How does a computer know where the planet is orbiting when one day changes on the calendar?
Add up a series of small rhythmic changes to get the true calculated position at that moment.
Planets and Small Solar System Objects
Planets not only move along a fixed ellipse, but disturbances from other celestial bodies also affect their positions. The original book uses a set of periodic terms to organize these motion laws into calculable heliocentric longitude, latitude and distance.
Guess first
Who is the origin of the heliocentric position?
Show answer and explanation
sun — The heliocentric coordinates have the sun as the origin, not the horizon coordinates to which the telescope is pointed.
Then connect experience and science
First make it clear that the origin is the sun
L, B, and R are the heliocentric longitude, yellow latitude, and heliocentric distance respectively. It is not the same as the azimuth angle seen on Earth.
The center of the sun is the origin, and the reference date is the mean ecliptic
Each periodic term varies with time
Add up the items retained in Appendix III of the original book. The eight major planets here all use the truncation table in the book, not the complete VSOP87 thousands of items.
Each term A cos(B+Cτ), the angular parameter is in radians
Different orders multiplied by different time powers
First find the sum of each order such as L0 and L1, then multiply it by the power of τ, and finally scale it by 10⁸. Here τ is used as the Julian millennium and should not be misused as the Julian century.
τ=(JDE−2451545)/365250
Specify reference system when accuracy is required
Small fix for FK5 with switchable 32.3. The geocentric direction, light travel time and observation conditions belong to the next step, not the heliocentric L, B, R itself.
Dynamic ecliptic → FK5: Add small angle correction
Expand the calculation
τ = (JDE − 2451545)/365250 Sⱼ(τ) = Σᵢ Aⱼᵢ cos(Bⱼᵢ + Cⱼᵢτ) L弧Spend = (S₀ + S₁τ + S₂τ² + S₃τ³ + S₄τ⁴ + S₅τ⁵)/10⁸ LSpend = L弧Spend × 180/π
B and R perform the same summation according to their respective preserved orders; B is also in radians, and R is AU. A, B, and C are taken from the table in Appendix III of this lesson item by item, and the internal phase angles are expressed in radians. The summation structure is expanded here, and the 2430 coefficient terms are not listed line by line, nor is it a complete VSOP87.
give it a try
First use the Venus 1992 book example, and then switch to other planets; the points on the screen and the values below come from the same periodic summation.
No need to memorize symbols
- L / B
- Heliocentric longitude and yellow latitude, degrees
- R
- Heliocentric distance, AU
- τ
- J2000 and the Julian Millennium
Algorithm, accuracy and scope of application
Complete summation of all retained L/B/R periodic terms of the eight planets in Appendix III of the original book, totaling 2430 terms. L/B internal radian, output degree; R is AU, the amplitude coefficient is divided by 10⁸. Default reference date mean kinetic ecliptic and equinox; FK5 mode uses equation 32.3 correction. Only used in the teaching date range 1900–2100. The truncation table in the book is not described as a complete VSOP87, and the JPL model on the homepage is not automatically called; this lesson does not include geocentric conversion, light travel time, light aberration, nutation, atmospheric refraction or actual observation schedule.
my understanding check
What is the unit of τ here?
Julian Millennium — Every 365,250 days is one τ unit.
Is Appendix III in the book the complete VSOP87 theory?
No, it's a truncation item — This lesson completely uses the items retained in the book, but it is not the original and complete VSOP87 series.
Can the heliocentric longitude be directly used as the local compass azimuth?
Can't — The origin and reference plane are different, requiring additional coordinate transformation.
The date input must be correct, the series unit must be correct, and the origin of the coordinates must be correct; only when these three conditions are combined, the position number will be meaningful.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 32 chapter
Planets and Small Solar System Objects · All interactive courses