The equilateral triangle at right tells the whole story:
If the bodies at A, B and C attract each other in direct proportion of their masses,
the so-called paralellogram law for
vector addition
does indicate that each body is subjected to a centripetal
acceleration toward O,
whose magnitude is proporttional to its distance to the common
center of mass O. (With a suitable scaling to
represent accelerations, the geometric construction of the center of mass
matches the parallelograms involved in vector addition, as depicted above.)
This means that the triangle ABC rotates rigidly
about its center of mass O.
Note that this much is true regardless of the dependence of forces on distance,
since the 3 bodies are at the same distance from each other.
Quantitatively, the square of angular velocity
w is the scaling factor of the above diagram:
To a distance R corresponds an acceleration
w2 R.
This remark allows the value of that scale to be obtained
geometrically in terms of Newton's
universal
gravitational constant (G) :
w
as a function of d = AB = AC = BC
|
w2 d 3
= G M
= G
( m A + m B + m C )
|
Proof :
In the diagram, we observe that the arrow extremities
divide each side (of length d) into three segments whose lengths are
proportional to the three masses (the coefficient
of proportionality being d/M).
Thus, an arrow toward B (from
A or C) translates (by scaling lengths into
accelerations) into the following component of
the acceleration, which is equated to its gravitational counterpart
(using Newton's inverse square law)
to yield the advertised relation.
w2
m B ( d / M )
= G m B / d 2
(This reduces to
Kepler's third law
when one body has negligible mass.)
The above can be applied to the case of two bodies
in circular orbit around each other: A third body of
negligible mass would follow their rotation rigidly if it's
in the plane of rotation and forms an equilateral triangle with those two bodies.
There are two such points (called L4 and L5).
These are stable locations (in the sense that they seem
to attract nearby test masses)
provided the ratio of the larger mass to the smaller one exceeds 24.96
or, more precisely:
½
( 25 + 3 Ö69 )
= 24.959935794377112278876394117361238...
L4 (the "Greek" triangular point) leads the
smaller body in its orbit around the larger one, while
L5 (the "Trojan" or "trailing" triangular point) lags behind.
L4 and L5 are sometimes collectively known as the "Trojan points".
Several asteroids which reside there in the Sun-Jupiter system have been
named after legendary heroes of the Trojan war.
The leading triangular point L4 is home to the
Greek
camp led by
588 Achilles
(discovered in 1906 by Max Wolf)
with 659 Nestor,
911 Agamemnon,
1143 Odysseus,
1404 Ajax,
1583 Antilochus,
1437 Diomedes
and 1647 Menelaus.
The trailing Trojan point L5 marks the
Trojan
camp where
884 Priamus,
1172 Aeneas,
1173 Anchises
and 1208 Troilus
reside.
Early naming has left only two so-called "spies"
(both discovered in 1907 by August Kopff)...
617 Patroclus
is the lone Greek in the Trojan camp.
624 Hector
is the lone Trojan among the Greeks.
In addition,
there are three unstable Lagrangian points
(aligned with the two orbiting bodies) where the centrifugal force
exactly balances gravity.
L1 (the inner Lagrangian point) is located
between the two orbiting bodies. L2 is outside those two
bodies, on the side of the lighter one, while
L3 is on the side of the heavier one.
