Fig.1: A schematic picture of a rotating and oscillating star as seen from an inertial frame.
This instability was discovered
by Chandrasekhar
(S. Chandrasekhar, Physical Review
Letters, 24, 611 1970), and subsequently considered by
Friedman
and Schutz (J.L.
Friedman and B.F. Schutz, The Astrophysical Journal, 222, 881 1978)in
1978 who have shown the generic nature of the instability. While a formal
proof is rather involved, qualitative arguments on the properties of the
instability can be given with a few illustrative examples.
Consider, therefore, a star with a sufficiently rapid rotation and which is also undergoing oscillations. For simplicity we will consider the simplest type of oscillation here, with mode numbers l=m=2 (see here for a more detailed discussion of the mode description and classification). Because the star is rotating, the mode properties can be considered both in a reference frame which is corotating with it (i.e. the corotating frame) or in a reference frame which is not rotating and is fixed with respect to, say, distant stars (i.e. the inertial frame).
In the animation below we show what
an observer in this latter frame would see (we are here imagining that
this observer is on the rotational axis of the star which is rotating
clockwise). The surface of the unperturbed star is here indicated with
a thin continuous line and the dot rotating clockwise is used to represent
the rotation and can be thought of a fluid element which has been marked
for this scope. The thick solid line, on the other hand, marks the surface
of the deformed oscillating star. Note that in the inertial frame the mode
is rotating in the same clockwise direction as the star and has
a lower angular velocity as compared to the stellar one (simply compare
the rotational speeds of the dots and of the deformed surface). Such a
mode is called prograde in
this frame.
Fig.1: A schematic picture of a rotating and oscillating star as seen from an inertial frame.
The oscillatory motion of the stellar matter creates gravitational waves which extract energy and the angular momentum from the oscillation and are indicate pictorialy with the concentric dashed circles emanating from the star (Note that the CFS instability is not triggered uniquely by gravitaional waves but will be produced by any mechanism that extracts energy and angular momentum.). The sign of the angular momentum carried away by the gravitational waves is the same as that of the angular momentum of the mode pattern as measured in the inertial frame (we recall that all measurements have a consistent definition when made in this frame). Because we have considered here a prograde mode, the gravitational wave will carry positive angular momentum away from the system and to infinity. As a result the star will be spun-down.
To appreciate the workings of the
CFS instability we need now to consider the same physical system but as
observed in the corotating frame , in which an observer is
rotating with the same angular speed as the background star. This is shown
in the animation below.
Fig.2: The same system seen from co-rotating observer with the star.
As the animation shows, in the corotating frame the marker dot does not vary its position, while the mode pattern is seen to counter-rotate and the mode is therefore in retrograde rotation. An observer in this frame would deduce that the mode has negative angular momentum as it decreases the total angular momentum of the star (the perturbed star has a smaller angular momentum than the unperturbed one and therefore spins a bit slower). Note therefore that a mode that is retrograde in the corotating frame can appear as prograde in the inertial. This change is simply due to rotation (an equivalent effect is present for linear momentum and is relevant in plasma physics) and has nothing to do with the general relativistic dragging of reference frames.
Because the star is emitting gravitational waves and loosing positive angular momentum, the observer in the corotating frame would see the angular momentum of the mode become increasingly negative. As a result, what started as perturbations with small negative angular momentum in the corotating frame, has been transformed (in the same frame) into large amplitude oscillations with a progressively larger negative angular momentum that will emit increasingly large amounts of gravitational waves.
It is important to bear in mind that in order to establish whether angular momentum has been carried away from the system, one needs the consider the measurements made in the inertial frame. The initial perturbation has negative angular momentum J0 < 0 and the oscillations will produce gravitational waves which carry away positive angular momentum jgw > 0. As a result, at any time the angular momentum of the perturbations will be J = J0 - jgw and this will become increasingly negative. In a way this is similar to someone's debts that get larger as new expenditures (with positive amounts of money) are made!