- The 5 L-points
of a (gravitationally bound)
2-finite-body system
are points
of mechanical equilibrium
for test particles
(i.e., astro-bodies which
are negligible in mass relative
to the masses of the
2 finite bodies) in the
a corotating reference frame
established by the 2 finite bodies.
In the special case of the animation,
the 2 bodies are
in astro-bodies
around their mutual center of mass (CM).
- To be clear,
the Lagrange points
corotate with
the 2 large
astro-bodies.
- Image 2 Caption: The 5 L-points
Sun-Earth
2-body system
and the trajectory
of the
Gaia spacecraft (mission 2013--2025?)
from Earth to the
L2 point
where it went into
Lissajous orbit
(which we describe below).
- The mechanical equilibria
are due to a combination of the
gravitational forces,
the centrifugal force,
and,
for the
L4 and L5 points,
the Coriolis force
(see Wikipedia: Lagrange point: Stability).
Lagrange points
are most easily understood for
circular orbits
for the 2 bodies are
in astro-bodies
and that is all we will consider in here.
- The L1,
L2,
and L3 points
are collinear with the
2 finite bodies.
The
L4 and L5 points
complete
equilateral triangles
with the
2 finite bodies.
This means they are 60° away from the line joining the
2 finite bodies as seen
from either of the 2 finite bodies.
- The 3
collinear
Lagrange points
were discovered by
Leonhard Euler (1707--1783)
and the other 2 by
Joseph-Louis Lagrange (1736--1813)
(see Wikipedia: Lagrange points:
History).
Both persons are famous
mathematicians and are the
eponyms of many
things in mathematics and
physics: e.g.,
Euler's formula,
Euler's number
(much more commonly known as the
exponential number e = 2.71828 ... ),
the Euler-Lagrange equation,
the
Euler-Mascheroni constant
γ = 0.5772156649 ... ,
Lagrangian mechanics,
and Lagrange multipliers.
- The
L1,
L2,
and L3 points are
unstable equilibria
(see Wikipedia: Lagrangian point:
Stability).
In principle,
any perturbation,
NO matter how small,
will cause
test particles to escape to
infinity.
- The
L4 and L5 points
are stable equilibria
due to the Coriolis force
(see Wikipedia: Lagrange point: Stability).
A stable equilibrium
is one where motions due sufficiently small
perturbation on a
test particle
placed at the
stable equilibrium
damp out and the
test particle average
position remains the
stable equilibrium.
The test particle NEVER escapes
to infinity.
- In fact, all physically real
stable equilibria
are metastable (i.e.,
a large enough perturbation will cause a
test particle to escape to
infinity)
and most (maybe all) physically real
unstable equilibria
are slightly metastable (i.e.,
a small enough perturbation will NOT cause a
test particle to escape to
infinity).
However, we customarily do NOT use the
term metastable state unless
the state is obviously NOT near the limits of absolute
stability
or instability.
- In planetary systems,
each planet
and its star
establishes
Lagrange points.
However, the
gravitational perturbations
of an
n-body system
with more than 2
astro-bodies
make the
Lagrange points less than ideal.
In general, the larger planet,
the larger its gravitational force,
and the more ideal its
Lagrange points.
For example, in the Solar System,
Jupiter's
L4 and L5 points
are particularly close to ideal, and therefore very
stable equilibria.
For that reason,
Jupiter's trojan asteroids
accumulated near its
L4 and L5 points
and tended to stay there over the
Solar System's evolution.
The word trojan
has come to mean any small
astro-body
(e.g., an asteroid) that
orbits near
L4 and L5 points.
- Now
test particles can
be kept at the
L1 point,
L2 point, and
L3 point by
orbital station-keeping
using orbital maneuvers (AKA burns)
using thruster burns
(see also
Wikipedia: Thrust,
Wikipedia: Rocket,
Wikipedia: Rocket engine,
and Wikipedia: Spacecraft propulsion).
- In fact, the
Earth's Lagrange points
Lagrange points
have become favorite locations for spacecraft
for special space missions
- In practice, it turns out that for some
space missions,
it is better to put
spacecraft into
Lissajous orbits
(or their special case halo orbits)
around L-points rather than
at L-points.
Lissajous orbits are
quasi-periodic orbits
around L-points
that are NOT in the
orbital plane
of the 2-body system.
In quasi-periodic orbits
the orbiting astro-body
is bound, but the orbital trajectory
never exactly repeats
(see also Wikipedia:
Orbit dynamics: asymptotically periodic orbit).
Lissajous orbits
are sort of like
quasi-periodic
epicycles come to life.
Lissajous orbits can
used for spacecraft at the
unstable
L1 point,
L2 point, and
L3 point
- The
Earth's Lagrange point
the L2 point
is currently a particular favorite
for important space missions.
This L2 point is on
Earth-Sun line
at 0.010 AU
(∼ 1.5*10**6 km) from the
Earth.
An astro-body
at that distance from the
Sun, NOT at an
L2 point would
have an orbital period
about the Sun
that is slightly longer than a
sidereal year = 365.256363004 days (J2000).
An astro-body
in Lissajous orbit
about L2
has, of course, an
orbital period
about the Sun of exactly a
sidereal year = 365.256363004 days (J2000).
- Note that Earth's
umbra
extends out to ∼ 0.009 AU
(∼ 1.4*10**6 km)
(see
Wikipedia:
Umbra, penumbra and antumbra: Umbra;
Richard Pogge:
Lecture 9: Eclipses of the Sun & Moon: scroll down ∼ 25 %).
So spacecraft
at the L2 point are
NEVER
in the umbra.
In fact, their Lissajous orbits
probably NEVER put them
in the penumbra
either.
Going into
penumbra would
decrease the solar power
that powers most
spacecraft.
- Examples of
spacecraft at the
Earth's Lagrange points:
- At the L2 point,
Gaia spacecraft (mission 2013--2025?)
(see also
gaia_2013_2025.html),
the
James Webb Space Telescope (JWST;
2021--2041?),
and
Euclid spacecraft (2023--2029?)
(see also
videos
Euclid in a nutshell (2023) | 1:14,
The Euclid Space Telescope:
tackling dark matter and dark energy mysteries (2023) | 14:54,
etc.).
- The 2
Lagrange spacecrafts (est. 2020s--2030s),
one at
L1 and
one at L5
(see Wikipedia:
Lagrange spacecrafts: Overview):
The European Space Agency (ESA)
is planning two spacecraft, the
Lagrange spacecrafts (est. 2020s--2030s),
which will study
solar weather
(and space weather generally)
and in particular monitor the Sun
in order to give early warning of
coronal mass ejections (CMEs).
How early?
Well hopefully days, but maybe sometime only hours
(see
Wikipedia:
Coronal mass ejection: Physical Properties).
Early warning is particularly necessary for
super coronal mass ejection (CMEs)
(e.g., the Solar storm of 1859
(AKA the Carrington event)
and the May 1921 geomagnetic storm
(AKA New York Railroad Storm of 1921))
which could burn out the
electrical power grids worldwide
which would be a major disaster.
Simply breaking the
electrical power grid
connections for a few hours while
a super CME
passes over could prevent said major disaster.
- Image 3 Caption: "An xample
of a Lissajous orbit
around
Sun-Earth
Lagrange L2 point."
(Slightly edited.)
- Image 4 Caption: "An animation
trajectory - Viewd from Earth
Wilkinson Microwave Anisotropy Probe
(WMAP, 2001--2010)
trajectory
as seen from Earth."
(Slightly edited.)
WMAP (2001--2010)
went into
Lissajous orbit
around
Sun-Earth
Lagrange L2 point.