The table only records once a day. Where does the number at that time of night come from?
Use several records before and after to fill in nearby moments that are not directly listed.
General Algorithms and Numerical Computing
Like looking at a record sheet for each day, the ephemeris may only give one number per day. Here are the Mars-Earth distances for three consecutive days from the book: If you want to look up the night before, that row is not in the table, and you need to use nearby changes to estimate it.
Guess first
If the ephemeris does not record the moment of night, is it necessarily accurate to just copy the number at midnight the next day?
Show answer and explanation
Not necessarily, it’s changing in the middle — Values at adjacent moments may be different; this needs to be estimated based on changes in nearby data and checked to see if interpolation is suitable.
Then connect experience and science
Select three equally spaced points
Using book Mars-Earth distances: November 7, 8, 9, 1992 0:00 TD. Center taken on the 8th.
y₁ = 0.884226，y₂ = 0.877366，y₃ = 0.870531 AU
Let’s first see how much changes every day
There is a slight difference in the changes from the previous day to the next, which tells us that the curve is not completely straight.
a = y₂ − y₁；b = y₃ − y₂
add a little bend
The second difference c = b − a describes the change in the rate of change. Three-point interpolation is equivalent to fitting a small parabola.
c = b − a = 0.000025 AU
Read results nearby
n = 0 is the central moment; negative n means earlier. Drag the slider to read the distance in between.
y = y₂ + n(a + b + nc)/2
Expand the calculation
a = y₂ − y₁；b = y₃ − y₂ c = b − a n = (t − t₂)/Δt y(t) = y₂ + n(a + b + nc)/2
First find the first-order difference between the two segments, and then find the second-order difference. n has no unit; this lesson Δt=1 day. When n=0, the correction term disappears and must return to y₂.
give it a try
First click "Existing time in the table", then click "12:00 TD the day before" and "06:00 TD today". The dotted record sheet shows the estimated results, and the original three records on the left will not change.
No need to memorize symbols
- y₁, y₂, y₃
- The distance between three adjacent days in AU (astronomical units)
- n
- Offset from center time, divided by table interval; in this case the interval is 1 day
- a, b, c
- First-order difference and second-order difference
Algorithm, accuracy and scope of application
Use table (3.2) and equation (3.3) in the book, printed page 24. TD is the way it is written in dynamics in the book; this lesson does not convert it to UT. The interval is restricted to −0.5 to +0.5 to keep the center point closest. There is very little curvature in the graph, which is the nature of the raw data itself. Scenario description: Keep the ground-to-fire distance and equation (3.3) in the original book table (3.2), in AU; the date is 0:00 TD on November 7-9, 1992. This lesson does not convert to local time zone or UT, the query range is limited to half a day before and after the center, and no true-scale planetary space orbits are drawn.
my understanding check
For the three original records in this lesson, how long is the interval between adjacent moments?
one day — Adjacent records are at midnight on consecutive dates, one day apart.
Is the middle number in the record sheet on the dotted line on the right a newly measured one?
No, it was obtained by interpolation — It comes from the calculation of the three records on the left, and the estimated value cannot be regarded as evidence of new measurement.
Can we safely predict decades after knowing three days of data?
cannot — Smooth interpolation in short intervals is not a complete law of motion and cannot be extrapolated to long distances at will.
Interpolation fills in gaps between nearby records. Know where the estimate comes from, but also know how far it cannot be pushed.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 3 chapter
General Algorithms and Numerical Computing · All interactive courses