How Would These Planets Affect Earth's Eccentricity?

Question

A cleaner, narrower-focused improvement of How This Alternate Solar System Influences the Milankovitch Cycle, so not a duplicate.

For further clarification, eccentricity is the shape of a planet's orbit, with the value of 0 being a perfect circle and 1 or more being a parabolic or hyperbolic oval. Milutin Milankovitch used Earth's eccentricity as one of three major factors responsible for the creation of ice ages, because the shape of Earth's orbit determines the length of each season. The sun's gravitational influence plays a part in shaping Earth's orbit, but it's not alone. The two gas giants--Jupiter and Saturn--also exert their share of influence regarding Earth's orbit. And Earth is the biggest of the inner planets, so the other three don't affect it as much. End result--an eccentricity varying between 0.000055 and 0.0679, with a logarithmic mean of 0.0019.

But in an alternate solar system, Earth, its moon and its place in the solar system are the exact same, but the others aren't. Here are the different bodies listed below:

1. A planet twice as wide and 8x as massive as Earth orbiting the sun from a distance of 1.13 million miles.
2. A planet 175% the width and 5.5x the mass of Earth orbiting the sun from a distance of 48.26 million miles.
3. A planet 2.7x as wide and 7x as massive as Earth orbiting the sun from a distance of 229 million miles.
4. A brown dwarf star twice as wide and 11x as massive as Jupiter orbiting the sun from a distance of 30 AUs.

Would these specified differences affect Earth's eccentricity in any way?

Answer 1

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.