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JEphem Ephemeris trail Truncating VSOP87 |

So we want to compute the position of the planets, and have a certain accuracy. We call

We introduce an intermediary variable :

Let's see the extreme cases we can meet :

S : position of the Sun
E : position of the Earth P : real position of the planet P' : position of the planet affected of err _{Max} |

In case 1, the distance Earth - planet is minimal ; in case 2, it is maximal.

In both cases we have : | tg(α_{Max}) = d / EP | (formula 1) |

And EP in case 2 > EP in case 1 ; this implies that :

The same α

If we take into account the fact that the orbits are elliptic, let's see this situation for an outer planet :

(I didn't check if it was necessary to consider eccentricities here).

If we note :

We have :

Then : | EP = a_{P}(1 - e_{P}) - a_{E}(1 +e_{E}) | (formula 2.1) |

A similar drawing for an inner planet would lead to : | EP = a_{E}(1 - e_{E}) - a_{P}(1 +e_{P}) | (formula 2.2) |

For the Earth, the formula is : | ES = a_{E}(1 - e_{E}) | (formula 2.3) |

Formula 1 and formulae 2 permit to calculate

But

If we note : , we have : .

If we tolerate the same error on X, Y or Z, we have : ΔX = ΔY = ΔZ = err

Then :

As we have :

(formula 3)

(formula 4.1), for an outer planet.

(formula 4.2), for an inner planet.

(formula 4.3), for the Earth.

Here we introduce a new angle :

With the notations of step 1, we have :

On figure 3, we see that, in the worst case, the incertitude of α

We call :

We now calculate

Then : (formula 5)

The worst case (α maximal) happens when ES is maximal and EP is minimal.

So in formula 5, we should take ES = a

(Notice that we'll never have at the same time ES maximal and EP minimal).

We now have to choose

Taking

The method

`BuildVSOP87.calcAlphaMax()`

was written for this purpose.
This leads to the following result :

| (Table 1) |

Once we have these values, we can select the terms. We have :

As , the contribution of each term to the sum is lower than .

The condition to retain a term is then |

In this formula, err

Planet | T_{Max} |

MER VEN EAR MAR | 4 |

JUP SAT | 2 |

URA NEP | 6 |

I listed here the points that could lead to this situation. Maybe you'll have better ideas.