Orion's belt is like a straight line, is it really not curved at all?
Using three lines of sight, check how straight the "line" is.
Sky coordinates and observations
When you look at Orion's Belt from the ground, the three stars are almost lined up. The sky is not a plane: the straight lines here should be understood as the great circles on the celestial sphere. We use the coordinates in the book to measure how far the middle star deviates from the great circles at both ends, and then see when Mars passes through the great circle of the other pair of stars.
Guess first
Is it enough to prove that the spheres are exactly collinear if they look aligned in the photo?
Show answer and explanation
Not enough, check the line of sight geometry — Both the projection method and the visual resolution may be misleading; it is necessary to determine whether the three lines of sight are on the same central plane.
Then connect experience and science
First use your eyes to find an approximate row
The three stars in Orion's Belt are very much in line, but "likely" is not exactly spherical collinearity.
The straight line in the sky corresponds to the great circle of the celestial sphere
Let the line of sight at both ends define a plane
Place the line of sight of the observer and the stars at both ends in the same plane. This plane passes through the center of the celestial sphere and intersects it to form a great circle.
Normal vector n = u₁ × u₃
Measure how far the middle line of sight is from this plane
If the center star is also in the plane, its angular distance from the great circle is zero; if it is not, it can deviate by a certain number of arc minutes.
ω = asin(|u₂·n| / |n|)
When planets pass by, you can look for collinear moments
Switch to the Mars example in the original book, use the coordinates of five consecutive days to calculate the coplanarity, and then interpolate to find zero. The condition is spherical geometry, not just finding a straight line on a plane.
u₃·(u₁×u₂)=0
u is the unit line of sight, × is the cross product, and · is the dot product. First use the first and third directions to find the normal vector of the great circle, and then find out how much the second direction deviates from the great circle; degenerate situations such as the overlap or reverse direction of the first two directions must be dealt with separately.
give it a try
The "original three stars" use the coordinates of the original book; the "imaginary moving closer to a straight line" is only for geometric comparison and does not actually move the stars. The Mars example can jump to the collinear instant obtained by interpolation.
No need to memorize symbols
- u
- unit direction vector from the observer
- n
- The normal vector of the great circle plane obtained by the cross product of two lines of sight (in the formula); the control n also represents the time offset
- ω
- The smaller angular distance of the third direction from the great circle
- TD
- When reading the original book dynamics, the local time zone is not converted.
Algorithm, accuracy and scope of application
Chapter 19 Equation (19.1), Example 19.a, and Orion’s Belt Example (printed pages 121–125). The collinearity of the sphere uses the mixed product of three unit directions; the distance from the great circle uses the equivalent stable form of vector dot product/cross product to avoid the singularity of direct tan near the celestial pole. In addition, the smallest included angle of 0-180° between the tangents of the two great arcs of the star is calculated and is not regarded as a substitute for the directional north angle of the original book in all circumstances. The collinear moment of Mars is zeroed according to the five daily sample interpolation of Equation (19.1); the coordinates are also interpolated separately during the display, and the two interpolation orders produce small residual errors. The great circle cannot be uniquely determined if the two ends coincide or are opposite, so null is returned. No lunar station parallax, arbitrary date ephemeris, or all collinear event scans are implemented.
my understanding check
What does "a straight line in the sky" correspond to in this lesson?
Great round celestial sphere — It only describes the directional relationship seen by earth observers, and does not explain the actual alignment of stars in straight space.
The star in Orion's belt is about 5.4 arc minutes from both ends of the great circle, what does it mean?
Very close, but not exactly collinear — The deviation is not zero, which means that they are only approximately in a row to the naked eye.
Does the moon also need to use geocentric coordinates to directly determine local collinearity?
Precise local observations take parallax into account — The lunar parallax can be very large, and the original book reminds you to use the station center position; this lesson does not deal with the moon.
Approximate alignment does not mean exact collinearity; starting from the observer, check whether the three lines of sight fall on the same great circle plane.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 19 chapter