Why is the ecliptic in the sky sometimes slanted and sometimes steeper?
The horizon follows where you stand, and the ecliptic follows the celestial sphere; as the earth rotates, the angle between the two changes.
Sky coordinates and observations
When you see the planets aligned along the ecliptic, sometimes they look like a sloping line, sometimes they're steeper. First draw the horizon, then let the local sidereal time change to see how the ecliptic intersects it.
Guess first
Is the angle between the ecliptic and the horizon the same path angle the sun will take when it rises today?
Show answer and explanation
no — The ecliptic describes the yearly reference path; the diurnal path is mainly caused by the Earth's rotation, and the two are separate.
Then connect experience and science
Start by standing on your ground level
The ecliptic is the Sun's reference path on the celestial sphere throughout the year; the horizon is determined by where you stand. They are not the same circle.
Two great circles: the ecliptic and the horizon
let the earth rotate
As local sidereal time changes, the tilt of the ecliptic before your eyes also changes; the two points of intersection are 180° apart in ecliptic longitude.
tan λ=−cosθ/(sinε tanφ+cosε sinθ)
Read the angle between two planes
Two complementary angles can be measured on both sides of the intersection. Here we give not only the original book I, but also a small angle for easy observation.
cos I=cosε sinφ−sinε cosφ sinθ
Don’t think of the ecliptic as the sun’s path throughout the day
The sun's path throughout the day comes primarily from the Earth's rotation; the angle at which it intersects the horizon is another matter. This is not the sunrise position or sunrise time.
Ecliptic-horizon angle ≠ Sunday track incident angle
θ is the local sidereal time, φ is the observer latitude, and ε is the ecliptic angle. First determine the correct quadrant of the yellow longitude of the intersection point, and then find the angle between the two great circles; the same great circle has a pair of intersection points that are 180° apart.
give it a try
Compare the two situations of 48° north latitude and sidereal time of 6h and 18h; then cut to 14.a of the original book. The intersection reading is the ecliptic longitude, not the ground azimuth.
No need to memorize symbols
- θ
- local sidereal time, hour
- λ
- zodiac node zodiacal longitude
- I
- The angle between the original book's directed planes is 0–180°
- ε / φ
- Yellow-red angle / observation latitude
Algorithm, accuracy and scope of application
Chapter 14 Printed Page 99 Type 14.2/14.3. The intersection point is in the equivalent atan2 form multiplied by cosφ to avoid the amplification of tanφ in the polar area and detect the coincidence of the two planes. In the original book, I is 0–180°, and the small included angle of min(I,180−I) is displayed at the same time. The supplementary angle difference is not regarded as a calculation error. The three-dimensional reference direction first turns from the ecliptic to the equator, and then is projected to the observer ENU. The tilt of the image does not replace the numerical angle.
my understanding check
At how many points does the ecliptic intersect with the horizon?
Two, 180° apart in longitude — Two great circles usually intersect in a pair of opposite directions.
Is θ here the local clock time?
No, it’s local sidereal time — Sidereal time describes the direction of the celestial sphere and cannot be directly used as a civilian time.
When two planes completely overlap, can a unique intersection point be specified?
cannot — At this point, the entire great circle is the same, and the concept of a unique intersection degenerates.
First distinguish which sky path it is, and then discuss its angle with the horizon; the sun's path in a year is not the same as its path in a day.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 14 chapter