Say where the sun is, why are there three numbers X, Y, and Z?
The same direction and distance from the earth to the sun are projected onto three coordinate rulers.
To describe the location in a room, you can say a few steps to the east, a few steps to the north, and a few steps up. Astronomical rectangular coordinates also split a space vector in this way; but the coordinate axis points to the reference direction of the celestial sphere, and the reference equinox of the year must be specified.
Guess first
If the same sun position is changed from J2000 to J2044 coordinate axis, is it calculated where the sun will be in 2044?
Show answer and explanation
No, just change the coordinate axis — The position of the sun is still 1992-10-13, and J2044 only marks the reference coordinate axis.
Then connect experience and science
Assign distance to direction
The origin of geocentric-solar coordinates is the center of the Earth. X, Y, and Z are components in three mutually perpendicular directions, not three different suns.
X² + Y² + Z² = R²
First explain where the three rulers are facing.
The current equator, the J2000 equator, and the B1950 equator are not the same set of direction rulers. Write down the reference points first, then the numbers can be compared.
X points to the equinox; Z points to the North Celestial Pole
At the same moment, only the coordinate axes are changed
First fix the position and date, and then switch the equinox. The length of the vector does not change due to rotation, but the components do. You can also change the position and date independently to see the actual direction of the sun change with time.
Formulas (26.1)–(26.3) and precession rotation
Confirm which algorithm the input comes from
The input is generated from the Earth period terms in Appendix III and Table 26.A according to the selected date, and then reversely obtained to obtain the geocentric sun vector. Do not mix numbers for the heliocentric Earth, the geocentric sun, or different equinoxes.
Input precision limits output precision
Expand the calculation
X = R cos β cos λ Y = R(cos β sin λ cos ε − sin β sin ε) Z = R(cos β sin λ sin ε + sin β cos ε) X² + Y² + Z² = R²
The above is the rectangular coordinate conversion from the ecliptic to the equator on the same date. λ and β are yellow longitude and yellow latitude, ε is the intersection angle, and R is AU; when selecting a fixed equinox, the rotation matrix specified by the course must also be applied.
give it a try
First compare the different coordinate axes based on the date of 1992-10-13 in the original book; then switch to the moving date in 2000. Location date and grid year are two independent choices.
No need to memorize symbols
- X,Y,Z
- Geocentric equatorial rectangular coordinates, unit AU
- R
- The distance from the center of the earth to the center of the sun
- FK5
- Equatorial reference system used in this lesson
- J2000/B1950/J2044
- Reference equinox, not current celestial body position date
Algorithm, accuracy and scope of application
Realize the ecliptic spherical coordinates of Equation 26.1 → date equatorial rectangular coordinates; generate date equinox input in real time according to the earth cycle term in Appendix III, and add FK5 correction to make it clear that nutation/optical aberration is not added (geometric coordinates are required in this chapter). The standard equinox route replaces the first term of L1, L2–L4, B1–B4 and removes L5 according to Table 26.A to obtain the VSOP J2000 ecliptic dynamics geocentric solar vector; press 26.3 to rotate to FK5 J2000, and can also rotate to any J equinox according to the B1950 matrix or precession. It has covered any course date in 2000 and calculation examples from 1992 in the original book, and no longer relies on the fixed intermediate vectors in the book; the truncated intermediate values in the book are still used as independent verification calculations. The B1950 output belongs to FK5 rather than FK4. It does not implement the complete VSOP87 thousands of items or modern numerical ephemeris, and is not called a precision observation tool; it does not call the homepage approximate planetary model.
my understanding check
Where is the origin of geocentric rectangular coordinates?
center of earth — The geocentric sun vector of Chapter 26 points from the center of the Earth to the Sun.
By just rotating the axes, will the actual length of the vector change?
Won't — Orthogonal rotation changes the components, not the vector length.
Is the periodic version of this lesson the complete VSOP87?
No, use the truncation terms in the book — Inputs can now be generated by date, but Appendix III and Table 26.A are still truncated versions and not the complete thousands of theories.
Three components describe the same position; first write down the origin, unit, coordinate system and equinoxes, then the numbers will make sense.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 26 chapter