How Would These Planets Affect Earth's Eccentricity?

The best tools for this job, I think, are perturbation theory and Laplace's planetary equations. You might know that the orbit of a planet can be described by six osculating elements $(a,e,i,\omega,\Omega,\bar{M_0})$. These are, respectively, the semimajor axis, eccentricity, inclination, argument of periapsis, longitude of the ascending node, and the mean anomaly at epoch averaged over one period (sometimes replaced by the true anomaly, denoted by $\nu$, or the orbital period).

To calculate the effects of a perturbing body on these elements, we construct a function $\mathcal{R}$ known as the disturbing function or disturbing potential. We can write down $\mathcal{R}$ as a function of the planet's orbital elements and time (as the position of the perturbing body depends on time). We then take derivatives $\frac{\partial\mathcal{R}}{\partial a},\frac{\partial\mathcal{R}}{\partial e},...$ and plug them into the equations (Eq. 4.41 - 4.46 in the above notes). For instance, the equation for the evolution of eccentricity is $$\frac{de}{dt}=\frac{1-e^2}{ena^2}\frac{\partial\mathcal{R}}{\partial\bar{M_0}}-\frac{\sqrt{1-e^2}}{ena^2}\frac{\partial\mathcal{R}}{\partial\omega}$$ To solve your problem, assume some initial values for all six of those osculating elements - say, the current values for Earth. Put a copy of Earth in your planetary system, along with the perturbers, and evolve it through time, numerically integrating Laplace's equations as you go. This will show how the other planets affect the eccentricity. They almost certainly will yield different figures from the changes we see our version of the Solar System.

This is not an easy task, but it's doable, if you want to.

posted almost 5 years ago

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