How much does the orbit of Mars deviate when it is in linear conjunction with Jupiter and Saturn? [closed]

Is the gravity of the planets Jupiter and Saturn strong enough to alter the orbit of Mars, or do they just give a small wiggle after which Mars is following his former orbit?

• Take a look at $F = \frac{GMm}{r^2}$ and see what you think. Oct 18 '17 at 13:33
• @CarlWitthoft I think g = G*M/r^2 would be easier. You also have to account for tidal forces cause Jupiter doesn't just pull on Mars it pulls on both Mars and the Sun. Oct 18 '17 at 22:49

To properly answer this question, you should apply the 3 body problem, which, gets very long and mathy. It's technically not solved but it can be approximated with very good accuracy.

Applying basic Newtonian formulas can give you an estimate.

The sun has about 1,048 times the mass of Jupiter, and if we line up Mars at it's Aphelion (1.67 AU) to Jupiter at it's Perihelion, (4.95 AU). Mars at it's furthest and Jupiter at it's closest, Mars is still about twice as close to the Sun as it is to Jupiter.

For simplicity, I'm going to call Mars at Aphelion an "MAU" - Mars Aphelion Unit, so Mars, furthest from the sun, 1 "MAU", Jupiter closest to Mars, about 2 "MAU" and Jupiter closest to the Sun, 3 "MAU"

So, in perfectly lined up conditions, Jupiter never has more than 1/4,000th the gravitational pull on Mars than the Sun does and we also have to factor in the tidal force. It gets tricky here, cause all 3 objects exert a force on the other (or stretch space time which tells the objects where to go if you like the relativity version), but I'm going to stick with forces.

Jupiter tugs on Mars and the Sun at the same time so how far it pulls Mars from the sun is more of an an applied tidal force. At closest point, Jupiter's tug on Mars from 2 "MAU" and it's tug on the Sun from 3 "MAU" means it tugs on Mars 2.25 times stronger. Take the inverse, the effective gravitational pull - at closest, is the inverse of that, or 44%, so you can only apply 56% of Jupiter's gravitational pull towards Mars relative to the sun. And that's at peak. As Mars moves closer to the sun and/or further from Jupiter that number drops.

The effect of Jupiter on Mars orbit relative to the sun is never more than about 1 part in 7200 of that of the sun.

As Mars moved towards Jupiter in it's orbit it speeds up and effectively moves away from the Sun, and as it moves away from Jupiter, this effect is largely cancelled out over time. In effect it's a slight stretching when moving towards Jupiter and squashing when moving away, of Mars' elliptical orbit every time around the sun. These are sometimes called orbital perturbations.

1 part in 7,200 works out to about 0.14% and that's the peak Jupiter effect. Rounding down, a .01% variation on Mars' orbit seems a plausible estimate and that might even be a little high. Applying that to the diameter of Mars' orbit, that works out to a stretching/squashing of about 20,000 to 25,000 km. Compared to Mars natural variation based on it's eccentricity of over 26 million miles, that's pretty insignificant. This stretching and squashing is also largely balanced out as Jupiter circles around the sun.

Now if we apply the acceleration formula, I get a somewhat different number. g = G*M/r^2.

Gravitational Constant: 6.67E-11 Mass of Jupiter 1.898E27 kg r = distance between the two. Closest pass is about 3.28 AU, or 4.91E11 meters

The peak gravitational acceleration Mars feels from Jupiter is about 5.26E-7 m/s^2. That sounds negligible, but Mars takes 187 days to orbit the sun and it spends about half that time closer to Jupiter than the Sun is to Jupiter.

As a (bad) estimate, because that's the maximum gravitational force at closest pass, if we estimate that over a Mars year, that works out to about 90 days of "falling" towards Jupiter and 90 days falling away. Obviously the period is longer than 90 days but the force is less when Mars is further away so that's a guestimate. Apply (d=1/2at^2) to 90 days or about 7.8 million seconds. The distance using that formula comes to about 16,000 km.

So, both estimates are pretty close, one of 16,000 the other of 20-25,000 km. This also works well with the 4 or 5 significant figures usually given for Mars' obit. 4 figures on distance works out to nearest 100,000 km, which wouldn't be affected by Jupiter's stretching and squashing of Mars' orbit.

Now as for Saturn, at 30% the mass of Jupiter and roughly twice the distance from Mars and the Sun, you're looking at less than 8% of the gravitational force and with tidal effects figured in, maybe just 4%-6% the orbital perturbations. Jupiter is the one that really matters.

Orbital stretching and squashing is what lead to the discovery of the planet Neptune, because astronomers noticed that Uranus orbit was effected in a similar way to how Jupiter affects Mars. Observing those perturbations told them about where to look for Neptune and it was found not long after.

I want to point out that over tens of thousands of orbits, these perturbations can lead to changes in Mars' orbit, primarily it's eccentricity. It's semi-major axis and period of the orbit remain mostly unchanged, but it's eccentricity operates on two cycles. Mars' eccentricity varies on a 96,000 year and a two million year cycle. Like Mars, Earth's eccentricity also has cycles driven by the other planets, 95,000, 125,000 and 413,000 years. Mars' eccentricity is prone to greater variation, presumably (I think) because it gets closer to Jupiter.

As a rule, these eccentricity oscillations move back and forth unless two planets end up in resonance with each other. When that happens the variations can continue to grow leading to significant movement of one or both of the planets over millions of years. This is more likely to happen shortly after the solar-system forms. Our solar system is thought to be long-term stable with no orbital resonances between planets now, and none expected for at least a couple billion years. There's a small chance that Jupiter and Mercury will fall into resonance in a few billion years. Here's a nice detailed article on orbital eccentricity variation and stability of the solar system.

That's a ballpark answer anyway. Corrections to my math are welcome and if anyone can do the 3 body problem, feel free. That would be much more accurate, but I think 16,000 - 25,000 km per orbit is in the ballpark.