Equilibrium points near two orbiting bodies
In
celestial mechanics
, the
Lagrange points
(
; also
Lagrangian points
or
libration points
) are points of
equilibrium
for small-mass objects under the
gravitational
influence of two massive
orbiting
bodies. Mathematically, this involves the solution of the
restricted three-body problem
.
[1]
Normally, the two massive bodies exert an unbalanced gravitational force at a point, altering the orbit of whatever is at that point. At the Lagrange points, the
gravitational
forces of the two large bodies and the
centrifugal force
balance each other.
[2]
This can make Lagrange points an excellent location for satellites, as
orbit corrections
, and hence fuel requirements, needed to maintain the desired orbit are kept at a minimum.
For any combination of two orbital bodies, there are five Lagrange points, L
1
to L
5
, all in the orbital plane of the two large bodies. There are five Lagrange points for the Sun?Earth system, and five
different
Lagrange points for the Earth?Moon system. L
1
, L
2
, and L
3
are on the line through the centers of the two large bodies, while L
4
and L
5
each act as the third
vertex
of an
equilateral triangle
formed with the centers of the two large bodies.
When the mass ratio of the two bodies is large enough, the L
4
and L
5
points are stable points, meaning that objects can orbit them and that they have a tendency to pull objects into them. Several planets have
trojan asteroids
near their L
4
and L
5
points with respect to the Sun;
Jupiter
has more than one million of these trojans.
Some Lagrange points are being used for space exploration. Two important Lagrange points in the Sun-Earth system are L
1
, between the Sun and Earth, and L
2
, on the same line at the opposite side of the Earth; both are well outside the Moon's orbit. Currently, an
artificial satellite
called the
Deep Space Climate Observatory
(DSCOVR) is located at L
1
to study solar wind coming toward Earth from the Sun and to monitor Earth's climate, by taking images and sending them back.
[3]
The
James Webb Space Telescope
, a powerful infrared space observatory, is located at L
2
.
[4]
This allows the satellite's large sunshield to protect the telescope from the light and heat of the Sun, Earth and Moon. The L
1
and L
2
Lagrange points are located about 1,500,000 km (930,000 mi) from earth.
The European Space Agency's earlier
Gaia
telescope, and its newly launched
Euclid
, also occupy orbits around L
2
. Gaia keeps a tighter
Lissajous orbit
around L
2
, while Euclid follows a
halo orbit
similar to JWST. Each of the space observatories benefit from being far enough from Earth's shadow to utilize solar panels for power, from not needing much power or propellant for station-keeping, from not being subjected to the Earth's magnetospheric effects, and from having direct line-of-sight to Earth for data transfer.
History
[
edit
]
The three collinear Lagrange points (L
1
, L
2
, L
3
) were discovered by the Swiss mathematician
Leonhard Euler
around 1750, a decade before the Italian-born
Joseph-Louis Lagrange
discovered the remaining two.
[5]
[6]
In 1772, Lagrange published an "Essay on the
three-body problem
". In the first chapter he considered the general three-body problem. From that, in the second chapter, he demonstrated two special
constant-pattern solutions
, the collinear and the equilateral, for any three masses, with
circular orbits
.
[7]
Lagrange points
[
edit
]
The five Lagrange points are labelled and defined as follows:
L
1
point
[
edit
]
The L
1
point lies on the line defined between the two large masses
M
1
and
M
2
. It is the point where the gravitational attraction of
M
2
and that of
M
1
combine to produce an equilibrium. An object that
orbits
the
Sun
more closely than
Earth
would typically have a shorter orbital period than Earth, but that ignores the effect of Earth's gravitational pull. If the object is directly between Earth and the Sun, then
Earth's gravity
counteracts some of the Sun's pull on the object, increasing the object's orbital period. The closer to Earth the object is, the greater this effect is. At the L
1
point, the object's orbital period becomes exactly equal to Earth's orbital period. L
1
is about 1.5 million kilometers, or 0.01
au
, from Earth in the direction of the Sun.
[1]
L
2
point
[
edit
]
The L
2
point lies on the line through the two large masses beyond the smaller of the two. Here, the combined gravitational forces of the two large masses balance the centrifugal force on a body at L
2
. On the opposite side of Earth from the Sun, the orbital period of an object would normally be greater than Earth's. The extra pull of Earth's gravity decreases the object's orbital period, and at the L
2
point, that orbital period becomes equal to Earth's. Like L
1
, L
2
is about 1.5 million kilometers or 0.01
au
from Earth (away from the sun). An example of a spacecraft designed to operate near the Earth?Sun L
2
is the
James Webb Space Telescope
.
[8]
Earlier examples include the
Wilkinson Microwave Anisotropy Probe
and its successor,
Planck
.
L
3
point
[
edit
]
The L
3
point lies on the line defined by the two large masses, beyond the larger of the two. Within the Sun?Earth system, the L
3
point exists on the opposite side of the Sun, a little outside Earth's orbit and slightly farther from the center of the Sun than Earth is. This placement occurs because the Sun is also affected by Earth's gravity and so orbits around the two bodies'
barycenter
, which is well inside the body of the Sun. An object at Earth's distance from the Sun would have an orbital period of one year if only the Sun's gravity is considered. But an object on the opposite side of the Sun from Earth and directly in line with both "feels" Earth's gravity adding slightly to the Sun's and therefore must orbit a little farther from the barycenter of Earth and Sun in order to have the same 1-year period. It is at the L
3
point that the combined pull of Earth and Sun causes the object to orbit with the same period as Earth, in effect orbiting an Earth+Sun mass with the Earth-Sun barycenter at one focus of its orbit.
L
4
and L
5
points
[
edit
]
The L
4
and L
5
points lie at the third vertices of the two
equilateral triangles
in the plane of orbit whose common base is the line between the centers of the two masses, such that the point lies 60° ahead of (L
4
) or behind (L
5
) the smaller mass with regard to its orbit around the larger mass.
Stability
[
edit
]
The triangular points (L
4
and L
5
) are stable equilibria, provided that the ratio of
M
1
/
M
2
is greater than 24.96.
[note 1]
This is the case for the Sun?Earth system, the Sun?Jupiter system, and, by a smaller margin, the Earth?Moon system. When a body at these points is perturbed, it moves away from the point, but the factor opposite of that which is increased or decreased by the perturbation (either gravity or angular momentum-induced speed) will also increase or decrease, bending the object's path into a stable,
kidney bean
-shaped orbit around the point (as seen in the corotating frame of reference).
[9]
The points L
1
, L
2
, and L
3
are positions of
unstable equilibrium
. Any object orbiting at L
1
, L
2
, or L
3
will tend to fall out of orbit; it is therefore rare to find natural objects there, and spacecraft inhabiting these areas must employ a small but critical amount of
station keeping
in order to maintain their position.
Natural objects at Lagrange points
[
edit
]
Due to the natural stability of L
4
and L
5
, it is common for natural objects to be found orbiting in those Lagrange points of planetary systems. Objects that inhabit those points are generically referred to as '
trojans
' or 'trojan asteroids'. The name derives from the names that were given to asteroids discovered orbiting at the Sun?
Jupiter
L
4
and L
5
points, which were taken from mythological characters appearing in
Homer
's
Iliad
, an
epic poem
set during the
Trojan War
. Asteroids at the L
4
point, ahead of Jupiter, are named after Greek characters in the
Iliad
and referred to as the "
Greek camp
". Those at the L
5
point are named after Trojan characters and referred to as the "
Trojan camp
". Both camps are considered to be types of trojan bodies.
As the Sun and Jupiter are the two most massive objects in the Solar System, there are more known Sun?Jupiter trojans than for any other pair of bodies. However, smaller numbers of objects are known at the Lagrange points of other orbital systems:
- The Sun?Earth L
4
and L
5
points contain interplanetary dust and at least two asteroids,
2010 TK
7
and
2020 XL
5
.
[10]
[11]
[12]
- The Earth?Moon L
4
and L
5
points contain concentrations of
interplanetary dust
, known as
Kordylewski clouds
.
[13]
[14]
Stability at these specific points is greatly complicated by solar gravitational influence.
[15]
- The Sun?
Neptune
L
4
and L
5
points contain several dozen known objects, the
Neptune trojans
.
[16]
- Mars
has four accepted
Mars trojans
:
5261 Eureka
,
1999 UJ
7
,
1998 VF
31
, and
2007 NS
2
.
- Saturn's moon
Tethys
has two smaller moons of Saturn in its L
4
and L
5
points,
Telesto
and
Calypso
. Another Saturn moon,
Dione
also has two Lagrange co-orbitals,
Helene
at its L
4
point and
Polydeuces
at L
5
. The moons wander
azimuthally
about the Lagrange points, with Polydeuces describing the largest deviations, moving up to 32° away from the Saturn?Dione L
5
point.
- One version of the
giant impact hypothesis
postulates that an object named
Theia
formed at the Sun?Earth L
4
or L
5
point and crashed into Earth after its orbit destabilized, forming the Moon.
[17]
- In
binary stars
, the
Roche lobe
has its apex located at L
1
; if one of the stars expands past its Roche lobe, then it will lose matter to its
companion star
, known as
Roche lobe overflow
.
[18]
Objects which are on
horseshoe orbits
are sometimes erroneously described as trojans, but do not occupy Lagrange points. Known objects on horseshoe orbits include
3753 Cruithne
with Earth, and Saturn's moons
Epimetheus
and
Janus
.
Physical and mathematical details
[
edit
]
Lagrange points are the constant-pattern solutions of the restricted
three-body problem
. For example, given two massive bodies in orbits around their common
barycenter
, there are five positions in space where a third body, of comparatively negligible
mass
, could be placed so as to maintain its position relative to the two massive bodies. This occurs because the combined gravitational forces of the two massive bodies provide the exact centripetal force required to maintain the
circular motion
that matches their orbital motion.
Alternatively, when seen in a
rotating reference frame
that matches the
angular velocity
of the two co-orbiting bodies, at the Lagrange points the combined
gravitational fields
of two massive bodies balance the
centrifugal pseudo-force
, allowing the smaller third body to remain stationary (in this frame) with respect to the first two.
The location of L
1
is the solution to the following equation, gravitation providing the centripetal force:
where
r
is the distance of the L
1
point from the smaller object,
R
is the distance between the two main objects, and
M
1
and
M
2
are the masses of the large and small object, respectively. The quantity in parentheses on the right is the distance of L
1
from the center of mass. The solution for
r
is the only
real
root
of the following
quintic function
where
is the mass fraction of
M
2
and
is the normalised distance. If the mass of the smaller object (
M
2
) is much smaller than the mass of the larger object (
M
1
) then L
1
and L
2
are at approximately equal distances
r
from the smaller object, equal to the radius of the
Hill sphere
, given by:
We may also write this as:
Since the
tidal
effect of a body is proportional to its mass divided by the distance cubed, this means that the tidal effect of the smaller body at the L
1
or at the L
2
point is about three times of that body. We may also write:
where
ρ
1
and
ρ
2
are the average densities of the two bodies and
d
1
and
d
2
are their diameters. The ratio of diameter to distance gives the angle subtended by the body, showing that viewed from these two Lagrange points, the apparent sizes of the two bodies will be similar, especially if the density of the smaller one is about thrice that of the larger, as in the case of the earth and the sun.
This distance can be described as being such that the
orbital period
, corresponding to a circular orbit with this distance as radius around
M
2
in the absence of
M
1
, is that of
M
2
around
M
1
, divided by
√
3
? 1.73:
The location of L
2
is the solution to the following equation, gravitation providing the centripetal force:
with parameters defined as for the L
1
case. The corresponding quintic equation is
Again, if the mass of the smaller object (
M
2
) is much smaller than the mass of the larger object (
M
1
) then L
2
is at approximately the radius of the
Hill sphere
, given by:
The same remarks about tidal influence and apparent size apply as for the L
1
point. For example, the angular radius of the sun as viewed from L
2
is arcsin(
695.5
×
10
3
/
151.1
×
10
6
) ? 0.264°, whereas that of the earth is arcsin(
6371
/
1.5
×
10
6
) ? 0.242°. Looking toward the sun from L
2
one sees an
annular eclipse
. It is necessary for a spacecraft, like
Gaia
, to follow a
Lissajous orbit
or a
halo orbit
around L
2
in order for its solar panels to get full sun.
The location of L
3
is the solution to the following equation, gravitation providing the centripetal force:
with parameters
M
1
,
M
2
, and
R
defined as for the L
1
and L
2
cases, and
r
being defined such that the distance of L
3
from the centre of the larger object is
R
?
r
. If the mass of the smaller object (
M
2
) is much smaller than the mass of the larger object (
M
1
), then:
[20]
Thus the distance from L
3
to the larger object is less than the separation of the two objects (although the distance between L
3
and the barycentre is greater than the distance between the smaller object and the barycentre).
L
4
and L
5
[
edit
]
The reason these points are in balance is that at L
4
and L
5
the distances to the two masses are equal. Accordingly, the gravitational forces from the two massive bodies are in the same ratio as the masses of the two bodies, and so the resultant force acts through the
barycenter
of the system. Additionally, the geometry of the triangle ensures that the
resultant
acceleration is to the distance from the barycenter in the same
ratio
as for the two massive bodies. The barycenter being both the
center of mass
and center of rotation of the three-body system, this resultant force is exactly that required to keep the smaller body at the Lagrange point in orbital
equilibrium
with the other two larger bodies of the system (indeed, the third body needs to have negligible mass). The general triangular configuration was discovered by Lagrange working on the
three-body problem
.
Radial acceleration
[
edit
]
The radial acceleration
a
of an object in orbit at a point along the line passing through both bodies is given by:
where
r
is the distance from the large body
M
1
,
R
is the distance between the two main objects, and sgn(
x
) is the
sign function
of
x
. The terms in this function represent respectively: force from
M
1
; force from
M
2
; and centripetal force. The points L
3
, L
1
, L
2
occur where the acceleration is zero ? see chart at right. Positive acceleration is acceleration towards the right of the chart and negative acceleration is towards the left; that is why acceleration has opposite signs on opposite sides of the gravity wells.
Stability
[
edit
]
Although the L
1
, L
2
, and L
3
points are nominally unstable, there are quasi-stable periodic orbits called
halo orbits
around these points in a three-body system. A full
n
-body
dynamical system
such as the
Solar System
does not contain these periodic orbits, but does contain quasi-periodic (i.e. bounded but not precisely repeating) orbits following
Lissajous-curve
trajectories. These quasi-periodic
Lissajous orbits
are what most of Lagrangian-point space missions have used until now. Although they are not perfectly stable, a modest effort of
station keeping
keeps a spacecraft in a desired Lissajous orbit for a long time.
For Sun?Earth-L
1
missions, it is preferable for the spacecraft to be in a large-amplitude (100,000?200,000 km or 62,000?124,000 mi) Lissajous orbit around L
1
than to stay at L
1
, because the line between Sun and Earth has increased solar
interference
on Earth?spacecraft communications. Similarly, a large-amplitude Lissajous orbit around L
2
keeps a probe out of Earth's shadow and therefore ensures continuous illumination of its solar panels.
The
L
4
and
L
5
points are stable provided that the mass of the primary body (e.g. the Earth) is at least 25
[note 1]
times the mass of the secondary body (e.g. the Moon),
[21]
[22]
The Earth is over 81 times the mass of the Moon (the Moon is 1.23% of the mass of the Earth
[23]
). Although the L
4
and L
5
points are found at the top of a "hill", as in the effective potential contour plot above, they are nonetheless stable. The reason for the stability is a second-order effect: as a body moves away from the exact Lagrange position,
Coriolis acceleration
(which depends on the velocity of an orbiting object and cannot be modeled as a contour map)
[22]
curves the trajectory into a path around (rather than away from) the point.
[22]
[24]
Because the source of stability is the Coriolis force, the resulting orbits can be stable, but generally are not planar, but "three-dimensional": they lie on a warped surface intersecting the ecliptic plane. The kidney-shaped orbits typically shown nested around L
4
and L
5
are the projections of the orbits on a plane (e.g. the ecliptic) and not the full 3-D orbits.
Solar System values
[
edit
]
This table lists sample values of L
1
, L
2
, and L
3
within the Solar System. Calculations assume the two bodies orbit in a perfect circle with separation equal to the semimajor axis and no other bodies are nearby. Distances are measured from the larger body's center of mass (but see
barycenter
especially in the case of Moon and Jupiter) with L
3
showing a negative direction. The percentage columns show the distance from the orbit compared to the semimajor axis. E.g. for the Moon, L
1
is
326
400
km
from Earth's center, which is 84.9% of the Earth?Moon distance or 15.1% "in front of" (Earthwards from) the Moon; L
2
is located
448
900
km
from Earth's center, which is 116.8% of the Earth?Moon distance or 16.8% beyond the Moon; and L
3
is located
?381
700
km
from Earth's center, which is 99.3% of the Earth?Moon distance or 0.7084% inside (Earthward) of the Moon's 'negative' position.
Lagrangian points in Solar System
Body pair
|
Semimajor axis, SMA (×10
9
m)
|
L
1
(×10
9
m)
|
1 ? L
1
/SMA (%)
|
L
2
(×10
9
m)
|
L
2
/SMA ? 1 (%)
|
L
3
(×10
9
m)
|
1 + L
3
/SMA (%)
|
Earth?Moon
|
0.3844
|
0.326
39
|
15.09
|
0.4489
|
16.78
|
?0.381
68
|
0.7084
|
Sun?Mercury
|
57.909
|
57.689
|
0.3806
|
58.13
|
0.3815
|
?57.909
|
0.000
009
683
|
Sun?Venus
|
108.21
|
107.2
|
0.9315
|
109.22
|
0.9373
|
?108.21
|
0.000
1428
|
Sun?Earth
|
149.598
|
148.11
|
0.997
|
151.1
|
1.004
|
?149.6
|
0.000
1752
|
Sun?Mars
|
227.94
|
226.86
|
0.4748
|
229.03
|
0.4763
|
?227.94
|
0.000
018
82
|
Sun?Jupiter
|
778.34
|
726.45
|
6.667
|
832.65
|
6.978
|
?777.91
|
0.055
63
|
Sun?Saturn
|
1
426
.7
|
1
362
.5
|
4.496
|
1
492
.8
|
4.635
|
?1
426
.4
|
0.016
67
|
Sun?Uranus
|
2
870
.7
|
2
801
.1
|
2.421
|
2
941
.3
|
2.461
|
?2
870
.6
|
0.002
546
|
Sun?Neptune
|
4
498
.4
|
4
383
.4
|
2.557
|
4
615
.4
|
2.602
|
?4
498
.3
|
0.003
004
|
Spaceflight applications
[
edit
]
Sun?Earth
[
edit
]
Sun?Earth L
1
is suited for making observations of the Sun?Earth system. Objects here are never shadowed by Earth or the Moon and, if observing Earth, always view the sunlit hemisphere. The first mission of this type was the 1978
International Sun Earth Explorer 3
(ISEE-3) mission used as an interplanetary early warning storm monitor for solar disturbances.
[25]
Since June 2015,
DSCOVR
has orbited the L
1
point. Conversely, it is also useful for space-based
solar telescopes
, because it provides an uninterrupted view of the Sun and any
space weather
(including the
solar wind
and
coronal mass ejections
) reaches L
1
up to an hour before Earth. Solar and heliospheric missions currently located around L
1
include the
Solar and Heliospheric Observatory
,
Wind
,
Aditya-L1 Mission
and the
Advanced Composition Explorer
. Planned missions include the
Interstellar Mapping and Acceleration Probe
(IMAP) and the
NEO Surveyor
.
Sun?Earth L
2
is a good spot for space-based observatories. Because an object around L
2
will maintain the same relative position with respect to the Sun and Earth, shielding and calibration are much simpler. It is, however, slightly beyond the reach of Earth's
umbra
,
[26]
so solar radiation is not completely blocked at L
2
. Spacecraft generally orbit around L
2
, avoiding partial eclipses of the Sun to maintain a constant temperature. From locations near L
2
, the Sun, Earth and Moon are relatively close together in the sky; this means that a large sunshade with the telescope on the dark-side can allow the telescope to cool passively to around 50 K ? this is especially helpful for
infrared astronomy
and observations of the
cosmic microwave background
. The
James Webb Space Telescope
was positioned in a halo orbit about L
2
on January 24, 2022.
Sun?Earth L
1
and L
2
are
saddle points
and exponentially unstable with
time constant
of roughly 23 days. Satellites at these points will wander off in a few months unless course corrections are made.
[9]
Sun?Earth L
3
was a popular place to put a "
Counter-Earth
" in
pulp
science fiction
and
comic books
, despite the fact that the existence of a planetary body in this location had been understood as an impossibility once orbital mechanics and the perturbations of planets upon each other's orbits came to be understood, long before the Space Age; the influence of an Earth-sized body on other planets would not have gone undetected, nor would the fact that the foci of Earth's orbital ellipse would not have been in their expected places, due to the mass of the counter-Earth. The Sun?Earth L
3
, however, is a weak saddle point and exponentially unstable with time constant of roughly 150 years.
[9]
Moreover, it could not contain a natural object, large or small, for very long because the gravitational forces of the other planets are stronger than that of Earth (for example,
Venus
comes within 0.3
AU
of this L
3
every 20 months).
[
citation needed
]
A spacecraft orbiting near Sun?Earth L
3
would be able to closely monitor the evolution of active sunspot regions before they rotate into a geoeffective position, so that a seven-day early warning could be issued by the
NOAA
Space Weather Prediction Center
. Moreover, a satellite near Sun?Earth L
3
would provide very important observations not only for Earth forecasts, but also for deep space support (Mars predictions and for crewed missions to
near-Earth asteroids
). In 2010, spacecraft transfer trajectories to Sun?Earth L
3
were studied and several designs were considered.
[27]
Earth?Moon
[
edit
]
Earth?Moon L
1
allows comparatively easy access to Lunar and Earth orbits with minimal change in velocity and this has as an advantage to position a habitable
space station
intended to help transport cargo and personnel to the Moon and back. The
SMART-1
Mission
[28]
passed through the L
1
Lagrangian Point on 11 November 2004 and passed into the area dominated by the Moon's
gravitational
influence.
Earth?Moon L
2
has been used for a
communications satellite
covering the Moon's far side, for example,
Queqiao
, launched in 2018,
[29]
and would be "an ideal location" for a
propellant depot
as part of the proposed depot-based space transportation architecture.
[30]
Earth?Moon L
4
and L
5
are the locations for the
Kordylewski dust clouds
.
[31]
The
L5 Society
's name comes from the L
4
and L
5
Lagrangian points in the Earth?Moon system proposed as locations for their huge rotating space habitats. Both positions are also proposed for communication satellites covering the Moon alike communication satellites in
geosynchronous orbit
cover the Earth.
[32]
[33]
Sun?Venus
[
edit
]
Scientists at the
B612 Foundation
were
[34]
planning to use
Venus
's L
3
point to position their planned
Sentinel telescope
, which aimed to look back towards Earth's orbit and compile a catalogue of
near-Earth asteroids
.
[35]
Sun?Mars
[
edit
]
In 2017, the idea of positioning a
magnetic dipole
shield at the Sun?Mars L
1
point for use as an artificial magnetosphere for Mars was discussed at a NASA conference.
[36]
The idea is that this would protect the planet's atmosphere from the Sun's radiation and solar winds.
See also
[
edit
]
Explanatory notes
[
edit
]
- ^
a
b
Actually
25 + 3
√
69
/
2
?
24.959
935
7944
(sequence
A230242
in the
OEIS
)
References
[
edit
]
- ^
a
b
Cornish, Neil J. (1998).
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(PDF)
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.
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Weisstein, Eric W.
"Lagrange Points"
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"DSCOVR: In-Depth"
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Lagrange, Joseph-Louis
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"Tome 6, Chapitre II: Essai sur le probleme des trois corps"
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L
2
is in deep space far away from any planetary surface and hence the thermal, micrometeoroid, and atomic oxygen environments are vastly superior to those in LEO. Thermodynamic stasis and extended hardware life are far easier to obtain without these punishing conditions seen in LEO. L
2
is not just a great gateway?it is a great place to store propellants. ... L
2
is an ideal location to store propellants and cargos: it is close, high energy, and cold. More importantly, it allows the continuous onward movement of propellants from LEO depots, thus suppressing their size and effectively minimizing the near-Earth boiloff penalties.
- ^
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External links
[
edit
]
J R Stockton - Includes translations of Lagrange's
Essai
and of two related papers by Euler