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