When two planets pass by each other, how do you find the closest moment?
Don’t directly connect the daily distances into a parabola; first find where they are at each moment.
Sky coordinates and observations
The two planets were photographed getting closer and farther each day, and the latest moment may fall between the two records. The Mercury-Saturn example in the original book illustrates: the seemingly reasonable "direct interpolation distance" will calculate the real small interval to be nearly eight times larger.
Guess first
Is calculating the daily angular distances first and then interpolating these angular distances necessarily as accurate as interpolating the coordinates first?
Show answer and explanation
Not necessarily. It’s especially easy to make a mistake when passing by at close range. — The angular distance is a nonlinear function of coordinates, and its transition cannot be replaced by widely spaced low-order polynomials during close passes.
Then connect experience and science
Keep the celestial coordinates first, instead of just remembering the distance
The original book gives the five-day right ascension and declination of Mercury and Saturn from September 12 to 16, 1978. Both bodies need to use the same time scale and reference system.
First interpolate δ₁, δ₂ and the right ascension difference
When approaching a pass, the distance curve is not a large parabola
The two points first approach and then separate, and the distance changes and turns in a very narrow range. Fitting daily distance values directly to a polynomial misses the truly small intervals.
Error demonstration: direct interpolation of angular distance
Find the minimum value of the two correct methods respectively
Calculate the spherical angular distance after five-point coordinate interpolation; or three-point local u/v interpolation, and iterate to make the change rate of the distance approach zero.
Δn=−(uu′+vv′)/(u′²+v′²)
Reported along with direction and accuracy
Minimum time, minimum angular distance, and position angle are different quantities. There will be a small difference between the approximation of three points and five points, and you cannot pretend to be accurate by displaying more decimals.
The position angle is measured from celestial north to east of the first celestial body
u and v represent the relative positions on the local tangent plane, and the apostrophe is the derivation of the interpolation parameter n. Interpolate the coordinates first and then find the closest moment. When the denominator is zero, Δn cannot be calculated directly; the angular distance still needs to be checked in the end.
give it a try
First switch the method to "Error Demonstration" to see why the orange curve deviates; then look at the five-point coordinate method and the three-point u/v method to distinguish the approximation errors between the two.
No need to memorize symbols
- n
- Daily offset relative to 1978-09-14 0h TD
- u / v
- Local relative coordinates of the original book, arc seconds
- P
- Position angle of Saturn relative to Mercury, 0–360° from north to east
- TD
- The original book Dynamics time; this lesson does not change the local time zone
Algorithm, accuracy and scope of application
Complete the main algorithm of the minimum distance in Chapter 17: The five-day Mercury/Saturn right ascension and declination of the original book are firstly made into five-point Lagrange interpolation and then used atan2 spherical angular distance; the extreme value is independently approximated using bounded golden minute search. In addition, the K/u/v expression, three-point differential derivative and Δn iteration on page 113 are fully implemented, retaining the three-point/five-point error difference. The daily angular distance of the same table is directly interpolated four times as a clear demonstration of the error. Including atan2 quadrant processing of position angle, same/reverse degradation protection, and haversine equivalence verification. The coordinates in the table are not mixed with star catalogs from another epoch; the minimum values for different modes come from different approximations and cannot claim the same accuracy.
my understanding check
What should be the priority for interpolation in this lesson?
coordinate — The angular distance is then calculated from the interpolated coordinates.
There is a little difference between the three-point u/v method and the five-point coordinate method. What does it mean?
Sampling is different from approximation methods and requires an explanation of accuracy — Results should be described together with the algorithm and data accuracy.
Is position angle the same as minimum angular distance?
Different, the former describes the relative orientation — The position angle is measured from celestial north to east at the reference celestial body, and the angular distance describes how far apart the two lines of sight are.
To find the closest moment of passing, first calculate the positions of the two celestial bodies, and then calculate the distance between them.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 17 chapter