> For the complete documentation index, see [llms.txt](https://kingsznhone.gitbook.io/vsop2013.net/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://kingsznhone.gitbook.io/vsop2013.net/vsop2013/original-readme/truncation-of-the-series.md).

# Truncation of the Series

The series VSOP2013 can be reduced with the elimination of terms less than a level of truncation ρ such as:

$$
(C^2+S^2)^{1/2}< ρ
$$

with the following units: au for a, radian for λ, without unit for the other variables k, h, q, p.

This table gives the largest differences, on the planetary heliocentric longitudes (L) in arcseconds, between a computation using the complete series and a computation with reduced series for 4 levels of truncation and 4 different time intervals.

Unit: arcsecond

## TIME INTERVALS

| Planet  | Truncation |   +1890 +2000 |   +0900 +3100 |   +0000 +4000 |   -4000 +8000 |
| ------- | ---------: | ------------: | ------------: | ------------: | ------------: |
| MERCURY |      10-08 | 0.67721"10-01 | 0.75801"10-01 | 0.17037"10 00 | 0.97520"10+01 |
| MERCURY |      10-10 | 0.15743"10-02 | 0.17826"10-02 | 0.38273"10-02 | 0.69811"10 00 |
| MERCURY |      10-12 | 0.48047"10-04 | 0.44625"10-04 | 0.14062"10-03 | 0.81597"10-01 |
| MERCURY |      10-14 | 0.81680"10-06 | 0.33789"10-05 | 0.10122"10-04 | 0.31690"10-01 |
| VENUS   |      10-08 | 0.53251"10-01 | 0.70522"10-01 | 0.12720"10 00 | 0.65818"10+01 |
| VENUS   |      10-10 | 0.14642"10-02 | 0.16925"10-02 | 0.46000"10-02 | 0.13975"10+01 |
| VENUS   |      10-12 | 0.41051"10-04 | 0.41504"10-04 | 0.16087"10-03 | 0.17887"10 00 |
| VENUS   |      10-14 | 0.19316"10-06 | 0.20458"10-05 | 0.79408"10-05 | 0.67518"10-01 |
| EMB     |      10-08 | 0.65292"10-01 | 0.70898"10-01 | 0.15865"10 00 | 0.29256"10+02 |
| EMB     |      10-10 | 0.19588"10-02 | 0.19376"10-02 | 0.62822"10-02 | 0.36086"10+02 |
| EMB     |      10-12 | 0.56497"10-04 | 0.63936"10-04 | 0.23802"10-03 | 0.24040"10 00 |
| EMB     |      10-14 | 0.24503"10-07 | 0.34754"10-06 | 0.97589"10-05 | 0.47410"10-01 |
| MARS    |      10-08 | 0.97685"10-01 | 0.14296"10 00 | 0.27820"10 00 | 0.75319"10+02 |
| MARS    |      10-10 | 0.24870"10-02 | 0.35107"10-02 | 0.11166"10-01 | 0.31709"10+02 |
| MARS    |      10-12 | 0.85882"10-04 | 0.89968"10-04 | 0.43410"10-03 | 0.78464"10 00 |
| MARS    |      10-14 | 0.27480"10-09 | 0.94669"10-07 | 0.36979"10-06 | 0.24848"10-02 |
| JUPITER |      10-08 | 0.72111"10-01 | 0.98043"10-01 | 0.43798"10 00 | 0.19935"10+03 |
| JUPITER |      10-10 | 0.23149"10-02 | 0.41402"10-02 | 0.18410"10-01 | 0.78025"10+02 |
| JUPITER |      10-12 | 0.52419"10-04 | 0.77432"10-04 | 0.95349"10-03 | 0.39318"10+02 |
| JUPITER |      10-14 | 0.18091"10-08 | 0.67189"10-07 | 0.12293"10-04 | 0.10433"10+02 |
| SATURN  |      10-08 | 0.85180"10-01 | 0.13503"10 00 | 0.83459"10 00 | 0.59209"10+03 |
| SATURN  |      10-10 | 0.44443"10-02 | 0.40293"10-02 | 0.36953"10-01 | 0.14434"10+03 |
| SATURN  |      10-12 | 0.58705"10-04 | 0.11242"10-03 | 0.99838"10-03 | 0.66771"10+02 |
| SATURN  |      10-14 | 0.68700"10-09 | 0.30549"10-07 | 0.15439"10-04 | 0.15582"10+02 |
| URANUS  |      10-08 | 0.12999"10 00 | 0.16070"10 00 | 0.41629"10 00 | 0.21533"10+03 |
| URANUS  |      10-10 | 0.28179"10-02 | 0.46278"10-02 | 0.15507"10-01 | 0.28820"10+02 |
| URANUS  |      10-12 | 0.56458"10-04 | 0.86700"10-04 | 0.54348"10-03 | 0.91941"10 00 |
| URANUS  |      10-14 | 0.15780"10-08 | 0.30961"10-07 | 0.29137"10-05 | 0.38149"10-01 |
| NEPTUNE |      10-08 | 0.11042"10 00 | 0.12789"10 00 | 0.24913"10 00 | 0.29114"10+02 |
| NEPTUNE |      10-10 | 0.24840"10-02 | 0.32288"10-02 | 0.91847"10-02 | 0.34946"10+01 |
| NEPTUNE |      10-12 | 0.60694"10-04 | 0.69311"10-04 | 0.44163"10-03 | 0.57928"10 00 |
| NEPTUNE |      10-14 | 0.86562"10-08 | 0.22378"10-06 | 0.99586"10-05 | 0.15162"10 00 |
| PLUTO   |      10-08 | 0.25573"10 00 | 0.29364"10 00 | 0.51874"10+01 | 0.93747"10+05 |
| PLUTO   |      10-10 | 0.26349"10-02 | 0.40748"10-02 | 0.27407"10 00 | 0.14000"10+05 |
| PLUTO   |      10-12 | 0.12252"10-04 | 0.30712"10-04 | 0.78821"10-02 | 0.36128"10+04 |
| PLUTO   |      10-14 | 0.45800"10-09 | 0.15325"10-06 | 0.19191"10-03 | 0.14328"10+03 |