The satellites of Uranus all have significant non-zero eccentricities and in the case of Miranda a significant inclination as well. The Voyager 2 encounter with Uranus in January 1986, provided more accurate estimates of the masses and sizes of the satellites. Orbital history and possible tidal heating were reexamined by using the Voyager data. The relevant orbital dynamics equations are described

Plescia, J. B.

English

NASA Technical Reports Server

OmHTAI[. DmMCS OF Tft& 
P l e s c i a ,  U.Se Geological  Survey, 2255 N. Gemini Drive, F l a g s t a f f ,  AZ 86001 
The s a t e l l i t e s  of Uranus a l l  have s i g n i f i c a n t  non-zero e c c e n t r i c i t i e s  and 
i n  t he  case of Miranda a  s i g n i f i c a n t  i n c l i n a t i o n  as w e l l  (Table I ) ,  Squyres 
e t  a l e  (1985) s tud ied  t h e  time s c a l e s  and poss ib l e  energy e f f e c t s  a s soc i a t ed  
wi th  the  o r b i t a l  e v o l u t i o n  of t hese  bodies ,  s p e c i f i c a l l y  t he  e c c e n t r i c i t i e s .  
However, because of t h e  unce r t a in ty  i n  t h e  s i z e s  and masses ( d a t a  were based 
on es t imates  of Brown e t  a l . ,  1982 and Brown and Cruikshank, 1983),  t h e r e  w a s  
a  corresponding u n c e r t a i n t y  i n  t h e  r e s u l t s .  The Voyager 2  encounter  wi th  
Uranus i n  January 1986 provided more accu ra t e  e s t ima te s  of t h e  masses and 
s i z e s  of the  s a t e l l i t e s  (Smith et a l . ,  1986). A s  a r e s u l t ,  i t  seems u s e f u l  t o  
reexamine t h e  o r b i t a l  h i s t o r y  and poss ib l e  t i d a l  hea t ing  by using t h e  Voyager 
da t a .  This  is p a r t i c u l a r l y  r e l evan t  because Niranda, which had t h e  l a r g e s t  
pre-Voyager s i z e  u n c e r t a i n t y ,  is a l s o  t h e  s a t e l l i t e  which appears  t o  have 
undergone the  most ex t ens ive  endogenic evolu t ion .  
Niranda and, t o  a  c e r t a i n  e x t e n t ,  A r i e l  have undergone s i g n i f i c a n t  
endogenic a c t i v i t y .  C r a t e r  frequency d a t a  f o r  A r i e l  ( P l e s c i a  and Boyce, 
1986a), i n  combination wi th  the  c r a t e r i n g  time s c a l e  pos tu l a t ed  by Smith e t  
a l e  (1986), i n d i c a t e  t h a t  Ariel 's  su r f ace  i s  >3.5 Gy old and t h a t  no major 
r e su r f ac ing  has occurred s ince .  Miranda, on the  o t h e r  hand, has  t he  least 
c r a t e r e d  s u r f a c e  of any s a t e l l i t e  i n  t h e  Uranian system ( P l e s c i a  and Boyce, 
1986b) and may have s u r f a c e s  a s  young a s  s e v e r a l  hundred m i l l i o n  yea r s  o ld .  
The r e l evan t  o r b i t a l  dynamics equat ions  (Pea le ,  1977; Pea l e  e t  a l . ,  1980) 
follow. 
The change i n  o r b i t a l  e c c e n t r i c i t y  ( e )  with  time (t) is 
where k 2  is  t h e  s a t e l l i t e  p o t e n t i a l  Love number, n is t h e  s a t e l l i t e  mean 
motion, m and M a r e  t h e  masses of t h e  s a t e l l i t e  and Uranus, r is  t h e  s a t e l l i t e  
r ad ius ,  a i s  the  s a t e l l i t e  semimajor a x i s  and Q is  t h e  s a t e l l i t e  d i s s i p a t i o n  
func t ion .  The equat ion  is  an approximation because t h e  term r e l a t i n g  t o  t h e  
e f f e c t  of d i s s i p a t i o n  w i t h i n  Uranus has not  been included. D i s s i p a t i o n  wi th in  
Uranus is n e g l i g i b l e  and d i s s i p a t i o n  w i t h i n  the  s a t e l l i t e  is  t h e  dominant 
term. 
The p o t e n t i a l  Love number (k2)  is  
where u is t h e  s a t e l l i t e  r i g i d i t y ,  p is  the  s a t e l l i t e  d e n s i t y ,  and g t h e  
s u r f a c e  g r a v i t y ,  
F i n a l l y ,  t he  homogeneous t i d a l  hea t ing  r a t e  (E) f o r  a  synchronously 
r o t a t i n g  body is approximated a s  
where o is t h e  s p i n  o b l i q u i t y :  t h e  angle  between the  s p i n  vec to r  and t h e  o r b i t  
normal. I f  t h e  s a t e l l i t e  is i n  Cass in i  s p i n  s t a t e  1 (o l e s s  than  the  o r b i t a l  
i n c l i n a t i o n ) ,  t h e r e  is  no obliquity-decay hea t ing .  I f ,  however, t h i s  is not 
https://ntrs.nasa.gov/search.jsp?R=19870013917 2020-03-20T10:37:35+00:00Z
the case, the obliquity term can become important, 
Assuming u = 4 X 10'~ dyne cmW2, a value typical for low-temperature ice, 
and Q = 20, the orbital eccentricity decay times and homogeneous tidal heating 
rates were calculated (Table I), These results are similar to those of 
Squyres et ale (1985) and indicate that the calculated rates are relatively 
insensitive to the uncertainties in mass and density of the satellites, The 
calculations indicate that the present eccentricities of Miranda and Ariel 
should decay on time scales of lo7 years, whereas those of Umbriel, Titania, 
9 and Oberon are stable for lo8 to 10 years. The original eccentricity (eo) 
is, however, unknown and could have been quite large, 
To make the present orbital eccentricity of Ariel and Miranda long lived, 
such that a late-stage event to pump up the eccentricity is not neccessary, 
requires that either u or Q be increased. Neither parameter is well 
constrained because the interior thermal conditions and compositions of the 
satellites are uncertain, A Q of >I00 would produce an eccentricity decay 
time of lo8--lo9 years for Ariel and Miranda, Similarly, increasing u would 
also lengthen the decay time, Rigidities of 4 X 10" d ne similar to 
rock rather than ice, would increase the decay time to 10'- years. 
When obliquity heating is not important, the tidal heating rates from 
eccentricity decay are about one to several orders of magnitude below that of 
the radioactive heating (Table 1). The low tidal heating and low radioactive 
heating of Ariel, Umbriel, Titania, and Oberon are consistent with their 
generally ancient age, Miranda, however, presents a significant problem 
because of the youthful appearance of its surface. 
The inclusion of the obliquity term in equation (3) has important 
implications for Miranda, If Miranda occupies Cassini state 2 (o greater than 
the orbital inclination), then the obliquity term in equation (3) becomes 
important. Assuming that all of the satellites have obliquities equal to 
their inclinations, the tidal energy produced (II) was calculated (Table 1). 
For Ariel, Umbriel, Titania, and Oberon, the term remains unimportant because 
the obliquity is small, However, inclusion of the term for Miranda increases 
the heating rate by 3 orders of magnitude, 
Alternatively, if there is no obliquity heating of Miranda, the 
eccentricity heating rate can only be raised to approximately that of 
radioactivity. Enchanced eccentricity heating will result, relative to the 
model in Table 1, by reducing either Q or u .  For & = 1 and u = 4 x 1 0 ~ ~  dyne 
the heatin rate is 3.03 X loL5 erg sec-l, corresponding to a heat flow 
of 0.45 er s cm-5 sec-l. For Q = 20 and u = 4 X 10' dyne cmW2, the rate is 
1.52 X 10' erg sec-\ corresponding to a heat flow of 0.20 erg e m 2  sec-l. 
mPEmHCES: Brown, R,H,, and Cruikshank, D,P,, 1983, The Uranian satellites: 
Surface compositions and opposition brightness surges: Icarus, v. 55, p, 83- 
92; Brown, R,H,, Cruikshank, D,P., and Morrison, Dm, 1982, Diameters and 
albedos of satellites of Uranus: Nature, v, 300, p. 423-425; Peale, S.J., 
1977, Rotation histories of the natural satellites, - in (Burns, JeAes ed.) 
Planetary Satellites, pa 87-112; Peale, S.J,, Cassen, P., and Reynolds, R.T,, 
1980, Tidal dissipation, orbital evolution, and the nature of Saturn's inner 
satellites: Icarus, v, 43, p, 65-72; Plescia, J.B., and Boyce, JmMP, 1986a, 
Crater frequencies on Ariel, this volume; Plescia, J.B., and Boyce, J , M . ,  
1986b, Miranda cratering history, this volume; Smith, B.A., Soderblom, L,A., 
et al., 1986, Voyager 2 in the Uranian system: Imaging science results: 
Science, v. 233, pa 43-64; Squyres, Sew*, Reynoldsp R a T - ,  and Lissauer, J.J., 
1985, The enigma of the Uranian satellites' orbital eccentricities: Icarus, v .  
61, p. 218-223. 
Radius (km) 2424-5 5804-5 5954-10 8054-5 7754-10 
Density (gm 1.27+0.39 1.63i.0.3 1.4r4-0.28 1.5%-0.09 1.5&0.10 
Mass (gm) 7.48-x 1.35-x 1.27-x 3.47-x 2.92-x 
Gravity (cm seem2) 8.52 26.74 23.94 35.76 32.48 
Semimajor axis (km) 129783 191239 265969 435844 582596 
Eccentricity 0.0027 0.0034 0.0050 0.0022 0.0008 
Inclination O 4.2 0.3 0.4 0,l 0.1 
tidal. heating 11.52X10'~ 1 . 0 2 X 1 0 ~ ~  1 . 7 0 ~ 1 0 ~ ~  8.18X10L3 8.37X101' 
erg se' (Q=20) I1 1.61 X 1017 1.36 X 1016 2.17 X 1015 8.91 X 1013 1.41 X 1012 
tidal energy 
heat flow I 0.02 0.24 0.04 0.001 0.00001 
erg cme2 se' 11 21.88 0.32 0.05 0.001 0.00002 
radioactive 
heat flow 0.15 0.93 0.64 
erg sec-' 
I does not include the obliquity term in equation (3) 
II includes the obliquity term in equation (3). 


Orbital dynamics of the Uranian satellites based on Voyager data

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server