How to calculate the time on the sunrise table step by step from the three-day ephemeris?
Put the movement of the celestial body and the rotation of the earth on the same time line to find the real rising, mid-setting and setting events.
Sky coordinates and observations
To predict when it will rise, it's not enough to look at a static picture of the sky. The moon and planets themselves are also moving; a three-day ephemeris, plus the observation location, time scale difference and standard altitude, can iterate the event moment of this day.
Guess first
To calculate when the moon rises, is it accurate enough to use only a fixed declination at midnight that day?
Show answer and explanation
Not enough, it's moving on its own — It is necessary to use adjacent ephemeris to describe its coordinate changes within a day.
Then connect experience and science
First get the real three-day coordinate table
Instead of just drawing a circle for one declination. The apparent right ascension and declination of 0h TT the day before, today and the next day are used to describe the movement of the celestial body during this day.
α₁,δ₁；α₂,δ₂；α₃,δ₃
First estimate the rising, midheaven and setting
The apparent sidereal time at 0h UT1 determines the orientation of the Earth. Combining the longitude and latitude with the standard height h₀, first estimate the daily fraction of the three events.
m₀=(α₂−East longitude−θ₀)/360; m rise and fall=m₀∓H₀/360
Aligning two sets of time scales using ΔT
The event time is UT1, and the coordinate table is sampled according to TT, so the interpolation variable is n=m+ΔT/86400. When the right ascension spans 0°, it must first be expanded into a continuous angle.
Use three-point interpolation to find α(n), δ(n)
Repeat corrections until the standard height is reached
The rise and fall are corrected by h−h₀, and the midheaven is corrected by the hour angle H. The output only pertains to the selected UT1 date; local clocks may fall on the previous day.
Δm Midheaven=−H/360; Δm rise and fall=(h−h₀)/(360cosδcosφsinH)
m is the UT1 daily score from 0:00 on the current day. Use ΔT to replace the ephemeris interpolation parameter n, and then iterate the height or hour angle residual; use different H symbols for ascending and setting. h₀ depends on the celestial body and model, and is not always 0°.
give it a try
First look at the original Boston Venus book example, and then change to the calculation table for the Sun/Moon. Standard heights can be tested individually; events are listed below on the same UT1 date, and the fall may precede the rise.
No need to memorize symbols
- m
- The daily score of UT1 on the current day
- n
- Interpolation factor for TT ephemeris
- h₀
- Geometric standard height, including selected empirical rise and fall definitions
- east longitude
- This site is in the east direction; the original book L is in the west direction, so the formula sign is opposite.
Algorithm, accuracy and scope of application
Completely implements 15.1/15.2 initial value, three-point coordinate interpolation, right ascension cross-circle expansion, ΔT and apparent sidereal time, and rise/set/midheaven difference correction. The book example is the three-day actual table of Venus in Boston on March 20, 1988. The apparent sidereal time is calculated by the existing algorithm; solar-coordinates, moon-position, and planet-light-time can be selected to generate three-day apparent positions for other TT dates. Enter the east longitude; all events are grouped into one UT1 day, and cross-day events are not quietly wrapped around. Automatic standard altitude: star/planet −0.5667°, sun −0.8333°, moon 0.7275π−0.5667° (π takes the central day), self-selected values are only for experiments. The report cannot give ordinary events in the near-polar region, tangent or when no initial value is available, and does not use static criteria to pretend to be actual all-day conclusions. The lunar altitude threshold is fixed within a day and does not include meteorological, topographic, lunar edge and ΔT actual measurement services. UT1 plus integer offset is just a local clock indication and cannot claim precise UTC or historical daylight saving time conversions.
my understanding check
Why do we need to add ΔT/86400 to the interpolation variable?
Align UT1 event time with TT ephemeris — The two sets of timescale labels need to be aligned first.
The right ascension changes from 359.8° to 0.2°. Is it appropriate to directly interpolate these numbers?
Not appropriate, expand the circle first — It should be regarded as around 359.8° to 360.2° to avoid inserting into the wrong 180°.
The fall of UT1 at 2:55 in this lesson, replaced by the local clock of about UT1−5 hours, may it belong to?
the night before — It must be converted together with the date, not just the numbers on the clock face.
The rising and setting moments are iterative events: first give three-day coordinates, and then deal with the time scale, earth rotation and standard altitude.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 15 chapter