# Does it even make sense to talk about Mercury's triangular libration points (L4, L5)?

The recent question How much radiation shielding would be required for a habitat at Mercury–Sun L5? got me thinking. There are a large number of disadvantages and challenges to building or putting a large artificial structure so much closer to the Sun than at the distance of Earth's orbit, and the only advantage I can think of is that you'd have a whole lot of solar power available to deal with those challenges.

But I also wondered if there is even any expectation that there is any meaningful benefit to putting something at one of Mercury's triangular libration points Sun-Mercury L4 or L5 as compared to just putting it in a heliocentric orbit and ignoring Mercury altogether, except of course for not getting hit by it.

So I chose a point in space that follows Mercury's orbit, except that it trails it by 1/6 of a period, the approximate temporal equivalent of trailing by 60° would be for a circular orbit. I then calculated the acceleration it would experience from Mercury, Venus, and Earth for five years, and it turns out that the "perturbations" from Venus and Earth are always stronger, and often much stronger than any guiding or stabilizing effects from Mercury.

Question: So I'm wondering, does it make sense to talk about Mercury's triangular libration points (L4, L5)? Besides the linked question, has there ever been any proposed missions or even serious discussion about these locations? Or are they really best thought of as orbital mechanical red herrings?

below: python script and results using the package Skyfield. Dots in last plot are at years = 2024.14 when Venus comes closer than Mercury.  class Ob(object):
def __init__(self, name):
self.name = name

import numpy as np
import matplotlib.pyplot as plt
from skyfield.api import Loader, Topos, EarthSatellite

days  = np.arange(365.2564*5)
times = ts.utc(2020, 1, days)
times_trailing = ts.utc(2020, 1, days-88./6)
years = 2020 + days/365.2564

names = ['sun', 'mercury', 'venus',
'earth barycenter', 'mars barycenter',
'jupiter barycenter', 'saturn barycenter',
'uranus barycenter', 'neptune barycenter']

obs = []
for name in names:
ob = Ob(name.split())
obs.append(ob)
ob.ob = data[name]

for ob in obs:
ob.pos = ob.ob.at(times).ecliptic_position().km
if ob.name == 'mercury':
ob.pos_trailing = ob.ob.at(times_trailing).ecliptic_position().km

sun, mercury, venus, earth, mars = obs[:5]
jupiter, saturn, uranus, neptune = obs[5:]

GMs = [1.32712440018E+20, 2.2032E+13, 3.24859E+14,
3.986004418E+14 + 4.9048695E+12,
4.282837E+13, 1.26686534E+17, 3.7931187E+16,
5.793939E+15, 6.836529E+15]

for ob, GM in zip(obs, GMs):
ob.GM = GM

for ob in obs:
rsq = ((ob.pos - mercury.pos_trailing)**2).sum(axis=0)
ob.F = ob.GM / rsq
ob.r = np.sqrt(rsq)

if 1 == 1:
fig = plt.figure()

for ob in obs[1:4]:
ax.plot(years, ob.F, label=ob.name)
# ax.legend()
ax.set_title('acceleration (m/s^2)', fontsize=16)
ax.get_xaxis().get_major_formatter().set_useOffset(False)
ax.text(2020.2, 0.01, 'Mercury')
ax.text(2020.2, 0.04, 'Earth')
ax.text(2020.2, 0.13, 'Venus')

for ob in obs[1:4]:
ax.plot(years, ob.r, label=ob.name)
# ax.legend()
ax.set_title('distance (m)', fontsize=16)
ax.get_xaxis().get_major_formatter().set_useOffset(False)
ax.text(2020.2, 0.4E+08, 'Mercury')
ax.text(2020.2, 0.7E+08, 'Venus')
ax.text(2020.2, 2.2E+08, 'Earth')

fig.suptitle("in Mercury's orbit trailing by 88/6 days", fontsize=16)

plt.show()

if 1 == 1:
fig = plt.figure()
for ob in obs[1:4]:
x, y, z = ob.pos
ax.plot(x, y)
i = np.argmax(venus.F)
for ob in obs[1:4]:
x, y, z = ob.pos
ax.plot(x[i:i+1], y[i:i+1], 'ok')
x, y, z = mercury.pos_trailing
ax.plot(x[i:i+1], y[i:i+1], 'or')
x, y, z = sun.pos
ax.plot(x, y, '-k', linewidth=4)
ax.set_xlim(-2E+08, 2E+08)
ax.set_ylim(-2E+08, 2E+08)

# ax.legend()
ax.set_title('ecliptic projection (m)', fontsize=16)

fig.suptitle("in Mercury's orbit trailing by 88/6 days", fontsize=16)

plt.show()

• If you run it for a period of several (simulated) years, how's the stability? – Russell Borogove Jul 26 '17 at 5:06
• @RussellBorogove that's a good question. Short answer; it's probably no more or less stable than a similar-sized heliocentric orbit that ignores Mercury completely, or even Mercury itself for that matter! Full answer; a calculation like that would be fun but this is not that, it's not a simulation. I put fixed a "dummy object" traveling the same exact path as the JPL ephemeris puts Mercury, but exactly 88/6 days behind it. It's really just a "sniffer" to check the relative strength of the forces. – uhoh Jul 26 '17 at 5:21
• @RussellBorogove The striking result is that the influence of Mercury on an independent body would be so much smaller than the other two planets that it calls the whole idea of there being a meaningful Lagrange point there into question. – uhoh Jul 26 '17 at 5:21

You're right, the Sun-Mercury libration points (all five of them) are merely mathematical curiosities of a hypothetical two-body system. As you've calculated, the actual gravitational effects of Venus and Earth (both much larger than Mercury, but also farther away) make the two-body approximation "less than useful" for any real world systems.

Even the Earth-Moon libration points (L5 being the most widely known) are not exactly useful: If you read discussions of O'Neill's High Frontier space colonies, you will note that they don't sit at (e.g.) L5, but rather follow dumbell-shaped orbits around it because of the Sun's gravity. Even that is an approximation because it doesn't take into account the gravity of the other planets, but their effect is largely swamped by the Sun's gravity and is generally left out of current discussions where the colonies are (presently) only design studies. Once there are actually multiple colonies at L5, though, you can bet their management is going to be taking every effect into account they can measure to ensure there aren't any collisions!

EDIT

The calculations Lagrange derived that bear his name are for a two body system - and as your own results indicate, Mercury and the Sun cannot be considered a two body system because of other nearby massive objects - Venus and the Earth. Hence, Lagrange's calculations aren't applicable, and as a result of that, L1 - L5 basically don't exist with respect to Mercury.

• I'm not sure that of the Earth-Moon libration points, L4 and L5 are the most widely known, considering there have been two actual missions to them and both were to EML2. I think you mean more widely known to readers of science fiction? – uhoh Jul 27 '17 at 15:34
• I mean most widely known to the general population - O'Neill's work did a lot to popularize L5, I suspect most people don't know about L4, come to think of it, I will tweak my answer. The advantage of L4 and L5 is they are gravitationally stable - something put there will fall back toward the libration point, but things at L1-L3 need active maintenance to deal with perturbations. – FKEinternet Jul 27 '17 at 15:54
• Thanks again for your answer. Since I'm an obsessive stickler for sourced answers and one has just been posted, I should accept it instead. – uhoh Feb 23 at 1:59
• Fair enough, I didn't publish my results, which would make it rather hard to cite them. :) – FKEinternet Feb 24 at 3:40

Reference 1, citing Reference 2, reports that hypothetical trojan-type asteroids are invariably unstable at the proposed Mercury L4 and L5 points, whereas stable Lagrange-point librations at least over millions of years are available at the L4/L5 points of both Venus and Earth(+Moon). The predictions were made in the 1990s. The discovery of Earth and Venus trojans, but not (as yet) any Mercury trojans, appears to corroborate the predictions.

If true, it means Mercury's L4 and L5 points are not of any real celestial significance as they would not be able to trap liberating asteroids.

References

1. R. Dvorak and J. Henrad, The Dynamical Behavior of our Planetary System: Proceedings of the Fourth International Alexander von Humboldt Colloquium on Celestial Mechanics, 17-23 March 1996 (Springer Science & Business Media, 1997), p. 160.

2. S. Mikkola and K. Innanen, "A numerical exploration of the evolution of Trojan-type asteroidal orbits". Astron. J., 104 (1992), 1641–1649.