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The accuracy of the prediction of a transit time depends on the method used to calculate it. Most planetarium programs, including Stellarium and Celestia, use a specific algorithm known as VSOP87 to calculate the positions of planets and other major bodies. It's a periodic series containing thousands of terms. (As I recall, Celestia truncates it to about a thousand terms in order to keep the calculation times reasonably short.) For details, see Its result is an approximation which gets worse and worse for times farther away from the present. Using a series approximation like this is necessary in order to minimize the amount of time used for the calculations.

A series approximation that's used by the space agencies is SPICE. It's appropriately calculated from "first principles" for each space mission and is much more accurate than VSOP. See Celestia can use it to position distant spacecraft within a few meters of their actual positions.

To "do it right" one would use a numeric integration of the gravitational effects of as many solar system objects as is practical, tracing their movements and gravitational interactions. This procedure is usually considered to be too "expensive", taking far longer than is reasonable for most situations.
Thanks for your comments. I checked a few of the dates that website gives for certain events. In some cases Stellarium's results are close, in some cases not close at all. So if I write a story which isn't consistent with what Stellarium shows, I hope you'll be tolerant?

I know we can write our own software program to use with Stellarium, but I haven't learned any current programming languages. Would I have to spend months learning the language before even beginning to write the program?

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