# 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.

| TIME    | INTERVALS  | +1890 +2000 | +0900 +3100   | +0000 +4000   | -4000 +8000   |
| ------- | ---------- | ----------- | ------------- | ------------- | ------------- |
| MERCURY | Truncation | 10-08       | 0.67721"10-01 | 0.75801"10-01 | 0.17037"10    |
|         | Truncation | 10-10       | 0.15743"10-02 | 0.17826"10-02 | 0.38273"10-02 |
|         | Truncation | 10-12       | 0.48047"10-04 | 0.44625"10-04 | 0.14062"10-03 |
|         | Truncation | 10-14       | 0.81680"10-06 | 0.33789"10-05 | 0.10122"10-04 |
| VENUS   | Truncation | 10-08       | 0.53251"10-01 | 0.70522"10-01 | 0.12720"10    |
|         | Truncation | 10-10       | 0.14642"10-02 | 0.16925"10-02 | 0.46000"10-02 |
|         | Truncation | 10-12       | 0.41051"10-04 | 0.41504"10-04 | 0.16087"10-03 |
|         | Truncation | 10-14       | 0.19316"10-06 | 0.20458"10-05 | 0.79408"10-05 |
| EMB     | Truncation | 10-08       | 0.65292"10-01 | 0.70898"10-01 | 0.15865"10    |
|         | Truncation | 10-10       | 0.19588"10-02 | 0.19376"10-02 | 0.62822"10-02 |
|         | Truncation | 10-12       | 0.56497"10-04 | 0.63936"10-04 | 0.23802"10-03 |
|         | Truncation | 10-14       | 0.24503"10-07 | 0.34754"10-06 | 0.97589"10-05 |
| MARS    | Truncation | 10-08       | 0.97685"10-01 | 0.14296"10    | 00            |
|         | Truncation | 10-10       | 0.24870"10-02 | 0.35107"10-02 | 0.11166"10-01 |
|         | Truncation | 10-12       | 0.85882"10-04 | 0.89968"10-04 | 0.43410"10-03 |
|         | Truncation | 10-14       | 0.27480"10-09 | 0.94669"10-07 | 0.36979"10-06 |
| JUPITER | Truncation | 10-08       | 0.72111"10-01 | 0.98043"10-01 | 0.43798"10    |
|         | Truncation | 10-10       | 0.23149"10-02 | 0.41402"10-02 | 0.18410"10-01 |
|         | Truncation | 10-12       | 0.52419"10-04 | 0.77432"10-04 | 0.95349"10-03 |
|         | Truncation | 10-14       | 0.18091"10-08 | 0.67189"10-07 | 0.12293"10-04 |
| SATURN  | Truncation | 10-08       | 0.85180"10-01 | 0.13503"10    | 00            |
|         | Truncation | 10-10       | 0.44443"10-02 | 0.40293"10-02 | 0.36953"10-01 |
|         | Truncation | 10-12       | 0.58705"10-04 | 0.11242"10-03 | 0.99838"10-03 |
|         | Truncation | 10-14       | 0.68700"10-09 | 0.30549"10-07 | 0.15439"10-04 |
| URANUS  | Truncation | 10-08       | 0.12999"10    | 00            | 0.16070"10    |
|         | Truncation | 10-10       | 0.28179"10-02 | 0.46278"10-02 | 0.15507"10-01 |
|         | Truncation | 10-12       | 0.56458"10-04 | 0.86700"10-04 | 0.54348"10-03 |
|         | Truncation | 10-14       | 0.15780"10-08 | 0.30961"10-07 | 0.29137"10-05 |
| NEPTUNE | Truncation | 10-08       | 0.11042"10    | 00            | 0.12789"10    |
|         | Truncation | 10-10       | 0.24840"10-02 | 0.32288"10-02 | 0.91847"10-02 |
|         | Truncation | 10-12       | 0.60694"10-04 | 0.69311"10-04 | 0.44163"10-03 |
|         | Truncation | 10-14       | 0.86562"10-08 | 0.22378"10-06 | 0.99586"10-05 |
| PLUTO   | Truncation | 10-08       | 0.25573"10    | 00            | 0.29364"10    |
|         | Truncation | 10-10       | 0.26349"10-02 | 0.40748"10-02 | 0.27407"10    |
|         | Truncation | 10-12       | 0.12252"10-04 | 0.30712"10-04 | 0.78821"10-02 |
|         | Truncation | 10-14       | 0.45800"10-09 | 0.15325"10-06 | 0.19191"10-03 |

Unit: arcsecond
