Why does the sunset go down diagonally and sometimes appear steeper?
Fix the declination of a celestial body and see how the path caused by the Earth's rotation crosses your horizon.
Sky coordinates and observations
When the sun is close to the horizon, it often does not set vertically, but sets diagonally lower. This tilt is related to the observation latitude and the area of the sky where the celestial body is located; it is not equal to the angle between the ecliptic and the horizon.
Guess first
Is the path angle here the angle between the sun's ecliptic and the horizon throughout the year?
Show answer and explanation
No, it’s the Sunday path caused by rotation — One-year routes and one-day routes cannot be mixed.
Then connect experience and science
See how it crosses the horizon
What is being compared here is the diurnal path of the same star to the horizon, the ecliptic, which is another curve that the Sun travels through throughout the year.
Fixed declination, ignore refraction
Different latitudes lead to different slopes of paths
The path is more upright near the equator; at higher latitudes, the ascent and descent paths are usually more inclined.
B=tanδ tanφ
Some paths don't go through at all
Arcpole stars, non-rising stars, or exactly tangent paths should not be drawn as they normally rise or set.
There is no ordinary rising and falling intersection when |B|>1
Understanding dwell time with true slope
Steeper paths stay longer near the horizon. However, the declination of the moon changes rapidly, so the fixed declination approximation in this lesson will have limitations.
tan J=√(1−B²) cosδ / |tanφ|
First find B and determine whether there is ordinary rise and fall, and then find the angle J between the trajectory and the horizon. Positions such as the poles and the equator that make the ratio strange are treated according to vector geometry, and tanφ cannot be eliminated directly.
give it a try
Compare the equinox sun and the solstice sun at 40° north latitude; the equatorial situation is closer to vertical. The picture only zooms in on a small section of the tangent line where it crosses.
No need to memorize symbols
- J
- The small angle between the path and the horizon
- φ / δ
- Observer latitude/celestial body declination
Algorithm, accuracy and scope of application
The B, C, and tanJ formulas at the end of Chapter 14 of the original book. Use atan2 and |tanφ| to output a small angle. The vertical path of the equator is 90°; the tangency is 0°. If there is no intersection or direction degradation, no false angle is output. Small errors are caused by changes in the sun's declination during the day, and the moon's error may exceed 1°. The original book boundaries are retained.
my understanding check
The declination is 0° and the latitude is 40°. The small path angle is approximately?
50° — J=90°−|φ|。
Does the arctic star cross the horizon every day?
Won't — If the trajectory does not cross the horizon, there will be no ordinary elevation and setting angles.
Is the fixed declination approximation always accurate for the Moon?
No, changes in the declination of the moon will cause errors — The book points out that the lunar error can exceed 1° and cannot be used for precise actual ascent and descent trajectories.
The slope of the rise and fall is determined by the path of the sun; first check whether it really crosses the horizon, and then talk about the angle of intersection.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 14 chapter