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I'm developing a C++ computer library with the formulas in the book "Practical Astronomy with your Calculator or Spreadsheet 4th Edition" but I have a problem with the formula 56, "Perturbations in a planet's orbit".

The book talks about using the Kepler's equation to calculate the eccentric anomaly, E. To do it I have implemented this routine:

double TheSun::RoutineR2(double meanAnomaly, double eccentricity)
{
    double E = 0.0;
    double aux = 0.0;
    double delta = 0.0;

    // Epsilon is the required accuracy (= 10^-6 radians).
    double epsilon = 0.000001;

    // 1. First guest, E = E0 = M.
    E = aux = meanAnomaly;
    
    do
    {
        // 2. Find the value of delta = E - esin(E) - M;
        delta = aux - eccentricity * std::sin(aux) - meanAnomaly;

        // 3. If delta gets enough accuracy, end here.
        if (std::abs(delta) <= epsilon)
            break;

        // 4. Find E growth.
        double growth = delta / (1 - eccentricity * std::cos(aux));

        // 5. Take the new value.
        aux = aux - growth;

    } while (true);

    return aux;
}

When I try to find the eccentric anomaly, E, for planet Jupiter on 22 November 2003, I get that E is equal to 8.7201007604944394 but, in the book, the value for E is 2.436915 radians.

The input value for RoutineR2 are:

meanAnomaly = 8.6884193953285500
eccentricity = 0.048906999999999999

The values above are the same than the book uses.

I have calculated the mean anomaly and I get the same value than the book. The eccentricity is the eccentricity of the orbit of Jupiter. I take it from the Table 8 in the book. This table has the values of Elements of the planetary orbits at epoch 2010.0.

Is the book wrong? Am I wrong?

Maybe my implementation of Kepler's equation is wrong or I'm not using the right value for eccentricity.

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    $\begingroup$ The answer for Eccentric Anomaly looks reasonable given the input for Mean Anomaly and the low eccentricity; Mean anomaly and eccentric anomaly are going to be pretty close in value under those circumstances. How did you get your value for Mean Anomaly on 22 November 2003? $\endgroup$
    – notovny
    Commented Dec 31, 2021 at 11:34

1 Answer 1

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The difference in values seems to be only $2\pi$, or one complete turn. So your value seems to be correct.

Here is my calculation:

Jupiter was at perihelion on March 17th 2011, which is 2672 days after your date. Jupiter has a period of 4330 days, so I calculate the mean anomaly to be $$(4330-2672)/4330 × 2\pi=2.4059$$

But it seems you are calculating the Mean Anomaly from the 1987 perihelion, so add $2\pi$ to give $M=8.689$ (that is very close to your value and the differences are probably because you've been more careful than me in rounding etc)

While I feel you should reduce this to the range 0-2pi, you should be able to solve Kepler's equation with this value, and your solution is correct (I checked by graphing M = E-e sin(E) online with GeoGebra). As expected with a nearly circular orbit (e=0.0489), $M\approx E=8.72$ I'd still reduce this to the range 0-2pi, to get the book value of E=2.43.

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  • $\begingroup$ Thanks for your answer. My answer for all of your question is yes. I have updated my question with more details. $\endgroup$
    – VansFannel
    Commented Dec 31, 2021 at 13:49
  • $\begingroup$ If I reduce the value of 8.6884193953285500 to 2.4052341153285512 (I subtract it 2pi), I get the same value than in the book. $\endgroup$
    – VansFannel
    Commented Dec 31, 2021 at 14:04
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    $\begingroup$ Then what is 5.527600? $\endgroup$
    – James K
    Commented Dec 31, 2021 at 14:28
  • $\begingroup$ My mistake..... $\endgroup$
    – VansFannel
    Commented Dec 31, 2021 at 15:49
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    $\begingroup$ Meeus Astronomical Algorithms is probably the best known reference, but getting hold of a copy now may be difficult. worldcat.org/title/astronomical-algorithms/oclc/1110649364 $\endgroup$
    – James K
    Commented Jan 4, 2022 at 18:14

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