Are the two stars that appear to be next to each other really neighbors?
First use your two lines of sight to measure: How much angle does it need to turn to look from one star to another?
Sky coordinates and observations
In the night sky, two stars seem to be next to each other; there are also two stars that you need to turn your eyes a long way to see in turn. Astronomy uses the term "angular distance" to describe this visual separation. It does not tell us how far apart two stars are in the universe.
Guess first
If two stars appear very close to each other in the night sky, can we be sure that they are also neighbors in the universe?
Show answer and explanation
Not sure — There are only two directions visible; one may be closer, and the other may be far behind.
Then connect experience and science
First draw two lines of sight
Points on the celestial sphere represent directions, not actual distances. Two stars may be many light years apart.
Angular distance = the angle between two lines of sight
Comparative right ascension difference
At the celestial equator, the longitude difference is equal to the short arc angular distance along the equator.
When δ = 0°, d = Δα (within the scope of this lesson)
Getting closer to the sky
Drag the declination and the two points gradually move closer. Even if the right ascension is still 90° off, the angle will decrease.
Meridians converge at the celestial pole
Calculate using unit vectors
The dot product of two unit lines of sight gives the cosine of the angle between them. The implementation is in the form of atan2, taking into account small angular separations close to 180°.
cos d = sin δ₁ sin δ₂ + cos δ₁ cos δ₂ cos Δα
Expand the calculation
Δα = α₂ − α₁ x = cos δ₂ sin Δα y = cos δ₁ sin δ₂ − sin δ₁ cos δ₂ cos Δα z = sin δ₁ sin δ₂ + cos δ₁ cos δ₂ cos Δα d = atan2(√(x²+y²), z)
This is a stable writing method used in calculations. Describes the same angle as the cosine formula, but is more suitable for numerical calculations between very close or far directions.
give it a try
First select "open two lines of sight", then select "move closer to the celestial pole". The two longitudes of the star chart are still 90° apart, but the actual line of sight you have to turn has become smaller.
No need to memorize symbols
- α
- Right ascension; interfaces are always expressed in degrees, 1 hour = 15°
- δ
- declination
- d
- Smaller angle between two lines of sight, 0° to 180°
Algorithm, accuracy and scope of application
Formula from Chapter 17; calculation uses the atan2 equivalent form of the vector. The diagram is an orthogonal projection of the celestial sphere, and the straight line lengths on the screen are not angular distances. The right ascension input of AA+ uses hours and the interface fee of this project. Book example 17.a of Arcturus and Spica is used for verification. Scenario description: The picture on the left corrects the planes of the two lines of sight and accurately draws the angle, but does not indicate the local horizon or celestial distance. The picture on the right is an orthogonal projection of the celestial sphere looking from the direction of the North Celestial Pole: the center is the North Celestial Pole, and the outer circle is the celestial equator; the screen spacing is not the angular distance. The two stars are adjustable imaginary directions, not a real star list that can be directly observed at a certain time.
my understanding check
What does angular distance measure?
The angle from one line of sight to another — It describes how much the directions are spread apart and does not directly describe the length between two stars.
What will happen if we move the two directions together toward the same celestial pole along two fixed star chart longitudes?
The angle of sight becomes smaller — The longitude lines will converge at the celestial pole, so the actual directions will get closer and closer.
Can the screen distance between the two points in the picture on the right be directly used as the sky angle distance using a ruler?
cannot — The picture on the right is a projection, the screen length is not the angle. Read the line of sight angle in the picture on the left, or calculate it using the formula.
Angular distance measures how much you need to turn your eyes; star map coordinate difference, screen spacing, and actual distance in the universe are three different things.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 17 chapter