THE RANGE OF INFLUENCE OF BLACK HOLES

Luis A. Aguilar

Inst. de Astronomıa, UNAM, Mexico.

(aguilar@astrosen.unam.mx)

1. Introduction

For a long time there has been evidence of large velocity dispersions at the center of many galaxies, whichsuggests the existence of massive black holes lurking at the center of these galaxies. Unfortunately, the lackof spatial resolution, left open the possibility of explaining the observed large velocity dispersions as a purelykinematic effect, if the central part of the galaxy is dominated by radial orbits, or by the existence of a verydense and massive star cluster. It is only within the last few years, that observations with increasing spatialresolution have pushed the constraints on alternative hypothesis to the point that, we now feel confidentthat the only explanation for the anomalous kinematics is, indeed, massive black holes (for a good reviewsee Kormendy and Richstone, 1995).

The boundary of a black hole is given by the point of no return, the Schwarzschild radius. This radius,however, is very small, even for very massive black holes: A 108 M⊙ black hole has a Scharzschild radius ofjust 2 astronomical units! A black hole, however, can perturb the local kinematics far beyond its boundary.The so called “radius of influence” (Binney and Tremaine, 1987, eq. 8–133) is given by:

rBH = GMBH

σ2, (1.1)

where σ is the local velocity dispersion of the surrounding stars. This radius can reach up to 100 pc and thusplay a non-negligible influence in the underlying galaxy. In physical units, this formula can be written as:

(rBH/pc) = 0.43(MBH/106 M⊙)

(σ/102 km/s)2(1.2)

Figure 1 shows the massof the central black hole againstthe central velocity dispersionin the host galaxy, for a sam-ple of 12 galaxies (the data istaken from Ferrarese and Mer-ritt, 2000). The dashed bluelines correspond to radii of in-fluence of 1, 10 and 100 pc. Thered line corresponds to a scal-ing proposed by Ferrarese andMerritt: MBH ∝ σ4.8±0.5. Itis clear that the influence ofblack holes at the centers ofthese galaxies can reach up tolenghts of hundreds of parsecs.

What fraction of stars inthe host galaxy is affected bythe black hole? The answerdepends on the orbital struc-ture of the galaxy.

Figure 1. Masses of black holes and central velocity dis-

persions of the host galaxies (Ferrarese and Merritt, 2000). The

blue dashed lines correspond to radii of influence of 1, 10 and

100 pc. The red line is a scaling proposed by these authors.

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If we have a disk galaxy where most of stars move on circular orbits, the answer is very simple: it is onlythose stars within the sphere of influence that will be affected. However, if we have a more complicated systemwhich is entirely, or at least in part, supported by thermal motions, the answer will be more complicated:we have to find at any particular radius, the fraction of orbits whose perigalactica is within the sphere ofinfluence. This is the problem that we address in this lecture.

We note that this is a problem that appears in many other contexts: for instance, when studying thepopulation of comets in the Solar System, we may be interested in finding out the fraction of comets inthe Oort cloud, whose perihelia lies within the inner Solar System. Or when studying globular clusters, wemay be interested in finding out what fraction of stars plunge to the core, whose dynamics is influenced bytwo–body relaxation. In every case the problem is the same: we want to know what fraction of a systemcrosses a given region within it. This is a complicated problem whose answer depends on the orbital structuresupported by the potential of the system, as well as its distribution function in phase space. For the caseof a system dominated by a spherical potential, however, there is a little known tool that is perfect for ourproblem: the Lindblad Diagram. But before we dwell into this, we must talk a little about the role of the socalled integrals of motion in shaping the orbital structure supported by a potential.

2. Integrals of Motion

The orbit of a star within a given potential is determined by Newton’s second law of motion. This is asecond order differential equation and its solution is thus completely determined by giving the star’s positionand velocity at some arbitrary time. This means that through a given point in physical space many orbitscan go through. However, if instead we consider phase space, which consists of position and velocity*, thenonly one orbit goes through a given point. This makes phase space the natural stage to describe orbits.

At the beginning of the XX century, the mathematician Emmy Noether proved that when the potentialof a system presents a symmetry (i.e. the functional form of the potential is unchanged by a spatial and/ortemporal transformation), there is a corresponding physical quantity that is conserved, when moving alongthe orbits that correspond to the potential (see Boccaletti and Pucacco, 1995). Examples of this are theconservation of energy, when the potential is invariant with respect to a temporal translation (e.g. Φ(t) =Φ(t + to)), the conservation of linear momentum, when the potential is invariant with respect to a spatialtranslation (e.g. Φ(x) = Φ(x + xo)), or the conservation of angular momentum, when the potential isinvariant with respect to rotation along a given axis (e.g. Φ(x) = Φ(x × e)).

A conserved quantity along orbital motion is very important in shaping orbits, let’s see why. Let usassume that I is conserved along the orbits of a particular system. Since the natural stage for orbits is phasespace, I must be a function of phase space:

I(x,v) = Io, (2.1)

where Io is the particular value along a given orbit. This equation represents a geometric restriction thatthe orbit must satisfy. In fact, it lowers by one the dimension of the region of phase space where the orbitmoves. This can be seen from the fact that equation (2.1) can be written in such a way, that one phase spacecoordinate is a function of the remaining phase space coordinates, so it is no longer independent. Anotherway of interpreting equation (2.1) is as a continuous hypersurface (manifold) in phase space. The “surface”in “hypersurface” is used to indicate that it is a region whose dimension is less, by one, that of the phasespace (in analogy to our familiar 3–dimensional space, where a region of dimension 2 is a sheet); whereasthe “hyper” is used to indicate that we are not talking, in general, about a 2–dimensional region.

If there is an additional conserved quantity, then the orbit must lie within the intersection of thecorresponding manifolds, which in turn, is a manifold whose dimension is 2 less than that of the original

* Technically, the phase space is a fiber bundle, whose base manifold is the configuration space and itsattached fibers are given by the tangent bundles (velocity space).

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phase space. In general, a potential with M symmetries, supports orbits along which M quantities areconserved and thus moves within a region of phase space whose dimension is 2N − M , where N is thenumber of degrees of freedom of the problem. Each conserved quantity allows us to perform one integrationof the equations of motion, this is why they are called integrals of motion.

Let us illustrate the previous discussion with a particular example: that of a spherical potential: Φ(x) =Φ(|x|). In this case, we know that the energy and the angular momentum are conserved. This represents4 restrictions like equation (2.1) (remember that the angular momentum is a 3–component vector). Sincethe problem has 3 degrees of freedom, the phase space has 6 dimensions and the orbits are constrained tomove within manifolds of dimension 6 − 4 = 2. Another way of looking at this situation is to express theangular momentum vector in spherical coordinates; the conservation of the 2 angular coordinates meansthat the orbit is restricted to a plane, the orbital plane, in physical space (L ≡ x × v = Lo = constant,=⇒ both x and v must be always on the plane othogonal to Lo). We can then look at the problem as onewith 2 degrees of freedom whose phase space is 4–dimensional. The conservation of energy and magnitudeof angular momentum then restrict the orbits to a 2–dimensional region.

E = 1/2 (v + v ) - 1/r0 r t2 2

rVt

Vr

Constant Energy

Surface

Figure 2–a. Constant energy surface in

phase space. The angular coordinate has not

been plotted to get a 3–space.

L = r vz t

r

Vt

Vr

Constant Angular

Momentum Surface

Figure 2–b. Constant angular momentum

in phase space. The same representation as in

figure 2–a is used.

Since it is not possible to represent 4–space in 3–space, we have to sacrifice one phase space coordinate.If we describe the orbits using polar coordinates: (r, θ), then the potential is independent of θ, and we caneliminate it in our plots, knowing that whatever we get in the (r, vr, vθ) representation, must be wrappedaround along θ. In figure (2) we show the constant energy and constant angular momentum magnitudemanifolds in our reduced phase space. The energy and angular momentum restrictions are given by:

Eo = (1/2)(v2r + v2

θ) + Φ(r), Lo = rvθ (2.2)

The first one represents a funnel of circular cross section and symmetry axis given by the r axis. The shapeof the funnel and its extent, along the r axis, depends on the potential function. The second restriction is awall parallel to the vr axis whose base on the r–vθ plane is a hyperbola.

Now, a given orbit must satisfy both restrictions at the same time and so, it is should lie then at theintersection of these two surfaces (figure 3). The intersection is an oval–shaped loop whose size is given bythe particular values of Eo and Lo. In fact, from this geometrical interpretation we can see at once that

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not all combinations of energy and angular momentum are possible: as we increase the angular momentum,keeping the energy fixed, the constant angular momentum wall moves away from the r–vr plane and theintersection with the constant energy funnel shrinks up to a point, beyond which no intersection is possible.So, at constant energy, the angular momentum of an orbit is constrained between zero (radial orbit) and amaximum value, which corresponds to the circular orbit.

E = 1/2 (v + v ) - 1/r0 r t2 2 L = r vz t

Constant Energy

Surface

Constant Angular

Momentum Surface

rVt

Vr

An orbit lies at the intersection

Figure 3. The orbit moves along the inter-

section of the constant energy and constant an-

gular momentum manifold. This is a 1–dimen-

sional loop.

Vr

Figure 4. If we wrap the solution around

the angular coordinate, we obtain a torus. In

this representation, physical space is the pro-

jection on the bottom plane.

The loop determines completely the orbit: vr and vθ depend on the value of r; the first velocity setsthe speed at which the radial coordinate is traversed, while the second velocity sets the rate at which theorbit wraps along the θ direction. To get the orbit in physical space, we now replace the vθ axis with theθ direction. This is shown in figure 4, where physical space appears as the bottom plane while the verticalaxis is vr. Wrapping the r–vr loop along the angular coordinate results in a torus: the so called orbital

torus. This is a very important result that is more general: all integrable orbits (i.e. orbits whith a numberof independent integrals of motion equal to the number of degrees of freedom) move on manifolds in phasespace which are diffeomorphic to an N–torus, where N is the number of degrees of freedom.

Projecting onto physical space, we see that the orbit is constrained to move within two extrema in r: thepericenter and the apocenter. The orbit describes a rosetta–like curve as it wraps around the two principaldirections on the orbital torus. If the angular frequencies along r and θ are commensurate (i.e. we can find arelation mωr +nωθ = 0, where ωr and ωθ are the angular frequencies and m and n are two integers), then theorbit will close; if not, it will never close and will cover densely the anular region between the pericenter andthe apocenter. These two frequencies vary in general within phase space and so a given potential can haveorbits that close and orbits that don’t close. The Kepler potential (Φ ∝ r−1) and the harmonic potential(Φ ∝ r2) are two very simple potentials with a unique property: their angular frequencies are commensurateover the whole phase space. In the first case, orbits close in θ after one cycle in r, in the second case, theyclose after two cycles in r.

In this section we have shown how the integrals of motion determine the orbits supported by a givenpotential*. In the particular case that interests us here, an arbitrary spherical potential, we see that the

* There are other type of orbits, the so called, non–integrable orbits, whose shape is not constrained to

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orbits are planar rosettas, whose particular shape is determined by the integrals of motion. Except for thespatial orientation in physical space, the energy and the magnitude of the angular momentum determineuniquely an orbit in a spherical potential.

3. The Lindblad Diagram

From the previous discussion, it is clear that looking at a plot of energy vs. angular momentum mustprovide us with a complete and non–amiguous catalog of all orbits supported by a spherical potential; thisis the Lindblad diagram (Lindblad 1933). Bertil Lindblad introduced this diagram and identified severallandmarks in it that help us to identify the general nature of orbits plotted in it. Although this diagramcan be constructed for any spherical potential, for the purpose of solving our problem, we will study oneparticular potential, one that produces a flat rotation curve.

3.1 The Galactic Model

For simplicity, we will use a finite size, singular, spherical model with a flat rotation curve. The densityprofile and cumulative mass function are given by:

ρG(r) =

(v2o/4πG)(1/r2), r < rT

0, r > rT

, (3.1)

Mr =

(v2o/G)r, r < rT

(v2o/G)rT , r > rT

, (3.2)

where vo is the circular velocity and rT the extent of the model. This model has a potential function givenby:

φG(r) =

−v2o [1 − ln(r/rT )], r < rT

−v2o(rT /r), r > rT

, (3.3)

and the force exerted by it is

FG(r) =

(v2o/r), r < rT

(v2o/r2)rT , r > rT .

(3.4)

The circular velocity, although constant within the model, decays in a Keplerian way beyond its edge:

v2circ =

v2o , r < rT

v2o(rT /r), r > rT .

(3.5)

3.2 Circular Orbits

At a given energy, we can have orbits with angular momentum between zero (radial orbits) and amaximum value given by the circular orbit of that energy (see section 2). The angular momentum of circularorbits, as a function of energy, should then provide us with an upper limit to the region populated by orbits

orbital torii. They have less integrals of motion than the number of degrees of freedom. Spherical potentialsdon’t have them

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in the Lindblad diagram. Let us then begin by charting the position of circular orbits in this diagram. Theenergy of a circular orbit is given by:

Ec = φG(r) +1

2v2

circ, (3.6)

and its angular momentum isLc = rvcirc. (3.7)

If we restrict ourselves to orbits within the galaxy (r < rT ), we can substitute r = Lc/vo in the expressionfor Ec and eliminate the radial coordinate to obtain the locus of all circular orbits within the galaxy in theLindblad diagram:

Ec = −v2o

[

1

2− ln

(

Lc

rT vo

)

.

]

(3.8)

It is convenient to have the inverse relation, given by:

Lc = rT vo exp

(

Ec

v2o

+1

2

)

. (3.9)

Let us now find the asymptotic behaviour of this envelope at the center and the edge of the model. Asthe radius of the circular orbit shrinks to zero, the angular momentum approaches zero (equation 3.7), andthe energy diverges (equation 3.8):

r → 0, =⇒ Lc → 0, =⇒ Ec → −∞, (3.10)

the corresponding values for the circular orbit at the edge of the galaxy are:

r → rT , =⇒ Lc → rT vo, =⇒ Ec → −(1/2)v2o. (3.11)

Turning now to circular orbits beyond rT , equations (3.8) and (3.9) should be replaced by:

Ec = −1

2

(

rT v2o

Lc

)2

, L2c = −1

2

(

r2T v4

o

Ec

)

; (3.12)

where in this case we have used the substitution r = L2c/(rT v2

o), obtained from equation (3.7) and the formof equation (3.5) valid for r > rT .

The asymptotic limits for very large orbits are:

r → ∞, =⇒ Lc → ∞, =⇒ Ec → 0. (3.13)

The location of the circular orbits defines a curve that incresases without bound in L as E grows andtends to 0. All possible orbits exist below these curve, since at a given energy, orbits of angular momentumbetween 0 (radial orbit) and Lc (circular orbit) exist. Conversely, at a given angular momentum, theminimum energy orbit is the circular orbit; all these results are summarized in figure (5).

It is convenient to introduce adimensional variables:

ζ ≡ r

rT

, E ≡ E

v2o

, L ≡ L

rT vo

, (3.14)

equations (3.8), (3.9) and (3.12) can then be written as:

Ec =

ln(Lc) − 1/2, Lc < 1

−1/(2L2c), Lc > 1,

(3.15)

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Figure 5. Lindblad diagram for the singular, finite, flat rotation curve, galaxy model.The horizontal axis is the dimensionless orbital energy, the vertical axis is the dimensionlessorbital angular momentum. A point in this diagram, except for spatial orientation, definesuniquely an orbit. The red curve is the location of circular orbits. Only points below this curverepresent physical orbits. The dot on the curve represents the circular orbit at the edge of themodel; the portion of the curve to the left of this point represents circular orbits within thegalaxy, the portion to the right represents circular orbits beyond it. Notice that the angularmomentum of circular orbits diverges as the energy approaches 0. The second horizontal axison the bottom represents the dimensionless radius of the circular orbit that corresponds to agiven energy in the first axis. Notice that beyond the edge of the model, this radius divergesvery quickly as the energy approaches 0.

Lc =

exp(Ec + 1/2), Ec < −1/2

1/√−2Ec, −1/2 < Ec < 0;

(3.16)

where the values in Lc and Ec that split the cases are the ones thet correspond to ζ = 1 (the edge of themodel).

3.3 The Characteristic Parabola

We will now introduce the concept of the characteristic parabola, which describes all orbits that touch,without crossing, a given radial position. Let us center our attention on those orbits that have zero radialvelocity at r = ro. The energy of these orbits is given by:

Eo = φG(ro) +L2

o

2r2o

, (3.17)

where we denote with Eo and Lo, the energy and angular momentum of these orbits. Susbstituting theappropiate expression for φG, we obtain:

Eo =

v2o [ln(ro/rT ) − 1] + (L2

o/2r2o), ro < rT

−v2o(rT /ro) + (L2

o/2r2o), ro > rT

(3.18)

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or the inverse relations:

L2o =

2r2o Eo − v2

o [ln(ro/rT ) − 1], ro < rT

2r2o [Eo + v2

o(rT /ro)], ro > rT .(3.19)

This is the equation of a parabola in the Lindblad diagram which opens in Lo as Eo increases. Theparabola intersects the horizontal axis at Eo = E∗

o , given by:

E∗o/v2

o =

ln(ro/rT ) − 1, ro < rT

−(rT /ro), ro > rT .(19)

Figure 6. Characteristic parabola for all the orbits that touch, without crossing, thepoint ro = 0.5 (green curve). Point A corresponds to the radial orbit that just reaches ro. Aswe move from point A toward B, the perigalacticon of the orbit increases and approaches rowhile maintaing fixed apogalacticon. Point B corresponds to the circular orbit ar ro. Movingbeyond point B toward point C, the perigalacticon is now fixed at ro while the apogalacticondiverges to infinity. Point C corresponds to a parabolic orbit with perigalacticon ro. TheLindblad diagram is split by this parabola in three regions: Region I corresponds to orbitsentirely contained within the region r=ro. Region II correspond to orbits that cross r = ro.Finally, region III corresponds to orbits outside the region r = ro.

This is the characteristic parabola and gives the location in the Lindblad diagram of all orbits that passthrough r = ro with zero radial velocity. One such parabola is shown in figure (6). It starts at E = E∗

o , L = 0with the radial orbit with apogalacticon equal to ro. It then moves to intersect the curve L = Lc(E), at thecircular orbit of radius ro. In between these two points, we have all elliptical orbits whose apogalacticon isat ro and whose perigalacticon is between 0 and ro. Finally, all points along L = Lo beyond its intersectionwith the circula orbits locus are the orbits whose perigalacticon is ro and whose apogalacticon goes beyond.The characteristic parabola splits the region of allowed orbits in three non–intersecting zones. The first zonelies to the left of it and represents all orbits which are entirely contained within r = ro. The second zone liesbelow and to the left of the characteristic parabola and represents orbits that cross the point r = ro. Finally,the third zone, above the characteristic parabola, represents all orbits that lie entirely outside r = ro.

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Figure 3. Ensemble of characteristic parabolae that define a grid of lines of constant perigalacticon (blue lines) andapogalacticon (green lines) in the Lindblad diagram. The values for the adimensional peri and apogalactica are indicatedwith small numbers on the left and bottom edges of the plot, respectively.

The adimensional forms of equations (17), (18) and (19) are given by:

Eo =

ln(ζo) + (1/2)(Lo/ζo)2 − 1, ζo < 1

−(1/ζo) + (1/2)(Lo/ζo)2, ζo > 1,

(20)

L2o =

2ζ2o [Eo − ln(ζo) + 1], ζo < 1

2ζ2o [Eo + (1/ζo)], ζo > 1,

(21)

E∗o =

ln(ζo) − 1, ζo < 1

−(1/ζo), ζo > 1.(22)

We can verify that at the circular orbit value for the angular momentum (equations 7 and 13):

Lo = Lc =

ζo, ζo < 1

√ζo ζo > 1,

(23)

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the characteristic parabola becomes

Eo(Lo = Lc) =

ln(ζo) − 1/2, ζo < 1

−1/(2ζo), ζo > 1,(24)

which lies on the curve that defines the position of circular orbits in the Lindblad diagram (equation 14).An ensemble of characteristic parabolae thus defines a grid of constant perigalactica y apogalactica in

the Lindblad diagram. Figure (3) presents one such grid.

3.3 Orbital parameters

As we have seen, characteristic parabolae define lines of constant apogalactica for energies between thoseof the radial and circular orbits. They also define lines of constant perigalactica for energies between thatof the circular orbit and 0. We can then use these relations to find the energy and angular momentum thatcorrespond to an orbit of a given peri and apogalacticon.

Finding the intersection of the characteristic parabolae (equation 20) that correspond to the peri andapogalacticon, we obtain:

Case I: (ζPeri < 1, ζApo < 1)

L2 = 2

(

1

ζ2Peri

− 1

ζApo2

)−1

ln

(

ζApo

ζPeri

)

, E =

(

ζ2Peri

ζ2Apo − ζ2

Peri

)

ln

(

ζApo

ζPeri

)

+ ln(ζApo) − 1; (25)

Case II: (ζPeri < 1, ζApo > 1)

L2 = 2

(

1

ζ2Apo

− 1

ζP eri2

)−1 [

ln(ζPeri) +1

ζ2Apo

]

E =

(

ζ2Peri

ζ2Peri − ζ2

Apo

)[

ln(ζPeri) +1

ζ2Apo

]

− 1

ζ2Apo

; (26)

Case III: (ζPeri > 1, ζApo > 1)

L2 = 2

(

ζPeriζApo

ζPeri + ζApo

)

E =1

ζApo

[(

ζPeriζApo

ζPeri + ζApo

)

− 1

]

(27)

4. The fraction of Orbits Affected by the Black Hole

Coming back to our original problem, it is clear that given an assumed one–integral (f(E)), or two–integral (f(E, L)) model for the galaxy, we should find the characteristic parabola that corresponds toro = rBH and then find the fraction of the model that lies within regions I and II (see figure 6), of thisparabola.

To accomplish this, there is is a last step that must be taken: we must find out the actual number(or fraction) of orbits that lie within a given region of the Lindblad diagram. Let us call the occupancy

number N(E, L)∆E∆L the number (or fraction) of orbits whose energy is between E and E + ∆E, andwhose angular momentum is between L and L + ∆L. One may think that this is just the local value of thephase space density times de corresponding area in the Lindblad diagram: f(E, L)∆E∆L, this, however, isincorrect as we now explain.

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f(E, L): is the fraction of stars per unit phase space volume d3r d3v, with energy between E and E + ∆E andangular momentum between L and L + ∆L.

whereas,

N(E, L): is the fraction of stars per unit interval in E and L, with energy between E and E + ∆E and angularmomentum between L and L + ∆L.

They sound very similar but are not the same. The key is in the normalization: the first has unitsof inverse phase space volume while the second has units of inverse energy and angular momentum. Moreimportantly, the first represents a local density value in phase space while the second is actually an integralof the first over the hypersurface that corresponds to the intersection of the constant energy and constantangular momentum manifolds that correspond to E and L (see figure 3).

In other words, we can write a relation between f and N as follows:

N(E, L) = f(E, L) × A(E, L) (28)

where A(E) has units of phase space volume divided by energy and angular momentum (i.e. length2), andis equal to the area of the 2–dimensional intersection of the corresponding energy and angular momentum3–dimensional manifolds in the original 4–dimensional phase space of the problem (See Section 2).

To clarify thing further, let us work out in detail the form of A for one–integral models of the formf(E). In this case, A(E) should have units of phase space volume divided by energy (i.e. length4/time).

If ΩE represents the region of phase space where the energy is equal to E, we can write:

N(E) dE =

∫

ΩE

f(r, v) d3r d3v = 16π2

∫

ΩE

f(E) r2dr v2dv,

where we have taken advantage of the fact the the system is spherically isotropic in both, position andvelocity space. We now want to introduce explicitely E as an integration variable. This can be accomplishedby using the Jacobian of the transformation from (r, v) to (r, E):

N(E) dE = 16π2

∫

ΩE

f(E) r2v2 ∂(r, v)

∂(r, E)dr dE.

To compute the Jacobian we notice that,

E = (1/2)v2 + φ(r) =⇒ v =√

2[E − φ(r)],

and so the Jacobian is simply,

∂(r, v)

∂(r, E)=

∣

∣

∣

∣

∂r∂r

∂v∂r

∂r∂E

∂v∂E

∣

∣

∣

∣

=

∣

∣

∣

∣

1 0∂r∂E

1v

∣

∣

∣

∣

=1

v.

Substituting back in our expression for the occupancy number, we get:

N(E) dE = 16π2 f(E)

∫

r2v drdE = f(E) × A(E),

where

A(E) ≡ 16π2

∫

φ<E

r2√

2[E − φ(r)] dr,

is the phase space “area” of the constant energy manifold.

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Our last step is now to choose a particular form for f(E), and compute the integral of the correspondingN(E), over the I and II regions of the characteristic parabola of the radius of influence of the black hole inthe Lindblad diagram.

Suggested Problems

1. Prove that for the case of two–integral models of the form f(E, L), the relation between phase spacedensity and occupancy number is given by (Problem 4.22 in Binney and Tremaine book):

N(E, L) = 8π2 L f(E, L)Pr(E, L),

where

Pr(E, L) = 2

∫ rapo

rperi

dr

vr

= 2

∫ rapo

rperi

dr√

2(E − φ) − (L/r)2,

is the radial orbital period of the orbit with energy E and angular momentum L.

References

Binney, J., and Tremaine, S., (1987) Galactic Dynamics, Princeton Series in Astrophysics.Boccaletti, D., Pucacco, G., (1995) Theory of Orbits, Springer–Verlag, section 1.6Ferrarese, L., and Merritt, D., (2000) ApJ, 539, L9.Innanen, K.A., Harris, W.E. and Webbink, R.F., (1983) AJ, 88, 338.Kormendy, J., Richstone, D. (1995) ARAA, 33, 581.Lindblad, B. (1933) Handbuch der Astrophysik (Springer, Berlin), Vol. 5, p. 937.Plummer, H., (1905) MNRAS, 76, 107.

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