## Interpolation

Note: Throughout this section, date means date, including the time of day.

An ephemeris of evenly spaced dates is required for all types of interpolation employed. The Ephemeris Interpolator will try and use a 3rd order interpolation method if there are enough ephemeris lines surrounding the required date.

## Determining the interpolation method

The method to be used is determined by locating the last ephemeris line with a date no greater than the required date, then

- If there are two or more ephemeris lines both before and after then a 3rd order interpolation method is used and will usually give the best accuracy.
- If there are two or more lines on one side but less than two on the other side, or just one line on either side then a 2nd order interpolation method is used.
- If there are just two ephemeris lines, with the required date between them then linear interpolation is used.
- Otherwise no interpolation is attempted and the error message "outside range of ephemeris" is displayed.

Note: When downloading MPES, NEOCP, Scout or Horizons ephemerides, the ephemeris start time and number of dates to be downloaded may be adjusted automatically to try and ensure interpolation via a 3rd order method.

The method being used is displayed in the Fit area of the Ephemeris Interpolator window and can include an estimate of the error in the calculated position due to the interpolation method, e.g. 3rd order~36".

## Estimated interpolation error

The estimated error is only given when 2nd or 3rd order interpolation methods are used, no error estimate can be provided for linear interpolation and for that case the Fit area will display "Linear" highlighted in red.

The error is given in arc-seconds ("), arc-minutes (') or degrees (°).

The estimated error provides an order of magnitude estimate of the size of error due to the interpolation process and is dependent on the ephemeris interval (step size) and the rates of change of RA and Dec.

The estimated error gives no indication of the actual precision of the ephemeris being used, the errors due to interpolation may be very small, but the ephemeris itself may not accurately represent the actual position of the object in the sky

When the estimated error is greater than one arc-minute the Fit area is highlighted in red, e.g. 2nd order~4.1'. Reduce the error by decreasing the ephemeris interval (step size).

If the Fit area indicates the interpolation is anything other than 3rd order, consider obtaining more ephemeris lines to better surround the required date (see Determining the interpolation method)

## Using astrometric lines to interpolate and extrapolate

If no ephemeris is to hand but some lines of astrometry are available then the astrometry (in MPC 80-column format) can be used in place of an ephemeris. This is only intended to be used:

- with astrometry obtained during a single night, over a short period of time
- with astrometry from one or a few nearby observatories, especially important for nearby objects that may have a large parallax
- to extrapolate only a short time outside of the range of dates covered by the astrometry

The first and last astrometry lines are used to linearly extrapolate the position to the required date, without taking into account any acceleration of motion and therefore will give increasingly inaccurate positions the further the required date is from the range of astrometric lines.

Ideally, if astrometric positions are available but no ephemeris is available, a program such as Find_Orb would be used to generate an ephemeris rather than extrapolating from astrometric positions.

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