Stability of Lissajous orbits around Sun-Venus L1

Question

How far is it from the Venus?

Does Mercury gives too big perturbations for a stable Lissajous orbit?

Answer 1

I get SVL1 as 1,007,927 kilometers from Venus' center. But that's too much precision. Since Venus' orbit isn't perfectly circular, you don't have that many significant digits. I usually say about a million kilometers.

I've made a spreadsheet calculating L1 and L2 distances from various bodies orbiting the sun: Sun plus Mercury, Venus, Earth, Mars, Ceres, Jupiter, Saturn, Uranus, Neptune, and Pluto.

The spreadsheet also gives various planet moon L1 and L2 distances.

The spreadsheet is based on equations found on pages 133 to 138 of Szebehely's Theory of Orbits - The Restricted 3 Body Problem.

I don't think Mercury is that much of an influence. I would guess pressure from sunlight would have a greater effect.

But even if the sun and venus and probe were a perfect circular 3-body system, L1 would not be stable. You would need station keeping propellent in any event.

Answer 2

Lagrange points move with the distance between primary and secondary body's barycenter, so the exact distance between Venus and $\text{SVL}_1$ (Sun-Venus Lagrange point 1) would depend on Cytherean distance to the Sun in its orbit. For mean distance though, we can simplify this to Cytherean semi-major axis ($108,208,000\ \text{km}$ or $0.723327\ \text{AU}$). Using following formula (refer to The Lagrangian Points, An Application of Linear Algebra, Hannah Rae Kerner, 2013 for how it is derived):

$$L_1 = R \left(1 -\left(\sqrt[3\ \ \ ]{\frac{\alpha}{3}}\right)\right)$$

Where $\alpha$ is the ratio of $M_2$ (secondary body's mass) to the combined mass of the system ($M_1 + M_2$), or $\frac{M_2}{M_1 + M_2}$.

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So in our case, since Cytherean mass is $0.815\ M_E$ and the Sun's mass $333,000\ M_E$ ($M_E$ is Earth's mass), this mass ratio $\alpha$ is $2.4474 \cdot 10^{-6}$ and the mean $\text{SVL}_1$ distance to Venus is $-0.009344\ R$ or, to make it fit our formula, $0.99057\ R$ from the Sun (primary body).

So, in kilometers and astronomical units (AU), mean distance of $\text{SVL}_1$ is then:

• $-1,011,090\ \text{km}$ or $-0.00676\ \text{AU}$ from Venus, or
• $107,196,910\ \text{km}$ or $0.71657\ \text{AU}$ from the Sun

I calculated this using more precise values without cutting decimals off, but to confirm it, let's cheat a bit and use one of the online Lagrange point calculators. It assumes circular orbit (so only mean distance will be somewhat precise, like the one I calculate), and the figures for $a$ of $108,208,000\ \text{km}$, $M_1 = 1\cdot M_\odot$ (one solar mass) and $M_2 = 1\cdot M♀$ (mass of Venus) come out as:

• $-1,007,985\ \text{km}$ or $-0.00674\ \text{AU}$ from Venus, or
• $107,200,015\ \text{km}$ or $0.71659\ \text{AU}$ from the Sun

So fairly close, but I'm unsure what values that calculator uses as input, or at which point it cuts off decimal places so some small-ish discrepancy was expected.

If we also apply Cytherean orbital eccentricity to our calculations ($e = 0.0067$), we get a movement of the $\text{SVL}_1$ distance to Venus from $-1,004,259\ \text{km}$ or $-0.006713\ \text{AU}$ at perihelion to $-1,017,920\ \text{km}$ or $-0.006804\ \text{AU}$ at aphelion.

Or simply $-1,011,090\ \text{km} \pm 0.67\%$.

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I'll let others describe $\text{SVL}_1$ stability and how much it would be perturbed by Mercury, but I suspect gravitational perturbations by Earth and Jupiter to be more severe. Of course, just that the distance of Lagrange point 1 from the primary and secondary body varies with their distance between each other during one orbit of the secondary body (Venus) around the primary one (Sun) due to being non-circular makes this $\text{SVL}_1$ saddle point rather unstable. Lissajous or indeed Halo orbits around these gravitationally flat points in space only make stationkeeping somewhat easier and these instabilities (and satellite's own due to precision burn requirements) easier to control. They don't cancel them out in any way, regardless how feeble they might be.

Answer 3

Lissajous orbits and Halo orbits around L1 or L2 are very closely related. For Lissajous orbits, the in-plane period (left-right) and out-of-plane period (up-down) are uncoupled. It will still oscillate around the Lagrange point, but in a kind of "oscilloscope squiggle".

But for some range of distances the two periods will become equal and you end up with an (essentially) closed halo orbit which is roughly a 3D-bent ellipse shape.

Halo orbits are then just a subclass of Lissajous orbits where the two periods become equal.

In a pure circular, restricted 3 body scenario (CR3BP), there are some halo orbits that are truly stable. See answers to Are some Halo Orbits actually Stable?, one of which links to this answer to Is it possible to have stable orbits around Lagrange point L1? in Physics SE. These would then be a sub-class of a sub-class of Lissajous orbits.

However, since gravity is a long-range force everything attracts everything (eventually) so even those "stable halo orbits" of the CR3BP are not going to be stable in the real solar system. For three-body orbits associated with the Sun and Venus, I think the major perturbers will be Earth and Jupiter, since Mercury's mass is so small. But then again periods of L1/L2 halo orbits are often roughly half of the planet's orbit, and that puts the synodic period of Venus + Mercury (145 days) on the same scale as the halo orbit's period (of order 112 days) offering the possibility of some resonant effects in some cases, so this will be hard to answer with certainty without a real, full-blown calculation.

Answer 4

I had tried to solve this problem in a different way.
- Sun-Mercury 240000km (L1&L2 distance from orbiting body center) - Sun-Venus no SVL1&SVL2.
- Sun-Earth 1640000km.
- Sun-Mars 1180000km.
- Sun-Jupiter 58250000km.
- Sun-Saturn 71380000km.
- Sun-Uranus 76830000km.
- Sun-Neptune 126900000km.
- Earth-Moon 65000km.

Sun-Venus never has L1&L2 because of "gravity field with different steering".

Answer 5

I had post my answer last year but seems no one interested about that.Anyway,I'll give an other relational eqution about L1 of all planets in solar system. I'll use the Sun-Earth L1 of Senbehely's answer(149.75 borrow from Hopdavid's spreadsheet) to caculate all L1 of planets.

(Units:ten thousand Kilometers)

MercuryL1=149.75 x 0.387AU x 0.381=22.08

VenusL1=149.75 x 0.73AU x 0.935=102.21

EarthL1=149.75 x 1AU x 1=149.75

MarsL1=149.75 x 1.523AU x 0.475=108.36

JupiterL1=149.75 x 5.205AU x 6.825=5319.7

SaturnL1=149.75 x 9.537AU x 4.564=6518.15

UranusL1=149.75 x 19.2AU x 2.44=7015.5

NeptuneL1=149.75 x 30.05AU x 2.575=11587.5

The following is Victor Szebehely's answer V.S. solution of relational eqution

Mercury:22.04 vs 22.08

Venus:100.79 vs 102.21

Earth:149.75 vs 149.75

Mars:108.25 vs 108.36

Jupiter:5191.1 vs 5319.7

Saturn:6426.1 vs 6518.15

Uranus:6951.9 vs 7015.5

Neptune:11520 vs 11587.5

(Victor Szebehely's answer borrow from Hopdavid's spreadsheet and thanks)

the relational eqution is:
Distance of planet's L1=Earth's L1 x Orbital radius ratio x Cubic root of mass ratio

This equation is simple and more accurate(If the L1 of Earth is accurate).Because it doesn't have errors due to excessive mass of the planet.

I have to apologize because I am swearing to answer in this forum.So that you should hardly understand what I said.