original equation 3.7/3.11, and the extreme value is zero for the derivative; the mathematical entrance to the fixed iteration of equation 3.6/3.10 is also retained for comparison. Do not treat failure to find a local root as zero, and do not accept extrapolated roots that are more than one interval away from the center. Interface query limit |n|≤0.5, event search check |n|≤1. For the moon's horizon parallax, use example 3.e; for Mercury's zero crossing at five o'clock, use the printed page 30 exercise; for the extreme values of Mars, use 3.b; the fourth table is a clearly marked self-programmed quartic function. The result time follows the TD or UT indicated in each table, and no mutual conversion is performed. All data are pre-converted to the same unit; there are also original right ascension data spanning 24 hours and solar longitude data spanning 0°, which are interpolated after expansion using the shortest continuous difference, and 359° to 0° is not regarded as 359° backwards. Five points is not more accurate in every situation. The original book's counterexample of distant sinusoidal extrema illustrates the risk of extrapolation and does not include unreliable extrapolation as a forecast function.
my understanding check
Are the conditions for finding the zero point of a function the same as finding the minimum?
different — The zero point requires y=0, and the extreme value requires the derivative to be zero.
Does n=0.25 always equal 6 hours?
No, it depends on the table interval — Only when the sampling interval is one day, the 0.25 interval is 6 hours.
With only three records, can it be directly used as a five-point method?
cannot — The five-point method requires five equally spaced values, and the program cannot make up the other two on its own.
First select the table near the event, then evaluate, zero or extreme values respectively; finally check the units, range and residuals.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 3 chapter
General Algorithms and Numerical Computing · All interactive courses