When does a crater on the moon get the sun?
Move the sunrise problem on the earth to the moon and use the direct point of the sun to find the boundary between light and dark.
When observing the lunar terrain, sunlight shining sideways can leave sharp shadows. But crater sunrise is not directly determined by what time it is on Earth. We pick a location on the moon and calculate how high the sun is there, as well as the nearby sunrise or sunset.
Guess first
Will the sun rise at the same moment at different places on the moon?
Show answer and explanation
Won't — The direct direction of the sun and the longitude and latitude of the location jointly determine the height of the sun.
Then connect experience and science
A full moon on the earth does not mean that it is noon everywhere on the moon
Locations on the moon at different latitudes and longitudes have different solar altitudes. First select a surface location and then calculate the direct sun point.
Moon surface solar coordinates l₀, b₀ and residual longitude c₀
The direct point of the sun determines the lighting hemisphere
At the direct point, the sun is overhead, and 90° away from it is the geometric light-dark boundary. As the day changes, the line between light and dark sweeps across the lunar surface.
c₀=90°−l₀ (normalized)
Find the height of the sun for a certain crater
By substituting the longitude and latitude of the moon, you can find how much the center of the sun is above the moon's horizon at that location. There are no lunar mountains blocking the way here.
sin h=sin b₀ sinφ+cos b₀ cosφ sin(c₀+λ)
Find sunrise or sunset from an approximate time
Corrections are started from the date close to the target event until the solar center altitude is close to zero. When the first guess is inappropriate, an error will be clearly reported and a convergence result will not be fabricated.
ΔJD=∓h/(12.19075cosφ)
λ and φ are the longitude and latitude of the moon surface location, and b₀ is the latitude of the sun's coordinates on the moon surface. First find the sun's height h, and then select the iteration time correction sign according to sunrise/sunset; when close to the lunar pole, cosφ is very small and cannot be extrapolated at will.
give it a try
First use the original Copernicus crater calculation example, and then switch to the first guess of sunset. For advanced conditions, you can enter your own lunar surface longitude and latitude.
No need to memorize symbols
- l₀ / b₀
- Longitude/latitude where the sun hits the moon directly
- c₀
- The remaining longitude of the sun and moon
- h
- Height of the moon at the center of the sun
- Longitude of the Moon
- According to the original book, the direction is towards Weihai.
Algorithm, accuracy and scope of application
Chapter 53: Longitude and latitude of the sun and moon, residual longitude, altitude of the sun at a designated location, and 53.4 iterations. The solar input comes from the truncated VSOP87 solarHigh of the existing solar-coordinates, without calling the home simplified ephemeris. The sun-earth distance conversion retains the AA+/149597970km/AU used in the original book era for comparison, and cannot be confused with the modern precise AU definition. The positive longitude of the moon is the direction toward the sea in the book; Copernicus −20°, 9.7°, Tycho −11°, −43.2° are taken from Table 53.A. The maximum number of iterations is 30 rounds, and it will definitely fail if there is a huge single step or the direction does not match; the actual terrain sunshine is not predicted.
my understanding check
The height of the sun on the moon is 0°. What does this mean?
The center of the sun reaches the ideal horizon — This model does not have the solar radius and lunar surface terrain.
The first guess is far from the target sunrise. What should you do when the iteration fails?
Report not converged — Iteration requires appropriate first guesses and stopping judgments.
What is the time scale of the output date?
TT — No local time zone or actual ΔT conversions are done.
First calculate the direct point of the sun on the moon, and then use the location to find the height; sunrise is the moment when the center of the sun crosses the local geometric horizon.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 53 chapter