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Which means that I can draw an imaginary plane that contains the major-axis of both Earth and Mars.

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    $\begingroup$ Hint: the major axes intersect at the Sun. $\endgroup$
    – PM 2Ring
    Aug 18 '21 at 6:21
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    $\begingroup$ @PM2Ring Thanks, I got it! $\endgroup$
    – LIdbioe
    Aug 18 '21 at 6:35
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Yes, as PM 2Ring hinted, the major axes of the planets intersect at the Sun (more precisely, at the solar system's center of mass). Therefore, given any two planets there is a plane that that contains their major-axes.

Of course, this statement assumes the approximation that orbits are perfect ellipses.


Edit: Prompted by M. A. Golding's answer, I will attempt to find this plane and figure out how it looks like.

I found the orbital elements of Earth and Mars in the paper Numerical expressions for precession formulae and mean elements for the Moon and the planets.

At page 675, we find the value of the longitude of perihelion $\varpi$ and inclination $i$ of the planets. $\varpi$ is the angle between the vernal equinox, the focus of the orbit and the perihelion of the orbit. $i$ is the angle between the plane of the orbit and the ecliptic at J2000. I report the data here, hoping that I didn't copy anything wrong.

Earth:

$\varpi = 102.93734808° + 11612.35290'' t + 53.27577'' t^2 - 0.14095'' t^3 + 0.11440'' t^4 + 0.00478'' t^5$

$i = 469.97289'' t - 3.35035'' t^2 - 0.12374'' t^3 + 0.00027'' t^4 - 0.00001'' t^5 + 0.00001'' t^6$

Mars:

$\varpi = 336.06023395° + 15980.45908'' t - 62.32800'' t^2 + 1.86464'' t^3 - 0.04603'' t^4 - 0.00164'' t^5$

$i = 1.84972648° -293.31722'' t - 8.11830'' t^2 - 0.10326'' t^3 -0.00153'' t^4 + 0.00048'' t^5$

In these formulas, $t$ is the time in thousands of Julian days from J2000: $$t = (\text{JD} - 2451545)/365250$$

First question: What are the values of the orbital parameters right now?

On 2021 January the first, the values where:

$$\varpi_{earth} = 103.0°, \varpi_{mars} = 336.2°$$ $$i_{earth} = 0.003°, i_{mars} = 1.848°$$

So we can say that the semi-major axes of Earth and Mars are not currently aligned, but have about 336-180-103 = 53° of separation.

a sketch of the orbit

This is just a sketch of the orbits to show the relative position of the semi-major axes. The ellipses are not accurate.

Are they going to align soon due to precession of the line of the apsis?

Precession of the longitude of perihelion

This figure shows that they are not going to be aligned for the next 20000 years or so. I have marked with red dashed lines the region where the error on the model is smaller than $1''$ (from 4000 BCE to 8000 CE). (At least, this is what I understood from the paper, I'm not 100% sure of this)

To complete the answer, let's calculate the inclination of the plane that contains both semi-major axes. The direction of the axis is defined by the vector that points from the Sun to the perihelion of the orbits: $$\vec{v} = (\cos(\varpi), \sin(\varpi), \sin(i))$$

The vector $\vec{w} = \vec{v}_{earth} \times \vec{v}_{mars}$ will be orthogonal to the plane that contains the semi-major axes. Its inclination will just be the angle between $\vec{w}$ and the z-axis.

$$\vec{w} = (0.0314, 0.0073, -0.8002)$$

The inclination is

$$i_{plane} = \arccos \left( \frac{\hat{z} \cdot \vec{w}}{|\vec{w}|} \right) = \arccos \left( \frac{w_z}{|\vec{w}|} \right) \approx 2.31°$$

This result shows that there is indeed a plane that contains both the semi-major axes of Earth and Mars, and it is currently close to the ecliptic plane and to Mars orbital plane. I don't know how this plane could be useful, but this was the OP's question.

The last figure is the evolution in time of the inclination of the plane: inclination of the plane as a function of time

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  • $\begingroup$ I think this answer is wrong and the planes of the orbits are tilted. Thus there is one line where the two planes interset. And two separate semi-major sexes can not exist in one 1-D line, unless by coincidence the longer axis passes through and beyond the shorter axis. Which would be a big coincidence. $\endgroup$ Aug 18 '21 at 22:11
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    $\begingroup$ I guess you can create a plane that contains both the semi-major axes - As you have stated yourself, it is possible to draw such plane. This plane does not coincide with either Earth's orbit plane or Mars' orbit plane, but the OP just asked if such plane existed. Without additional information I cannot judge how this imaginary plane will be useful for the OP $\endgroup$
    – Prallax
    Aug 19 '21 at 5:37
  • $\begingroup$ @MAGolding sorry, forgot to notify you $\endgroup$
    – Prallax
    Aug 19 '21 at 5:40
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An semi-major axis of a planet is a one dimensional line contained within the two dimensional plane of the panet's orbt. A planet's orbital plane is a 2 dimensional plane in three dimensional space.

Since the orbits of the planets are all tilted slightly with respect to each other, the orbital planes of any two planets are aligned differently in 3 demensional space. Thus the orbital planes of two planets will intersect in a one dimensional line.

Since the semimajor axis of one planet, like Earth, is entirely within the plane of the planet's orbit, and the semi-major axis of another planet, like Mars, is entirely within the different plane of the second planet's orbit, the two semi-major axes can not be co planer -at least not within the different planes of their planetary orbits.

And since the two lines containing the semi-major axes do intersect, I guess you can create a plane that contains both the semi-major axes. But that plane will not be identical to the orbital pane of either planet, let along both orbital planes. And as far as I can see, that third plane should be almost at a right angle to the two very close, but different, planes of the planetary orbits.

Imagine that the planes of the orbits of Earth and Mars were parellel and separated by about a million miles. If their orbital major axes were lined pointing in the same direction, a plane passing thorugh both the orbital major axes would be perpendicular to the two planes of the planetary orbits.

And in real life the situation is similar.

The plane of Earth's orbit is almost exactly the same as the plane of Mars's orbit. But they are titled slightly. Thus on one side the plane of Earth's orbit will be "above" the plane of plane of Mars's orbit, and on the other side the plane of Mars's orbit will be "above" the plane of Earth's orbit.

Thus a third plane containing lines in both of those separate orbital planes will have be angled at close to a right aangle to either of those two separate orbital planes.

So if you try to draw the two planetary orbits within a plane which contrains both major axes, both planetary orbits will be very narrow ellipses, close to straight lines, since that plane will be angled at almost 90 degrees to the two planes in which the two planets orbit. The shapes of the orbits of the two planets would be depicted very inaccurately.

It would be far more accurate to pretend the two planetary orbits are in the same plane and draw them that way.

And this discussion ignores the directions of the major axes of the orbits of the Earth and Mars. There is little reason to suppose that they happen to be lined up at any particula moment.

In fact I believe that due to gravitational perturbations by other planets, the line of the major axis of every planet precesses.

In celestial mechanics, apsidal precession (or apsidal advance)1 is the precession (gradual rotation) of the line connecting the apsides (line of apsides) of an astronomical body's orbit. The apsides are the orbital points closest (periapsis) and farthest (apoapsis) from its primary body. The apsidal precession is the first time derivative of the argument of periapsis, one of the six main orbital elements of an orbit. Apsidal precession is considered positive when the orbit's axis rotates in the same direction as the orbital motion. An apsidal period is the time interval required for an orbit to precess through 360°.[2]

https://en.wikipedia.org/wiki/Apsidal_precession[1]

So as the major axes of the orbits of Mars and Earth precess around the Sun or the barycenter of the Solar System, the direction between them constantly changes and thus the orientation of any plane which could contain both those lines should also change drastically.

Added 08-21-21. Prallax's revised answer show that at the present time the plane containng the major axes of the orbits of Mars and Earth is inclined only a couple of degreess from the planes of their orbit. But thousands of years in the future, when the major axes are lined up, that plane should be close to perpendicular to the two orbital planes.

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    $\begingroup$ that third plane should be almost at a right angle to the two very close - I do not agree. Assuming this picture is correct, I would say that the common plane of the axes intersects the ecliptic with a small angle, not close to 90 degrees. images.app.goo.gl/UkqYkYbLniVqiGyKA it would be interesting to do the proper calculation and settle the matter $\endgroup$
    – Prallax
    Aug 19 '21 at 5:29
  • $\begingroup$ @Prallax I have added several paragraphs to my answer. $\endgroup$ Aug 20 '21 at 17:18
  • $\begingroup$ @M.A.Golding I have edited my answer too. You had made me curious about this and so I have decided to do the proper calculation. Let me know what you think $\endgroup$
    – Prallax
    Aug 20 '21 at 21:04
  • $\begingroup$ @Prallax. I guess you are right. but I expect that the angle of thire plane containing the two major axes may change from time to time as the axes precess. I think that when the axes are lined up the third plane should be almost perpendicular to the orbitial planes,. . $\endgroup$ Aug 21 '21 at 18:10
  • $\begingroup$ I totally agree with you on this. Indeed the question in the OP was a bit peculiar $\endgroup$
    – Prallax
    Aug 21 '21 at 18:48

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