# Are Lagrangian points associated only with the smaller body?

Diagrams of Lagrangian points I've seen always show the points near the smaller object, following its orbit about the larger one. For instance:

from Wikipedia.

But in fact the smaller object actually orbits about their common center of gravity, not about the larger object.

So even though they might be too close to be "useful", are there also similar Lagrangian points associated with the larger object in its orbit about the smaller?

Consider the Sun/Earth system as an extreme example, the Earth/Moon as moderate, and two objects with similar mass as the other extreme.

To simplify the question, temporarily ignore the effects of any external gravity (e.g. Jupiter) that would make the answer messy.

• Now that I've posted this and reread it a few times, I think I realize why it might actually be a "stupid question". But unless it gets a lot of downvotes and no answer, I'll leave it here in case someone else later has the same blindspot. Jun 18, 2019 at 19:03
• No. it's a great question, please keep it!
– uhoh
Jun 19, 2019 at 1:20
• @uho, it's amazing how easy it is to see the solution to a problem only after asking someone else. As I reread my question, it became so obvious that I still can't understand why I would ever need to ask: as the masses are made more equal, L3 moves closer, L2 moves farther away, L1 moves toward the midpoint, and L4 and L5 just follow along; the points associated with the smaller body are the points associated with the larger. Jun 19, 2019 at 1:53

The 5 points you show are the only ones in the Sun-Earth system, there is no other set "with respect to Sun". The set of points seems to be Earth-centric, just because the points are closer to the smaller of the bodies and we choose Sun as the center of rotation here. You can take the exact same drawing and revolve it around Earth or around the common barycenter - no matter which reference system you choose, you will end up with the same Lagrange points.

For L4/L5 you can directly see the symmetry, because Sun-L4-Earth is an equilateral triangle that doesn't change if you swap Sun and Earth. For the other three: These are points were the gravitational pull of the two bodies just adds up in the right way to create an orbit with the correct period. Again, there is a point between the two bodies, and one off to either side - and these do not change if you swap the two bodies. L1/L2 will always be closer to the lighter body, as its influence vanishes more quickly compared to the heavier body.

In the Circular Restricted Three-Body Problem or CR3BP (where the Lagrange points are defined) there are some conventions to make the math easier. With $$m_1 + m_2$$, the distance between $$m_1$$ and $$m_2$$, and the rotation rate all being unity (1.0) the reduced mass is defined as

$$\mu = \frac{m_2}{m_1 + m_2}.$$

So for the Earth-Moon system that's about 0.012, and for the Sun-Earth system it's about 3E-06. For equal-mass bodies, it's 0.5 or one-half.

It's most convenient to solve CR3BP problems by using a rotating frame, so we keep the positions of the objects fixed and in addition to the normal gravitational force terms we add pseudo-forces like centrifugal and Coriolis. The Coriolis force depends on velocity in the rotating frame and since we are looking for stationary points (another name for the Lagrange points) we can ignore it.

Expressed in terms of $$\mu$$ the x positions and masses of the two objects are then:

$$m_1 = 1 - \mu$$ $$m_2 = \mu$$ $$x_1 = -\mu$$ $$x_2 = 1 - \mu$$

The distance from a particle at $$x, y$$ to the two bodies is then

$$r_1 = \sqrt{(x + \mu)^2 + y^2}$$ $$r_2 = \sqrt{(x + \mu - 1)^2 + y^2}$$

and the acceleration components on the particle in the rotating fame (assuming zero velocity) are then

$$\ddot{x} = -(1-\mu) \frac{x+\mu}{r_1^3} - \mu \frac{x+\mu-1}{r_2^3} + x$$ $$\ddot{y} = -(1-\mu) \frac{y}{r_1^3} - \mu \frac{y}{r_2^3} + y$$

with those last terms being the centrifugal (pseudo)forces in the rotating frame, and the Coriolis force terms dropped becasue we are looking for stationary points.

The total acceleration $$\sqrt{\ddot{x}^2 + \ddot{y}^2}$$ will be drop to zero at the stationary or Lagrange points, so we just have to search for the five zeros in the total acceleration field.

In the plot below I show six cases from $$\mu=0.01$$ for an asymmetric system to $$\mu=0.5$$ for a pair of equal-mass bodies. The Lagrange points move slightly between the six cases but in a very predictable way; with L1 going to the center of mass of the two bodes for the equal mass case.

You can read further about the math in these (no particular order): 1, 2, 3, 4, 5, 6.

Python script:

class Lpoint(object):
def __init__(self, name, X0, mus, opts, tol):
self.name = name
self.X0   = X0
self.mus  = mus
self.mu0  = self.mus[0]
self.opts = opts
self.tol  = tol
self.Xs   = []
self.res  = []

self.solve_mus()
self.X    = np.vstack(self.Xs)
self.oks  = [res['success'] for res in self.res]
self.ok   = all(self.oks)

def solve_mus(self):
for mu in self.mus:
options=self.opts, tol=self.tol,
args=(mu))
X      = results['x']
self.X0 = X
self.Xs.append(X)
self.res.append(results)

def H(self, X, mu):
x, y   = X
r1 = np.sqrt((x + mu)**2 + (y)**2)
r2 = np.sqrt((x + mu - 1)**2 + (y)**2)
# zero velocity forces  # m1, m2 = 1-mu, mu
xddot = -( (1-mu) * (x+mu) / r1**3 + mu * (x+mu-1) / r2**3 ) + x
yddot = -( (1-mu) *    y   / r1**3 + mu *     y    / r2**3 ) + y
return np.sqrt(xddot**2 + yddot**2)

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import minimize

mus = [0.01, 0.02, 0.05, 0.1, 0.2, 0.5]

X0s   = ((0.7, 0), (1.2, 0), (-1, 0), (0.5, 1), (0.5, -1)) # arbitrary starting point
X0s   = [np.array(X0) for X0 in X0s]
names = ('L1', 'L2', 'L3', 'L4', 'L5')

opts  = {'xtol': 1E-12, 'disp': False, 'maxiter':1000, 'maxfev':2000}

Lpts = []
for (X0, name) in zip(X0s, names):
Lpt = Lpoint(name, X0, mus, opts, tol=1E-06)
Lpts.append(Lpt)

L1, L2, L3, L4, L5 = Lpts

for L in Lpts:
L.solve_mus()

all_oks = sum([L.oks for L in Lpts], [])

print ('All okay: ', all(all_oks), str(sum(all_oks)) + '/' + str(sum(all_oks)))

if True:
plt.figure()
for i, mu in enumerate(mus):
plt.subplot(3, 2, i+1)
x1, x2 = -mu, 1-mu
m1, m2 = 1-mu, mu
for L in Lpts:
x, y = L.X[i]
plt.plot([x], [y], '.k')
plt.plot([x1], [0], 'or', markersize = max(16 * m1, 6))
plt.plot([x2], [0], 'ob', markersize = max(16 * m2, 6))
plt.xlim(-1.5, 1.5)
plt.ylim(-1.0, 1.0)
plt.text(-1.4, 0.8, 'mu='+str(mu), fontsize=16)
hw = 0.1
plt.plot([-hw, hw], [0, 0], '-k', linewidth=0.5)
plt.plot([0, 0], [-hw, hw], '-k', linewidth=0.5)
plt.show()