Why do some stars never set all night long?
Standing on the ground and looking at a circle of stars, the horizon is the boundary between "visible" and "blocked".
Sky coordinates and observations
The Earth rotates, allowing us to see the stars slowly turning. When taking pictures of star trails, some stars draw a complete circle, while some only draw a section. The ground under your feet will block half of the sky, and the trajectories of different stars are different: some will be exposed in a circle, and some will disappear below the horizon.
Guess first
Will every star rise and set every day where you are?
Show answer and explanation
uncertain — Some tracks are completely above the horizon, some completely below. First see if it crosses the horizon.
Then connect experience and science
Use a line to represent the horizon
The curve represents the geometric height of the star over an entire sidereal day. The blue dot is the current hour angle you selected.
h = 0°: geometric horizon
See if you can wear it
The curve intersects the horizon, corresponding to rising and falling. Increasing the hour angle from negative to positive is equivalent to making the earth rotate.
cos H₀ = −tan φ tan δ
Find the Upper Midian and Lower Midian
H = 0° is the superior midheaven, and H = ±180° is the inferior midheaven. They give the highest and lowest position during the day.
The highest altitude is on the superior meridian, and the lowest altitude is on the inferior meridian.
No intersection is also an answer
The entire curve is an arcuate pole at the top and a never-rising pole at the bottom. Exactly tangential and stopping at the horizon throughout the day are also treated separately.
Not crossing the horizon ≠ Algorithm failed
Expand the calculation
C = (sin h₀ − sin φ sin δ)/(cos φ cos δ) H₀ = acos(C) above the horizon时长 = 2H₀/15 Hour
In this lesson h₀=0°. Only −1<C<1 has ordinary rise and fall; the equal sign is tangent. When the denominator is zero, make a polar or celestial judgment first and do not divide by zero directly.
give it a try
Compare φ = 70°, δ = +60° and δ = −60°. Let φ = 90° and δ = 0° again and look at the special situation in the polar regions.
No need to memorize symbols
- φ / δ
- Observer latitude / stellar declination
- H₀
- The absolute value of the angle of rise and fall, only applicable during normal crossing
- sidereal hour
- A 15° rotation of the Earth corresponds to 1 sidereal hour
- h
- Geometric height, excluding refraction and terrain
Algorithm, accuracy and scope of application
Demonstrate the geometric rise and fall relationship of h₀=0° starting in Chapter 15, fix the star’s declination, and do not seek the sunrise time in civil dates. Visible duration pressing 2H₀/15 gives sidereal hours; not solar hours. Does not include stellar standard refraction −0.5667°, solar radius, moving object interpolation, or weather. The full rise and fall algorithm of the original book and AA+ is more complex; here it does not pretend to be a full ephemeris predictor through independent height sampling/root finding, polar and tangent boundary checks.
my understanding check
The bright solid line goes around a full circle without touching the horizon. What does it mean?
This star never falls — It's still circling, but its trajectory is always above your horizon.
If the same star is moved to a more southerly location, will the rising and setting conditions change?
may change — If the orientation of your ground plane changes, so will the trajectory you can see.
What does the ghost shadow of stars under the ground in the picture mean?
Line of sight is blocked by the earth — It marks the invisible geometric direction, not where the object actually enters the Earth.
Whether a star will set depends on whether its orbit crosses the horizon in front of you.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 15 chapter