![FIG. I.-The positions of the most important resonances for fixed m »I. We only show resonances with 1/- ml s; 1 since the perturbing potential from the satellite (cf. eq. [5]) is oc ell-ml, The symbols 0, C, and I denote outer Lindblad resonances, corotation resonances, and inner Lindblad resonances, respectively.](/figures/fig-i-the-positions-of-the-most-important-resonances-for-31p8wmnb.webp)
1980ApJ...241..425G
THE
AsTROPHYSICAL
JouRNAL, 241:425-441, 1980 October I
©
1980.
The American Astronomical Society.
All
rights reserved. Printed in U.S.A.
DISK-SATELLITE
INTERACTIONS
PETER
GOLDREICH
California Institute
of
Technology
AND
ScoTT
TREMAINE
Institute for Advanced Study, Princeton, New Jersey
Received 1980 January 7; accepted 1980 April 9
ABSTRACT
We calculate the rate
at
which angular momentum
and
energy are transferred between a disk
and
a satellite which orbit the same central mass. A satellite which moves
on
a circular orbit exerts
a torque
on
the disk only in the immediate vicinity
of
its Lindblad resonances. The direction
of
angular momentum transport
is
outward, from disk material inside the satellite's orbit to the
satellite
and
from the satellite to disk material outside its orbit. A satellite with an eccentric orbit
exerts a torque
on
the disk
at
corotation resonances as well as
at
Lindblad resonances. The angular
momentum and energy transfer
at
Lindblad resonances tends
to
increase the satellite's orbit
eccentricity whereas the transfer
at
corotation resonances tends to decrease it.
In
a Keplerian disk,
to lowest order in eccentricity
and
in the absence
of
nonlinear effects, the corotation resonances
dominate by a slight margin
and
the eccentricity damps. However,
if
the strongest corotation
resonances saturate due
to
particle trapping, then the eccentricity grows.
We present an illustrative application
of
our
results to the interaction between Jupiter
and
the
protoplanetary disk. The angular momentum transfer is shown
to
be so rapid
that
substantial
changes in
both
the structure
of
the disk
and
the orbit
of
Jupiter must have taken place
on
a time
scale
of
a
few
thousand years.
Subject headings: hydrodynamics - planets: Jupiter - planets: satellites -
solar system: general
I.
INTRODUCTION
The main purpose
of
this paper is to evaluate the transfer
of
angular momentum
and
energy between a disk
and
a
satellite in order to determine their
mutual
evolution.
Our
results are applicable
to
a variety
of
systems: the rings
of
Saturn (Goldreich
and
Tremaine 1978b, henceforth
GTl),
the rings
of
Uranus (Goldreich
and
Tremaine 1979a),
accretion disks in close binary systems (Lin
and
Papaloizou 1979),
and
the protoplanetary nebula (cf. §VI).
The plan
of
the paper is as follows. In § II we calculate the angular momentum
and
energy transfer due to the
torques which the satellite exerts
on
the disk
at
Lindblad
and
corotation resonances. The orbital evolution
of
a
satellite
and
a neighboring narrow ring is explicitly evaluated. Section III contains
an
alternate derivation
of
the
results obtained
in§
II, based
on
a single close encounter between the satellite
and
each ring particle. The cutoff in
the torque
at
Lindblad resonances which occurs close to the satellite is accurately computed
in§
IV. Next,
in§
V we
describe additional features
of
disk-satellite interactions which are relevant in applications to planetary rings.
Section VI includes
an
illustrative application to the
mutual
evolution
of
Jupiter's orbit
and
the protoplanetary gas
disk. Finally,§ VII contains a summary
and
guide
to
the most
important
equations.
From
time
to
time
we
will refer to Goldreich
and
Tremaine (1978c) as GT2,
and
to
Goldreich
and
Tremaine
(1979b) as
GT3.
II. STEADY-STATE INTERACTIONS AT RESONANCES
a)
The Disk
For
our
purposes, it suffices
to
consider a two-dimensional disk which lies in the equatorial plane
of
a cylindrical
coordinate system (r,
(),
z).
The unperturbed disk is azimuthally symmetric
and
rotates with angular velocity
n(r)
>
0.
Oort's parameters A(r),
B(r)
and
the epicyclic frequency K(r) are defined by
rdn
2
1 d
2 2
A(r)
=
"2
dr ' B(r) = O(r) +
A(r),
K (r) = ,
3
dr
[r
O(r)] = 4B(r)O(r). (1)
425
©
American
Astronomical Society •
Provided
by
the
NASA Astrophysics
Data
System
1980ApJ...241..425G
426
GOLDREICH
AND
TREMAINE
Vol.
241
The
validity
of
most
of
our
results
does
not
depend
upon
the
nature
and
composition
of
the
disk material.
It
may
be a fluid, a collisionless gas,
or
a collection
of
macroscopic particles. However,
the
magnitude
of
the
typical
random
particle velocity, denoted
by
c,
is assumed
to
be
much
smaller
than
the
circular velocity, c
«Or,
as is observed
in
planetary
rings. Also, the surface mass density
~
is constrained
by
G~
« 0
2
r,
which implies
that
the
disk
makes
a
negligible
contribution
to
the
unperturbed
gravity field.
Thus,
we consider only disks which
orbit
some
central
rigid
body, whose mass we
denote
by
MP.
Some
other
restrictions
on
the
validity
of
our
results
are
discussed
in§
V.
The
most
important
special case is
the
nearly Keplerian disk for which 0
2
(r)
::::::
G
MP/r
3
,
A/0
::::::
- i,
B/0
::::::
i,
and
K/0::::::
1.
b)
The
Satellite
The
satellite
orbit
is characterized
by
the
elements a
and
e
and
is assumed
to
lie
in
the
disk plane. We define a such
that
the
instantaneous
angular
velocity is
equal
to
O(a) when the satellite crosses r
=a.
Note
that
for the Kepler
problem
a differs
from
the
semimajor axis
in
order
e
2
.
The
eccentricity e = (rmax-
rmin)/2a.
Fore«
1,
the
satellite
makes
an
epicyclic oscillation
at
angular
frequency K
8
=
K(a)
about
a guiding center which revolves
at
the
rate
o.
= O(a).
To
first
order
in e, we
may
write
r.
=
a(l-
ecos
K
8
t),
(2)
(Chandrasekhar
1960).
The
apse precession
rate
is given
by
dw
dt
=
Os-
Ks.
(3)
The
perturbation
potential
due
to
a satellite
of
mass
M.
reads
ri.S(
8 ) GM.
M.
2
'f'
r,
't
=
--1
- I + M 0 (r)r.·
r.
r r. P
(4)
The
second
term
is
the
indirect
part
of
the
potential which arises because
the
coordinate
origin is
attached
to
the
central
mass.
It
is convenient
to
expand
¢•
in a
Fourier
series:
00 00
cp•(r,
8,
t) = L L
c/Jf.m(r)
cos
{m8-
[mO.+
(I-
m)K.]t} .
(5)
1=-oom=O
Fore«
1,
the
largest term in
c/Jtm
is
proportional
to
ell-mi.
The
pattern
speed
of
the
/,
m
potential
component
is
(1-m)
Olm
= 0
8
+ K
8
•
. m
(6)
It
is straightforward
to
calculate
c/Jtm
from
equations
(2)
and
(4).
To
first
order
in
e,
the
only nonvanishing
components
are
(7)
(8)
• GM.
[(1
mo.
f3
d)
(3
2B.
o.)
J
c/Jm-l.m
=
-~
e(2
-
Om.o)
2-----;::
+ l
d/3
b'f12
-
f/3
2-
{l.-
Ks
om.l
·
(9)
Here
f = 02;a
3
/GMP,
{3
= rja,
om,n
is
the
Kronecker
delta function,
and
b'f
12
({3)
is
the
Laplace coefficient,
2 (" cos
mcpd¢
b'ft2({3)
=
~
Jo
(1
-
2{3
cos¢
+ /32)1/2 .
(10)
© American Astronomical Society • Provided by the NASA Astrophysics
Data
System
1980ApJ...241..425G
No. I, 1980 DISK-SATELLITE INTERACTIONS
427
The
terms in
¢f.m
proportional
to
bm,l
arise
from
the indirect
part
of
the
perturbation
potential.
Equations
(7)-(1 0)
are
valid for all
{3.
c) Torques
at
Resonances
Torques
are
exerted
on
the
disk by
¢i.m
only
in
the
immediate vicinity
of
Lindblad
and
corotation
resonances.
The
former occur where
K(r)
Q(r)
+
t-
=
n,
m '
m .
t=±l,
m>O;
(11)
and
the latter where
Q(r)
=
nl,m'
m>O.
(12)
We
ignore m = 0
perturbations
since they
are
axisymmetric
and
exert
no
torque.
At
the Lindblad resonance the
epicyclic
motion
of
a particle
in
a circular
orbit
is strongly excited, since the
perturbation
frequency felt by the
particle is
equal
to
its epicyclic frequency.
At
the
corotation
resonance the
angular
momentum
of
a particle in a
circular
orbit
undergoes large changes, since the particle feels a slowly varying azimuthal force. However, the
particle's epicyclic
motion
is
not
excited. A Keplerian disk
of
infinite extent has one inner
(t
=
-1)
and
one
outer
(t
= +
1)
Lindblad resonance
and
a single
corotation
resonance for each Q
1
,m > 0
and
m >
1.
Analytic expressions
for
the
torque
are
derived
in
GT3.
At
a Lindblad resonance r =
ru
L
2
[
(rdD)-
1
(rd¢f.m 2Q s )
2
]
Tl,m
= -
mn
~
dr
~
+ n -
nl,m
¢1.m
rL
' (13)
where
D = K
2
- m
2
(Q
- Q
1
,m)
2
•
Note
that
the
sign
of
Ttm
is opposite
to
that
of
dD/dr. Thus,
angular
momentum
is
removed from the disk
at
an
inner Lindblad resonance
and
added
to
it
at
an
outer
Lindblad resonance (Lynden-Bell
and
Kalnajs 1972).
At
a
corotation
resonance r = r
0
Tf.m
=
m;2
[(~~rt:,
(~)<¢f.m)21c
(14)
Note
that
the
sign
of
Tf.m
is
that
of
the gradient
of
vorticity
per
unit
surface density
if
dn/dr <
0.
The
torque
formulae
are
valid in the limit
that
c «
Qr,
G~
« Q
2
r,
and
m «
Qrjc.
The
implications
of
the violation
of
the last
inequality are examined
in§
IV.
Given a disk
and
a satellite, the
apparatus
we have assembled enables
us
to
locate
the
resonances
and
to
calculate
the
torques
exerted
on
the disk. This procedure was applied
to
Saturn's
rings
in
GTl
to
provide
an
explanation for
the
formation
of
the Cassini division. Here
our
goal is slightly different. We are primarily concerned with those
resonances which
are
close
to
the
satellite, i.e., those for which
lrL
- al « a
and
lr c - al « a
or,
what
is equivalent,
m »
1.
In
this limit the positions
of
the resonances
are
located as shown in
Figure
1.
Two
features
are
worth
commenting on. First, some
of
the
resonances are
at
r =
a.
Clearly, the linear
perturbation
theory
used
to
calculate
r
X=
m
+I
R = m
R.
= m
-I
t t t
K 0
a(
1
+
IAim)
---------------------•--
0 c
a(l
+
2l~m)
--------------•------+-
o c I
a
------
•------•-------e--
K C I
a(1-ii'Aj";,)
------•------•--------
I
0 (1-
IA~m)
------•---------------
FIG.
I.-The
positions
of
the most important resonances for
fixed
m
»I.
We
only show resonances with
1/-
ml
s; 1 since the perturbing
potential from the satellite
(cf.
eq. [5])
is
oc
ell-ml,
The symbols
0,
C, and I denote outer Lindblad resonances, corotation resonances,
and inner Lindblad resonances, respectively.
© American Astronomical Society • Provided by the NASA Astrophysics
Data
System
1980ApJ...241..425G
428
GOLDREICH
AND
TREMAINE
Vol.
241
the torques given by equations (13)
and
(14) is
an
inadequate
tool for these resonances. Second, there is
an
infinite
sequence
of
resonances
of
each type l =
m,
l = m ±
1,
and
each sequence has
an
accumulation
point
at
r =
a.
The
high density
of
resonances
near
a leads
us
to
introduce the average
torque
per radial interval,
or
torque
density,
which we denote by
dTT,m/dr
or
dTf.m/dr.
To
compute
the
torque
densities we use equations (7)-(10), (13),
and
(14)
and
Figure
1.
To
evaluate
bT
1
if3)
for
11
-
/31
« 1
and
m » 1, we note
that
most
of
the
contribution
to
the integral
in
equation
(1
0) comes from 8 «
1.
Thus,
we replace cos
e by 1 - 8
2
/2, extend the
upper
limit
to
infinity'
and
set
f3
= 1 except where it
appears
in the
combination 1 -
{3.
This procedure yields
bT;z(/3)
:::::
~
K
0
(mll
-
{31)
,
n
where
Kv
denotes the modified Bessel function
of
order
v.
Similarly,
dbT;z(f3)
:::::
sgn
(1
-
{3)
2m
Kl(mll-
{31)'
df3
n
and
d
2
bT
12
({3)
2m
2
[
1 J
df3
2
:::::
n K0
(mll
-
{31)
+
mil
_
f3l
K
1
(mil
-
{31)
·
The
resulting
torque
densities are
(15)
(16)
(17)
dTL K
2
rL
(GM
)
2
~
= sgn
(r-
a)
34
( s
4
{(2!1/K)K
0
(K/2IAI) + K
1
(K/21AI)}
2
,
(18)
dr
2 A
a-
r)
e) Orbital Variations
Next
we investigate the rates
of
change
of
a
and
e due
to
the interaction between a satellite
and
a circular ring
of
radial width
!J.r
and
mass Mr. The mean radius
and
radial width
of
the ring are subject
to
the constraints
c
Ia
- rl
Ia
- rl
2
!J.r
Ia
- rl
-«~~«1
«-«~~-
(21)
Or a ' a
2
a a
The lower limit
on
the mean separation between the ring
and
the satellite
orbit
is equivalent
to
the requirement
that
m «
Qrjc
for those l = m
and
l = m ± 1 resonances which lie within the ring.
Our
torque
equations (13)
and
(14) are
only valid in this limit. The lower limit
on
the ring width ensures
that
many
resonances
of
each type fall within the
ring boundaries so
that
the
torque
density is a meaningful concept. The effects
of
a wide ring
on
the satellite
orbit
may
be determined by summing the effects
of
many
narrow
rings.
A simple derivation
of
the
perturbation
equations for a
and
e starts with
the
integrals
of
the
unperturbed
satellite
orbit. These are the angular
momentum
(22)
and
the
energy
(23)
where
<I>
is the
unperturbed
gravitational potential.
The
forms
of
the integrals follow immediately from the
definitions
of
a
and
e
adopted
in§
lib. The expression for
His
exact whereas
that
for E is valid to
order
e
2
•
For
each
ring
torque
component
Tr
with
pattern
speed
QP
there is a reaction
torque
on
the satellite which changes
Hand
E
according
to
dH
dt
= -
T"
(24)
dE
df
=
-QpTr.
(25)
© American Astronomical Society • Provided by
the
NASA Astrophysics
Data
System
1980ApJ...241..425G
No. I, 1980
DISK-SATELLITE INTERACTIONS
429
From
equations (22)-(25)
and
the identities
aQ
2
=
d(J>jdr
and
K
2
-
30
2
= d
2
(J>jdr
2
,
we find
da
2QT,
dt
=-
aK
2
Ms'
(26)
de=
-[(n
_
Q)
_
2
e
2
n(l
+ d ln
K)J
T, .
dt
P d ln r M.e(aK)
2
(27)
Here
n,
K,
and
d ln
K/d
ln
rare
to
be evaluated
at
r
=a.
The leading contribution to dafdt is due
to
dT';,,m/dr
(eq. [18]). We find
da
QG
2
M M
dt
= sgn
(a
- r)
4
(
s
')
4
[(20/K)K
0
(K/2IAI)
+ K
1
(K/2IAI)]
2
,
8nA a
a-
r
(28)
where
M,
= 2nLrLlr.
The
computation
of
dejdt is slightly more subtle. The effect
of
dT~
,jdr
is smaller
than
that
of
dT~±
1
,jdr
by
~
Ia
- rlfa «
1.
To
ascertain the contribution
of
dP
±
1
/dr, we integrate over the ring holding constant everything
but
L
and
Ia-
rl-
5
, the most rapidly
varyingm
factors. We find
that
dT~±
1
,,/dr
and
dr;,±
1
,,/dr make
comparable contributions
to
dejdt. The
end
result is
1
de
K
2
G
2
M M {
-;
dt =
8
niAI5ala•-
~~
5
([1
+ (20/K)
2
]K
0
(K/IAI)
+
[IAI/K
+
40/K]K
1
(K/IAI))
2
IAI
}
-
2
B ((20/K)K
0
(K/2IAI)
+ K
1
(K/2IAI))
2
•
(29)
The positive first term
and
the negative second term come from the Lindblad
and
corotation resonances,
respectively.
A numerical evaluation
of
a-
1
dajdt
and
e-
1
defdt for the Keplerian disk yields
1
da
M.M,
( a )
4
--
=
0.798--
--
Qsgn(a-
r),
a dt
M;
a-
r
(30)
1
de
M.M,
( a )
5
-;
dt =
-0.0739
M;
Ia-
rl
n.
(31)
Applications
of
these results are presented in§§ V and VI. Here we merely note
that
the satellite is repelled by the
ring
and
its orbital eccentricity is damped. The latter conclusion is dependent
on
the assumption
that
the resonances
are
not
saturated (cf. § Ve).
III. INTERACTIONS DURING A CLOSE ENCOUNTER
In
this section we rederive equations (28)
and
(29) without reference
to
individual resonances. This alternate
derivation helps
to
clarify the mechanisms
of
angular momentum
transport
between a satellite
and
a differentially
rotating disk. The approximation we use here was applied previously
by
Julian
and
Toomre (1966)
and
by Lin
and
Papaloizou (1979).
It
was originally devised by Hill for his
lunar
theory.
a)
Basic Model
We introduce a local coordinate system with origin
at
r = R which revolves with n = Q(R). The x axis points
radially outward,
and
they
axis points in the direction
of
increasing
e.
For
x/
R « 1
andy/
R «
1,
the equations
of
motion
of
a particle
of
mass m subject
to
a central potential
$(r)
and
a perturbation potentialqyP read (Spitzer
and
Schwarzschild 1953)
.X
+ 4QAx -
2Qy
= -
oqyp/ox
,
.Y
+
2nx
=-
o4Jp/oy,
where A = A(R). The unperturbed
(qyp
= 0) motion
of
a particle is given by
X =
rx
- f COS
(Kt
+
<5)
, y =
2Arxt
+
')'
+
20
f sin
(Kt
+
<5).
K
Here
rx,
f,
y,
and
<5
are constants
and
K = K(R).
© American Astronomical Society • Provided by the NASA Astrophysics
Data
System
(32)
(33)
(34)