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An interpretation of titius-bode law
2 (1978) 183-192
© P e r g a m o n Press. Printed in Great Britain
ActaA8tr, Sinica 17 (1975) 123-130
Chinese Astronomy
0146-6364/78/1201-0183-$07.50/O
AN INTERPRETATION OF TITIUS-BODE LAW
Dai
Wen-sai
Department of Astronomy, NankingUniversi~ (Received 1975 April 7)
ABSTRACT The distance of a planet or a regular satellite from the central body is related to the mass of the planet or regular satellite. The boundary between two adjacent planetary regions (or regular satellite regions) is related to the mass ratio of the two planets (or regular satellites). When the boundary is properly chosen, it is found that the width AP of the planetary (regular satellite) region is almost proportional to the size of the gravitational region 2x = 2(m/3M)I/3r, the ratio AP/2~ decreases outwards.
This result is a strong support of the view that
planets (solid planetary cores in the case of Jupiter and Saturn) and regular satellites are formed by the accumulation of planetesimals, and that they did not go through the stage of huge proto-planets and protosatellites.
(1) In 1766, a German mathematics teacher, Titius, discovered that the distances of the planets of the solar system formed the series 4, 4 + 3, 4 + 6, 4 + 12, . . . .
A few years later, the
Director of Berlin Observatory, Bode, further described this law of planetary distances. Subsequently, this law was called the "Titlus-Bode Law".
rn
It can be expressed as
(I)
= c I + 02.2n'2
where n is the serial number of planet according to increasing distance from the Sun, n = 1 for Mercury, n = 2 for Venus, and so on, r
is the orbital seml-major axis of the n-th planet, n
and Cl and c 2 are two constants. ~I = 0.4 and c 2 = 0.3.
If the astronomical unit is taken as unit distance, then
For n = 1, the exponent n - 2 should be modified to read -=.
value did not fit the observed value for Jupiter, while the r6-value did. led to the discovery of the asteroids at the beginning of the 19th century. calculated value of ~ in
The r 5-
The gap at n = 5 For n = 8~ the
was very close to the observed value for the planet Uranus, discovered
1781, but for n = 9 and 10, the calculated values differ greatly from the observed values
for Neptune and Plauto. T1tius-Bode type of laws also exist for some of the satellites of Jupiter, some of the
1 84
Titius-Bode Law
satellites of Saturn and all the satellites of Uranus.
If we take 10 0 0 0 k m as unlt distance,
then for J-V, J-I, J-ll, J-Ill, J-IV, we have o I = 19, c~ = 22; for S-I through S-V, we have o 2 = 18.6, and o2 = 4.8; for the satellites of Uranus, 01 = 14 and o 2 = 6. of the planets, when n = i, the exponent n - 2 should be changed to -=.
As in the case
In TABLE I, we list
for comparison the observed values of P
and the values calculated according to (I) for n J-V,I, II,III,IV, S-I through S-VII and U-V,I,II,III,IV. For the asteroids, the observed r
Table I
Planet
1-n(calc.1
Co~par.ison between the C.alculate.d and Observed r ~z for Planets and Parts of Satellite Systems ~(obs.)
Satellite ~ ( c a l c . )
~(obs.)
Satellite ~ ( c a l c . 1
Pn(Obs.1
Mercury
0.4
0.387
J-V
19
18.1
S-I
18.6
18.6
Venus
0.7
0.723
J-I
41
42.2
S-II
23.4
23.8
Earth
1.0
1.000
J-II
63
67.1
S-III
28.2
29.5
1.523
J-III
107
107.1
S-IV
37.8
37.7
J-IV
195
188.3
S-V
57.0
52.7
Mars
1.6
Asteroids
2.8
(2.71 5.203
U-V
1~
]3.0
S-VI
95.4
122.2
9.52
U-I
20
19.2
S-VII
172~2;
148.3
U-If
26
26.7
U-Ill
38
43.8
U-IV
62
58.6
Jupiter
5.2
Saturn
10.0
Uranus
19.6
Neptune
38.8
30.2
Pluto
77.2
39.5
i s t h e mean f o r t h e f i r s t and S - V I I d i f f e r
n
19.2
three asteroids.
g r e a t l y from t h e i r
The v a l u e s c a l c u l a t e d a c c o r d i n g to ( I ) f o r S-VI
observed values.
A n o t h e r form o f T i t i u s - B o d e Law i s
~ + 1 / rn
=
S
(2)
In this expression, B is roughly a constant for the planets, while for the satellite systems, it is also some constant between I and 29 apart from a few exceptions.
In many theories of
the origin of the solar system~ it is the form (2) rather than the form (11 that is used when interpreting the Titius-Bode Law.
The observed values of B for the planets and for
the satellites of Jupiter, Saturn and Uranus are given in TABLE 2.
In constructing this
table, we regard J-VI, J-VII and J-X as one satellite and J-VIII and J-IX as another. For the satellites of Jupiter and Saturn, we distinguish between "regular" and "irregular" satellites.
Those satellites whose orbits have small eccentricities and are inclined at
small angles to the primary's equator are called "regular" satellites; those with large eccentricltes and inclinations are called "irregular" satellites. Of the 13 satellites of Jupiter, only 5 (J-V,I,II,III,IVI are regular satellites; of the 10 Saturnian satellites, S-VIII and S-IX are irregulars. all 5 satellites of Uranus are regular satellites. involving the irregulars are all abnormal.
According to the above definitionsp
From TABLE 2, we see that the B-values
Even among the 5 regulars of Jupiter and the
8 regulars of Saturn, the B-value is not a constant.
The 8-value for J-V and J-I is as
large as 2.33, and that for S-V and S-VI is as large as 2.32, and both pairs consist of objects that differ greatly in their mass, J-V being 4 orders of magnitude less massive than
T1tius-Bode Law
Observe d Values of ~+2/_~r of Planets and Satellites
Table 2
rn+l/l" n
Planet
Satellite
Mercury 1.87 Venus 1.38 Earth 1.52 Mars (Asteroids) (1.77) Jupiter (1.92) 1.83 Saturn 2.02 Uranus Neptune Pluto
J-I,
1 85
,rn+i/rn
J-V J-I J-II J-III J-IV J-XIII
2.33 1.59 1.59 1.76 5.42
S-X S-I S-II S-III S-IV S-V
1.14 J-VI,VII,X 1.95 J-XII,XI,VIII,IX U-V ~'' 1.48 U-I 1.39 U-II 1.64 U-II1 1.34 U-IV
1.57 1.31
8-value is particularly
of a planet
from i t s
(or satellite)
Many a t h e o r y o f t h e s o l a r
3.64 S-IX
regarding the origin
small.
The m a s s o f P l u t o i s 1/154
A l l t h e s e show t h a t
body i s r e l a t e d
have been generally
regard the distance
p r o p o s e an i n t e r p r e t a t i o n
2.32 1.21 2.40
to its
of the Titius-Bode
regarded as satisfactory.
t o be r e l a t e d
to the mass.
o f T i t l u s - B o d e Law and we s h a l l
of the solar
the distance
mass.
system has given this or that interpretation
Law; none o f t h e i n t e r p r e t a t i o n s of the interpretations
central
1.17 1.28 1.24 1.28 1.24
S-VI S-VII S-VIII
and S-V b e i n g 2 o r d e r s o f m a g n i t u d e l e s s m a s s i v e t h a n S-VI.
t h a t o f N e p t u n e , and t h e i r
~+1/~
Satellite
Only a few
I n t h i s p a p e r , we
d e r i v e from i t
certain
conclusions
system.
(n) I n 1951, F e s s e n k o v [1] m a i n t a i n e d t h a t t h e m a i n f a c t o r planet
f r o m t h e Sun i s t i d a l
instability
circum-solar nebular disk is greater can condense o u t . was formed f i r s t ,
The f u r t h e r
out a planet
is,
between the planets
t o t h e Sun t h a t p l a n e t s
t h e s m a l l e r i s t h e Roche d e n s i t y ;
Mercury.
of a
i s o n l y when t h e m a t t e r d e n s i t y i n t h e
t h a n t h e Poche d e n s i t y r e l a t i v e
t h e n N e p t u n e , and l a s t l y ,
and t h e m u t u a l d i s t a n c e s the tides
and t h a t i t
in determining the distance
The d i s t a n c e
are therefore
of a planet
hence Pluto
f r o m t h e Sun
d e t e r m i n e d by t h e s o l a r
t i d e and
r a i s e d by t h e a l r e a d y - f o r m e d o u t e r n e i g h b o u r s .
m,, g ( r . - - r,._O s ' where K i s a c o n s t a n t .
Me r~,'
(3)
Prom ( 3 ) , we h a v e
(4) rm--i Comparing with (2), we have
p=
l
+
.
Thus, according t o Fessenkov's argument, 8 is a function of the planetary masses.
(5)
In order
to make the calculated and observed values of I- agree, Fessenkov made several unverified n assumptions: the first is to multiply the masses of the 4 terrestrial planets each by 30,
18 6
Titius-Bode Law
regarding their fohner selves as being massive proto-planets and the gas that was their bulk content as having evaporated; the second is to set K = 18 at first and later to change to K = 13.25; the third is to change the ~
in (4) into ~ j n - 1
=
(mn÷mn-])/2"
Because
there are no sufficint grounds for these assumptions and because the statements that the further out a planet, the earlier is its formation and that the planets were at one time huge proto-planets have not proved to be convincing, Fessenkov's interpretation of TitiusBode Law has not been accepted.
However, his point that the planetary distance is related
to the planetary mass is corregt. In a paper pulbished in 1949, Kuiper [2] gave an empirical relation between the distance of a planet or satellite from its central body and its mass.
Let m be the mass of the n-th n planet or satellite in units of the mass of its central body, and let •
1
(rt +
~gmm
r2)
(6) 2
2 then the relation was ~s
where C is a constant, roughly equal to 10 -~.
C
1
(7)
In another paper in 1951, Kuiper [3] found (8)
m . ' ~ - -'O ' O 0M 0l(~)
here, the A is not the same as in (6) and (7), it is equal to P 2 - r]" is basically the same as (7).
The expression (8)
Kuiper maintained that because of gravitational instability,
the nebular disk collapsed into a number of huge proto-planets at locations where the density was high and exceeded the Roche density.
The p~oto-planets later changed into planets
Kuiper laid stress on the planets being huge proto-planets at a former time, hence his theory has been called the '=proto-planet theory".
In his paper [3] and later papers, he
took m to be the mass of the proto-planet. We have repeated Kuiper's 1949 calculatlons to see whether or not ~/A 3 is a constant, and if not, how large the dispersion is.
New values of the planetary masses have been used.
The values of m and ~/A 3 are shown in TABLE 3.
t~ 1/17
The values of ~/A 3 ranges from 30 times
times 10 -~, the largest being 520 times the smallest.
Table 3 Planet
Values of m, r, ~ / 4 ' , z , At/a= of Planets
m
X
(g)
(cm)
3.332 X 10u
5.791X 10*=
5 . 8 9 X 10 "~
Mercury Venus
4 . 8 7 0 X 10"
1.0821 X lO ts
8.23X 10-j
Earth Mars
5.976X 10zv 6.421X10 u
1 . 4 9 6 0 X 10=s
2.33 X 10-s 3.65X 10-.4 3. OSX I ~ 3 5.46X 10-4 5.55X10-~
Jupiter Saturn Uranus
Neptune Pluto
1.8997 X 10 m
5.6879X 10*9 8.7424X 109* 1.0292X 10" 6.632X lOa*
2.2794X 10=3 7.753X10 t9 1.427X10 =* 2 . ~ 9 6 X 10'4 4.4966X 10 '4 5.900X 10=`
1.32 X 1~3
(cm)
"r/2x
2.21X I0 l° 1.01Xl0 u
96.0
1 . 5 0 X l O t'
1 . 1 6 X I O a3
19.2 39.9 5.19 7.44 11.2 6.71
2 . 8 4 X 1 0 t*
13.2
1 . 0 8 X 10 u 5 . 3 1 X IO ta 6 . 5 2 X 1 0 t* 7 . 0 2 X lO t*
23.8
Titlus-Bode Law
18 7
The other formula of Kuiper's can be written in the form m==K-3(~)s M extracting
'
(9)
t h e cube r o o t s o f b o t h s i d e s and m u l t i p l y i n g by r , we have A r .= K
r.
The radius ~ of the "gravitational region" of a planet with respect to the Sun can be expressed by the formula x--
(11)
r.
This formula is derived in the circular model of the restricted problem of three bodies, taking as the gravitational region with respect to the Son the Roche equipotential surface enclosing the planet and passing through the first Lagrangian point (cf. [4,5]).
Inside
the gravitational region, the gravitational influence of the planet is greater than that of the Son.
We do not take take AP to be P2 - PI' rather, we take it to be the width of the
planetary region; from (10) and (11), we obtain A r - - 3~ K - ' K u 2x 2 If Kuiper's empirical relation directly
is precise,
(12)
t h e n t h e w i d t h o f a p l a n e t a r y r e g i o n w i l l be
proportional to the size of the gravitational
region relative
t o t h e Sun.
Usually,
t h e i n n e r b o u n d a r y o f t h e n - t h p l a n e t a r y r e g i o n i s t a k e n t o be ( ~ _ ] l " n ) 1 / 2 , and t h e o u t e r b o u n d a r y t o be
(z,nz,n+l)1/2.
We s h a l l
follow this practice.
The v a l u e o f z and At/2= so
c a l c u l a t e d a r e g i v e n i n Columns 5 and 6 o f TABLE 3.
For t h e i n n e r m o s t Mercury and t h e
o u t e r m o s t P l u t o and t h e two p l a n e t s Mars and J u p i t e r
on e i t h e r
s u p p o s e e q u a l o u t e r and i n n e r h a l f w i d t h s .
t h e d i s p e r s i o n h e r e i s t h u s s m a l l e r t h a n i n t h e ~/A 3 v a l u e s . made f o r t h e r e g u l a r s a t e l l i t e s
of Jupiter
side of the asteroids,
we
The l a r g e s t ~w/2= i s 18.5 t i m e s t h e s m a l l e s t , S i m i l a r c a l c u l a t i o n s were
and S a t u r n , and t h e s a t e l l i t e s
o f Uranus, t h e
results are shown in Col,--- 4 of TABLE 5, the ratio of the largest to the smallest is 70 for the Jovlans, 13 for the Saturnians, and 5 for the Uranlans. The calculated values of AP/2zapproxlmatemore but the dispersion is still considerable; lightweights, Mercury, Mars and J-V.
to a constant than do the values of ~/A 3,
the values of AP/2z are especially large for the
This makes us realise that it is not appropriate to
define the boundary of a planetary region by the geometrlcalmean of the distances of the planet and its nelghbour.
Since AP increases with increasing m, the boundary of a planetary
region must also be related to m.
Suppose mn+ I > mn, then the llne of demarcation between
the (n+1)-th and the n-th regions m u s t be closer to the n-th planet.
We shall take the
demarcation line to be where the tidal forces of the two nelghbours are equal, that is, we divide the l e n g t h ~ + 2 - 2 , n into two parts in the ratlomn]/3 a more reasonable boondarybetween
the two neighbouring regions.
: mn+2 I/3 and thereby fix For the four planets,
Mercury, Pluto, Mars and Jupiter, we still suppose the inner and outer two halves of each region to have the same width.
The planetary regions so calculated are given in Column 2 of
18 8
Titius-Bode Law
TABLE 4.
Using the new boundaries, the r e v i s e d
values of AP/2x for the planets are given
in Column 3, the dispersion is much smaller than before.
For the regular satellites of
Jupiter and Saturn and the satellites of Uranus, the revised values of Ar/2.z calculated by the same method are given in the last column of TABLE 5; here, too, the dispersions are much less than before. Tab,le 4
Limits ofPl,anetayy Regions_..and Revised Values of Ar/2x Limits of Planetary Region (AU) ....
Planet Mercury Venus Earth Mars Jupiter
0.287 0.486 0.857 1.355 2.60 -
Saturn Uranus Neptune Pluto
Table 5
7.80 15.82 24.47 37.97
Values of ~ m
Satellite
(g)
-
0.486 0.857 1.355 1.693 7.80
15.82 24.47 37.97 40.91
Revised Value of ~" / 2 z 66.0
27.5 24.9 23.3 7.31
9.20 9.22 8.69 7.76
pj AP/2x of Regular Satellites ff
(10 '° cm)
=r/2x
3.42X10 't
1.81
7.89X10" 4.764X10" 1.535X102~ 9.113X10"
4.22 6.71 10.70 18.83
1.6X10 '~ 3.75X 10'2
1.59 1.86
67.2 36.9
34.6 36.4
S-II S-Ill
8.42X10'* 6.228X 10'3
2.38 2.95
S-IV S-V
1.160XI0 u 1.SXI0 u 1.4003X 10" 1.1 X 10~3
3.77
31.3 16.2 16.9
28.0 17.6 17.3
5.27 12.22 14.83
33.2 5.12 23.0
19.8 7.55 3.71
1.30 1.92
31.1 10.4
19.9
2.67 4.38
17.2 7.36
5.86
6.35
J-V J-I J-ll J-Ill J-IV S-X S-I
S-VI S-VII U-V U-I U-II U-Ill U-IV
8.7X10" 1.31XI0 u 5.25X 10'3 4.37 X 10" 2.54X 10*4
625 12.6 11.6
=r/2s revised
8.92
9.77
53.6 18.1 10.1 10.6 7.83
13.3 13.1 8.72 5.39
The calculated results show that the distance of a planet (or regular satellite) from its central body is related to its mass, that the width of a planetary (regular satellite) region is roughly proportional to its gravitational region, and since the latter is proportional to m 1/3, the width of the planetary region increases with the planetary mass. The size of the gravitational region is also proportional to the distance of the planet (satellite) from its central body, but Form (2) of Titius-Bode Law a ~ L ~ p ~ e ~ e ~ only one aspect that the width of a planetary region increases with increasing distance, it does not
Titius-Bode Law
express the relation between the width and the mass.
189
There is another result of the calculation
that should be noted: the further out the planet (satellite), the smaller is its A1-/2x. (In) To explain Titius-Bode Law, we must answer the following three questions: (I) Why are the widths of the planetary regions proportional to the diameters of the gravitational regions ? What problem does this elucidate ? (2) Why is it that the regular satellites possess the same type of law of distances as the planets do ? (3) Why are the Ar/2x values smaller at larger distances from the central body ? There have already been over twenty nebular theories of the origin of the solar system that maintain that both the planets and the satellites were formed from disk surrounding the Sun.
matter in a nebular
As regards how the planets were formed out of this matter, there
have been two quite different views.
According to one, because of gravitational instability,
the nebular matter first formed a number of very large proto-planets.
Inside the proto-planets,
the gas condensed into the solid particles, which sank to the centre and there formed a solid core, the gas in the outer part then left thp proto-planet through evaporation and through being driven away by the various solar radiations, so that only the solid core remianed behind (for Jupiter and Saturn, with part of the gas in the inner part added on), and so the proto-planets were transformed into planets.
The other view is that, because of
their relative movements, the solid particles inside the nebular disk underwent mutual collisions at which the smaller particle was often swallowed up by the large one, the two joining together to become one. called planetesimals.
In this way,.small particles grew into large solid peices
When a planetesimal became sufficiently massive, it began to "accrete"
other approaching planetesimals without the latter colliding with it.
Some of the larger
planetesimals thus became planetary embryos, and when the embryos grew to certain sizes, they would begin to accrete gas, if there was gas about.
Jupiter and Saturn were probably
formed in this manner, for their solid cores are only a small part of their total masses, most of which is gaseous or liquid.
The embryos grew gradually into planets.
We shall now answer the first question.
The fact that the widths of the planetary regions
are directly proportional to the masses shows that the second view is probably the correct view.
If the planets were fromed from proto-planets,
then this would not demand that the
widths be proportional to the masses, whereas if the planets were formed through the accumulation of material particles, then this feature would be demanded.
The gravitational
region of an embryo grows with the embryo, although even at the end of the formation process, the dlamter of the gravitational region is only a fraction of the width of the planetary region, the relative movement among the planetesimals means that the accretion region of an embryo is much greater than its gravitational region.
Because of their mutual perturbations
the orbital eccentricities and inclinations of the planeteslmals were constantly changing, thus producing their relative movements. the gravitational region of an embryo.
Planetesimals would thus be constantly entering If the velocity of entry is less than the velocity of
escape a t t h e s u r f a c e of t h e g r a v i t a t i o n a l a c c r e t e d by t h e embryo.
This v e l o c i t y
r e g i o n (2GM/~) ½ , t h e n t h e p l a n e t e s i m a l w i l l be
f o r the i n c i p i e n t Mercury, E a r t h , J u p i t e r and Neptune
is respectively equal to 0.45, 0.73, 2.18 and 0.34 km/sec.
If the planetesimal enters the
19 0
Titlus-Bode Law
gravitational sphere with a greater velocity, then so long as its impact parameter * is less than x, or if it is going to directly hit the embryo, then the planetesimal will still be accreted.
During the period of formation of the planets, planeteslmals in general have
small velocities, so that as soon as one enters a gravitational sphere, it will generally be accreted.
In this manner, the larger the gravitational region, the greater is the
possibility of accreting planeteslmels, therefore the observed fact that the width of the p l a n e t a r y
region
is roughly proportional to the diameter of the gravitational
region shows that the planets were formed through the accumulation of planetesimals and not from proto-planets. The second question was: "Why do the systems of regular satellites have the same type of distance laws as the system of planets has ?". were formed in the same way as the planets.
The answer is that the regular satellites
In the Jupiter and Saturn region of the
nebular disk, the temperature was lower than in the terrestrial region and so a large part of the volatile gases (mainly hydrogen and helium) remined there.
Furthermore, the gas
t h e r e d i d n o t e s c a p e a s d i d t h e g a s i n t h e r e g i o n o f U r a n u s , Neptune and P l u t o , where the velocity
o f e s c a p e was s m a l l b e c a u s e o f t h e weaker s o l a r g r a v i t a t i o n .
formation process of Jupiter planets.
core which, after first
reaching a certain
formed a l a r g e f l a t
the shell,
their
eccentricities orbits portion
and S a t u r n was n o t q u i t e
Each b e g a n w i t h s m a l l s o l i d p a r t i c l e s
velocity
shell
t h e same a s t h a t o f t h e o t h e r
and i c y p i e c e s a c c u m u l a t l n g i n t o a s o l i d
mass, began to accrete
around the solid core.
gas gravitationally.
the planets
to the p r o t o - p l a n e t s
p o i n t i s f a v o u r a b l e to the a c c r e t i o n
According to Kuiper's
500 t i m a s t h e p r e s e n t E a r t h ' s
present value, sufficient
o r 68 k m / s e c .
in their
theory of the proto-planets,
m a s s , so t h a t
distances
proto-planets,
i n e x p l a i n i n g how t h e e x c e s s m a s s e s l e f t
o f t h e d e n s i t y now, t h e n t h e v e l o c i t y
was lower s t i l l , proto-planets.
t h e m a s s o f P r o t o - E a r t h was
o f e s c a p e a t t h e s u r f a c e would be 6 . 0 7 t i m e s t h e
The t e m p e r a t u r e o f t h e g a s on t h e P r o t o - E a r t h would n o t be
and i t would be more d l f f l c u l t If the sate11ites
still
For t h e J o v i a n p l a n e t s ,
would f a l l the temperature
f o r the e x c e s s mass to leave the
were a l s o t r a n s f o r m e d f r o m h u g e p r o t o - s a t e l l i t e s ,
i t would be e v e n more d i f f l c u l t
t o e x p l a i n how t h e e x c e s s m a s s was g o t r i d o f ,
central
systems - the planets
bodies of the satellite
t h e n we
i f we s u p p o s e t h e d e n s i t y t h e n t o be ] / S t h a t
t o d r i v e away 4 9 9 / 5 0 0 o f t h e m a s s .
l l k e t h e Sun d i d .
as
the proto-
f o r t h e g a s t o e v a p o r a t e away and a l s o t h e v a r i o u s s o l a r r a d i a t i o n s
by a l a r g e f a c t o r
A large
h y p o t h e s i s and u n f a v o u r a b l e
I f t h e p l a n e t s were o r i g l n a l l y
dlfficulty
sate11ites.
became t h e m i d d l e and o u t e r l a y e r s o f t h e p l a n e t .
h a v e t h e same t y p e o f r e g u l a r i t y
hypothesis.
would h a v e t h e g r e a t e s t planets.
later
satellites
do, and t h i s
orbltal
would a l s o be r e d u c e d so a s t o r e v o l v e i n r o u g h l y c l r c u l a r
of the gas of the shell the regular
entered
o f t h e g a s , and t h e i r
a r o u n d t h e s o l l d c o r e and s u b s e q u e n t l y t o g a t h e r i n t o r e g u l a r
Therefore,
The g a s
When t h e p l a n e t e s i m a l s
would be r e d u c e d by t h e r e s i s t a n c e
and i n c l l n a t i o n s
Hence t h e
then
since the
- d i d n o t e m i t any s t r o n g r a d i a t i o n s
Recently, it has been discovered that,
slmilar
t o what i s on t h e
s u r f a c e o f t h e Noon, t h e r e a r e n u m e r o u s c r a t e r s
o f a l l s i z e s on N e r u c r y which h a s p r a c t i c a l l y
~rhe formula for the impact parameter is,
b ° = b [I
denote respectlvely p l a n e t e s l m a l when i t
[6],
+ 2 ( ~ / ~ v 2 ] ~ , where m and b
t h e m a s s and r a d i u s o f t h e embryo, and v i s t h e v e l o c i t y i s f a r f r o m t h e embryo.
of the
791
Titius-Bode Law
no atmosphere and on Mars which has only a very thin atmosphere, which shows that there had been planetesimals of all sizes falling on these bodies.
The irregular satellites are
probably objects captured by the planets, though they were also formed by the accumulation of solid particles, they did not enter the extended gaseous shells surrounding the planets, or the planets concerned did not have such shells, hence they have in general large orbital eccentricities and inclinations. The third question was "Why were the values of Ar/2x smaller at larger distances from the central body ?".
The width of a planetary region is not only related to the size of the
gravitational region, but also to the relative velocities of the planetesimals.
We have
already mentioned that the mutual perturbations of the planetesimals will change their initially circular orbits into ellipses of various values of e and i, giving rise to mutual relative velocities.
Suppose at a distance P from the Sun, not only does the
circular orbit of a planetary embryo pass, but also pass the perihelion point of a planetesimal orbit of eccentricity e I and the aphelion point of another orbit of eccentricity e 2. Suppose further that the two planetesimal orbits have zero inclinations and semi-major axes u I and u~ respectively.
We then have s,(1
-
e,)
, -
.,O
+
e,) ~
(13)
,.
Using the vis viva integral, it is easily shown that (14)
~ v 2 - - vo - - v , ffi
(1 -- V~
-- e,);
(is)
in these, V° = (GM/r) ~ is the velocity of the embryo in its circular orbit. V O % 30 km/sec, and if AV ffi I km/sec, then e I = 0.068 and e2 = 0.066. maximum
For the Earth,
Here AV is the
relative velocity of a planetesimal falling into the Earth embryo at the final
stage of the Earth's formation.
At this time, the width of the accreting zone is roughly
equal to
[at(1 + et) - - r] + [r - - s , ( 1 - - e2)]. Substituting from (13), we have
2(e1+ e~)
or 0.268r.
(16)
This is 13.4 times the diameter
of the gravitational region, and is 54% of the Earth region of 0.498r.
A slight increase
in the value of the maximum relative velocity of the planetesimal will make the Earth's accreting region at the final stage of the Earth's formation as wide as the Earth region. The nebular disk surrounding the Sun is mainly gas, and because the z component of the solar gravitation does not fall with increasing distance as fast as does the force of presuure gradient, the disk is thin in the inner part and thick in the outer.
The small
solid particles and icy pieces, in virtue of the z component of the solar gravitation, slowly overcome the viscous force of the gas and settle to the neighbourhood of the plane of symmetry of the disk, and ther form a thin absorbing layer.
This absorbing layer, also
because of the outward decrease in the z component, is also thin in the inner part and thick in the outer, which means that the large the seml-major axis, the greater is the average inclination.
The larger the inclination, the smaller is the eccentricity projected
on the plane of symmetry, and hence the smaller is the projected relative velocity so that the width of the accreting zone, this is, the width of the planetary region AI• will
19 2
Titius-Bode
be correspondingly we h a v e e x p l a i n e d ~/2m.
smaller. why i t
In our calculation
masses, oreder
A similar
is that,
of the m-values
but a more reasonable of magnitude
This explains
interpretation
procedure
down on t h e i r
why o u r v a l u e s
We h a v e now a n s w e r e d a l l
situation
the further
entire
Law
holds
of Jupiter
of the Titius-Bode
the regular
and Saturn,
satellites.
body,
bodies),
questions
the m-values
we u s e d t h e i r
and Saturn
should
Thus
the smaller
w o u l d be t o u s e t h e m a s s e s o f t h e i r
of Ar/2x for Jupiter three
for
away f r o m t h e c e n t r a l
solid
is
total cores
then be smaller.
are somewhat too small.
and have thus given a rather
satisfactory
Law.
REFERENCES
[1] [2] [3]
Fessenkov, V.G., Astr. gh. 28 (1951), 492-517. Kuiper, G.P., Astz~hZ~S. J. 109(1949), 308-313. Kulper, G.P., "Origin of the Solar System", in "Astrophysics" edited by J.A.Hynek, Chapter 8, pp.357-424, partlcularly p.380 (1951).
[4] [5]
Lyttleton, R.A., t4on.No"c.R,astz,on.,F,oe.15~(1972) 463-484. Houlton, F.R., "Celestial Hechanics" (1914). H a r t m a n n , W . K . , Astz,ophys. J. ] 5 2 ( 1 9 6 8 ) 3 3 9 - 3 4 2 .
[6]
(one
© P e r g a m o n Press. Printed in Great Britain
ActaA8tr, Sinica 17 (1975) 123-130
Chinese Astronomy
0146-6364/78/1201-0183-$07.50/O
AN INTERPRETATION OF TITIUS-BODE LAW
Dai
Wen-sai
Department of Astronomy, NankingUniversi~ (Received 1975 April 7)
ABSTRACT The distance of a planet or a regular satellite from the central body is related to the mass of the planet or regular satellite. The boundary between two adjacent planetary regions (or regular satellite regions) is related to the mass ratio of the two planets (or regular satellites). When the boundary is properly chosen, it is found that the width AP of the planetary (regular satellite) region is almost proportional to the size of the gravitational region 2x = 2(m/3M)I/3r, the ratio AP/2~ decreases outwards.
This result is a strong support of the view that
planets (solid planetary cores in the case of Jupiter and Saturn) and regular satellites are formed by the accumulation of planetesimals, and that they did not go through the stage of huge proto-planets and protosatellites.
(1) In 1766, a German mathematics teacher, Titius, discovered that the distances of the planets of the solar system formed the series 4, 4 + 3, 4 + 6, 4 + 12, . . . .
A few years later, the
Director of Berlin Observatory, Bode, further described this law of planetary distances. Subsequently, this law was called the "Titlus-Bode Law".
rn
It can be expressed as
(I)
= c I + 02.2n'2
where n is the serial number of planet according to increasing distance from the Sun, n = 1 for Mercury, n = 2 for Venus, and so on, r
is the orbital seml-major axis of the n-th planet, n
and Cl and c 2 are two constants. ~I = 0.4 and c 2 = 0.3.
If the astronomical unit is taken as unit distance, then
For n = 1, the exponent n - 2 should be modified to read -=.
value did not fit the observed value for Jupiter, while the r6-value did. led to the discovery of the asteroids at the beginning of the 19th century. calculated value of ~ in
The r 5-
The gap at n = 5 For n = 8~ the
was very close to the observed value for the planet Uranus, discovered
1781, but for n = 9 and 10, the calculated values differ greatly from the observed values
for Neptune and Plauto. T1tius-Bode type of laws also exist for some of the satellites of Jupiter, some of the
1 84
Titius-Bode Law
satellites of Saturn and all the satellites of Uranus.
If we take 10 0 0 0 k m as unlt distance,
then for J-V, J-I, J-ll, J-Ill, J-IV, we have o I = 19, c~ = 22; for S-I through S-V, we have o 2 = 18.6, and o2 = 4.8; for the satellites of Uranus, 01 = 14 and o 2 = 6. of the planets, when n = i, the exponent n - 2 should be changed to -=.
As in the case
In TABLE I, we list
for comparison the observed values of P
and the values calculated according to (I) for n J-V,I, II,III,IV, S-I through S-VII and U-V,I,II,III,IV. For the asteroids, the observed r
Table I
Planet
1-n(calc.1
Co~par.ison between the C.alculate.d and Observed r ~z for Planets and Parts of Satellite Systems ~(obs.)
Satellite ~ ( c a l c . )
~(obs.)
Satellite ~ ( c a l c . 1
Pn(Obs.1
Mercury
0.4
0.387
J-V
19
18.1
S-I
18.6
18.6
Venus
0.7
0.723
J-I
41
42.2
S-II
23.4
23.8
Earth
1.0
1.000
J-II
63
67.1
S-III
28.2
29.5
1.523
J-III
107
107.1
S-IV
37.8
37.7
J-IV
195
188.3
S-V
57.0
52.7
Mars
1.6
Asteroids
2.8
(2.71 5.203
U-V
1~
]3.0
S-VI
95.4
122.2
9.52
U-I
20
19.2
S-VII
172~2;
148.3
U-If
26
26.7
U-Ill
38
43.8
U-IV
62
58.6
Jupiter
5.2
Saturn
10.0
Uranus
19.6
Neptune
38.8
30.2
Pluto
77.2
39.5
i s t h e mean f o r t h e f i r s t and S - V I I d i f f e r
n
19.2
three asteroids.
g r e a t l y from t h e i r
The v a l u e s c a l c u l a t e d a c c o r d i n g to ( I ) f o r S-VI
observed values.
A n o t h e r form o f T i t i u s - B o d e Law i s
~ + 1 / rn
=
S
(2)
In this expression, B is roughly a constant for the planets, while for the satellite systems, it is also some constant between I and 29 apart from a few exceptions.
In many theories of
the origin of the solar system~ it is the form (2) rather than the form (11 that is used when interpreting the Titius-Bode Law.
The observed values of B for the planets and for
the satellites of Jupiter, Saturn and Uranus are given in TABLE 2.
In constructing this
table, we regard J-VI, J-VII and J-X as one satellite and J-VIII and J-IX as another. For the satellites of Jupiter and Saturn, we distinguish between "regular" and "irregular" satellites.
Those satellites whose orbits have small eccentricities and are inclined at
small angles to the primary's equator are called "regular" satellites; those with large eccentricltes and inclinations are called "irregular" satellites. Of the 13 satellites of Jupiter, only 5 (J-V,I,II,III,IVI are regular satellites; of the 10 Saturnian satellites, S-VIII and S-IX are irregulars. all 5 satellites of Uranus are regular satellites. involving the irregulars are all abnormal.
According to the above definitionsp
From TABLE 2, we see that the B-values
Even among the 5 regulars of Jupiter and the
8 regulars of Saturn, the B-value is not a constant.
The 8-value for J-V and J-I is as
large as 2.33, and that for S-V and S-VI is as large as 2.32, and both pairs consist of objects that differ greatly in their mass, J-V being 4 orders of magnitude less massive than
T1tius-Bode Law
Observe d Values of ~+2/_~r of Planets and Satellites
Table 2
rn+l/l" n
Planet
Satellite
Mercury 1.87 Venus 1.38 Earth 1.52 Mars (Asteroids) (1.77) Jupiter (1.92) 1.83 Saturn 2.02 Uranus Neptune Pluto
J-I,
1 85
,rn+i/rn
J-V J-I J-II J-III J-IV J-XIII
2.33 1.59 1.59 1.76 5.42
S-X S-I S-II S-III S-IV S-V
1.14 J-VI,VII,X 1.95 J-XII,XI,VIII,IX U-V ~'' 1.48 U-I 1.39 U-II 1.64 U-II1 1.34 U-IV
1.57 1.31
8-value is particularly
of a planet
from i t s
(or satellite)
Many a t h e o r y o f t h e s o l a r
3.64 S-IX
regarding the origin
small.
The m a s s o f P l u t o i s 1/154
A l l t h e s e show t h a t
body i s r e l a t e d
have been generally
regard the distance
p r o p o s e an i n t e r p r e t a t i o n
2.32 1.21 2.40
to its
of the Titius-Bode
regarded as satisfactory.
t o be r e l a t e d
to the mass.
o f T i t l u s - B o d e Law and we s h a l l
of the solar
the distance
mass.
system has given this or that interpretation
Law; none o f t h e i n t e r p r e t a t i o n s of the interpretations
central
1.17 1.28 1.24 1.28 1.24
S-VI S-VII S-VIII
and S-V b e i n g 2 o r d e r s o f m a g n i t u d e l e s s m a s s i v e t h a n S-VI.
t h a t o f N e p t u n e , and t h e i r
~+1/~
Satellite
Only a few
I n t h i s p a p e r , we
d e r i v e from i t
certain
conclusions
system.
(n) I n 1951, F e s s e n k o v [1] m a i n t a i n e d t h a t t h e m a i n f a c t o r planet
f r o m t h e Sun i s t i d a l
instability
circum-solar nebular disk is greater can condense o u t . was formed f i r s t ,
The f u r t h e r
out a planet
is,
between the planets
t o t h e Sun t h a t p l a n e t s
t h e s m a l l e r i s t h e Roche d e n s i t y ;
Mercury.
of a
i s o n l y when t h e m a t t e r d e n s i t y i n t h e
t h a n t h e Poche d e n s i t y r e l a t i v e
t h e n N e p t u n e , and l a s t l y ,
and t h e m u t u a l d i s t a n c e s the tides
and t h a t i t
in determining the distance
The d i s t a n c e
are therefore
of a planet
hence Pluto
f r o m t h e Sun
d e t e r m i n e d by t h e s o l a r
t i d e and
r a i s e d by t h e a l r e a d y - f o r m e d o u t e r n e i g h b o u r s .
m,, g ( r . - - r,._O s ' where K i s a c o n s t a n t .
Me r~,'
(3)
Prom ( 3 ) , we h a v e
(4) rm--i Comparing with (2), we have
p=
l
+
.
Thus, according t o Fessenkov's argument, 8 is a function of the planetary masses.
(5)
In order
to make the calculated and observed values of I- agree, Fessenkov made several unverified n assumptions: the first is to multiply the masses of the 4 terrestrial planets each by 30,
18 6
Titius-Bode Law
regarding their fohner selves as being massive proto-planets and the gas that was their bulk content as having evaporated; the second is to set K = 18 at first and later to change to K = 13.25; the third is to change the ~
in (4) into ~ j n - 1
=
(mn÷mn-])/2"
Because
there are no sufficint grounds for these assumptions and because the statements that the further out a planet, the earlier is its formation and that the planets were at one time huge proto-planets have not proved to be convincing, Fessenkov's interpretation of TitiusBode Law has not been accepted.
However, his point that the planetary distance is related
to the planetary mass is corregt. In a paper pulbished in 1949, Kuiper [2] gave an empirical relation between the distance of a planet or satellite from its central body and its mass.
Let m be the mass of the n-th n planet or satellite in units of the mass of its central body, and let •
1
(rt +
~gmm
r2)
(6) 2
2 then the relation was ~s
where C is a constant, roughly equal to 10 -~.
C
1
(7)
In another paper in 1951, Kuiper [3] found (8)
m . ' ~ - -'O ' O 0M 0l(~)
here, the A is not the same as in (6) and (7), it is equal to P 2 - r]" is basically the same as (7).
The expression (8)
Kuiper maintained that because of gravitational instability,
the nebular disk collapsed into a number of huge proto-planets at locations where the density was high and exceeded the Roche density.
The p~oto-planets later changed into planets
Kuiper laid stress on the planets being huge proto-planets at a former time, hence his theory has been called the '=proto-planet theory".
In his paper [3] and later papers, he
took m to be the mass of the proto-planet. We have repeated Kuiper's 1949 calculatlons to see whether or not ~/A 3 is a constant, and if not, how large the dispersion is.
New values of the planetary masses have been used.
The values of m and ~/A 3 are shown in TABLE 3.
t~ 1/17
The values of ~/A 3 ranges from 30 times
times 10 -~, the largest being 520 times the smallest.
Table 3 Planet
Values of m, r, ~ / 4 ' , z , At/a= of Planets
m
X
(g)
(cm)
3.332 X 10u
5.791X 10*=
5 . 8 9 X 10 "~
Mercury Venus
4 . 8 7 0 X 10"
1.0821 X lO ts
8.23X 10-j
Earth Mars
5.976X 10zv 6.421X10 u
1 . 4 9 6 0 X 10=s
2.33 X 10-s 3.65X 10-.4 3. OSX I ~ 3 5.46X 10-4 5.55X10-~
Jupiter Saturn Uranus
Neptune Pluto
1.8997 X 10 m
5.6879X 10*9 8.7424X 109* 1.0292X 10" 6.632X lOa*
2.2794X 10=3 7.753X10 t9 1.427X10 =* 2 . ~ 9 6 X 10'4 4.4966X 10 '4 5.900X 10=`
1.32 X 1~3
(cm)
"r/2x
2.21X I0 l° 1.01Xl0 u
96.0
1 . 5 0 X l O t'
1 . 1 6 X I O a3
19.2 39.9 5.19 7.44 11.2 6.71
2 . 8 4 X 1 0 t*
13.2
1 . 0 8 X 10 u 5 . 3 1 X IO ta 6 . 5 2 X 1 0 t* 7 . 0 2 X lO t*
23.8
Titlus-Bode Law
18 7
The other formula of Kuiper's can be written in the form m==K-3(~)s M extracting
'
(9)
t h e cube r o o t s o f b o t h s i d e s and m u l t i p l y i n g by r , we have A r .= K
r.
The radius ~ of the "gravitational region" of a planet with respect to the Sun can be expressed by the formula x--
(11)
r.
This formula is derived in the circular model of the restricted problem of three bodies, taking as the gravitational region with respect to the Son the Roche equipotential surface enclosing the planet and passing through the first Lagrangian point (cf. [4,5]).
Inside
the gravitational region, the gravitational influence of the planet is greater than that of the Son.
We do not take take AP to be P2 - PI' rather, we take it to be the width of the
planetary region; from (10) and (11), we obtain A r - - 3~ K - ' K u 2x 2 If Kuiper's empirical relation directly
is precise,
(12)
t h e n t h e w i d t h o f a p l a n e t a r y r e g i o n w i l l be
proportional to the size of the gravitational
region relative
t o t h e Sun.
Usually,
t h e i n n e r b o u n d a r y o f t h e n - t h p l a n e t a r y r e g i o n i s t a k e n t o be ( ~ _ ] l " n ) 1 / 2 , and t h e o u t e r b o u n d a r y t o be
(z,nz,n+l)1/2.
We s h a l l
follow this practice.
The v a l u e o f z and At/2= so
c a l c u l a t e d a r e g i v e n i n Columns 5 and 6 o f TABLE 3.
For t h e i n n e r m o s t Mercury and t h e
o u t e r m o s t P l u t o and t h e two p l a n e t s Mars and J u p i t e r
on e i t h e r
s u p p o s e e q u a l o u t e r and i n n e r h a l f w i d t h s .
t h e d i s p e r s i o n h e r e i s t h u s s m a l l e r t h a n i n t h e ~/A 3 v a l u e s . made f o r t h e r e g u l a r s a t e l l i t e s
of Jupiter
side of the asteroids,
we
The l a r g e s t ~w/2= i s 18.5 t i m e s t h e s m a l l e s t , S i m i l a r c a l c u l a t i o n s were
and S a t u r n , and t h e s a t e l l i t e s
o f Uranus, t h e
results are shown in Col,--- 4 of TABLE 5, the ratio of the largest to the smallest is 70 for the Jovlans, 13 for the Saturnians, and 5 for the Uranlans. The calculated values of AP/2zapproxlmatemore but the dispersion is still considerable; lightweights, Mercury, Mars and J-V.
to a constant than do the values of ~/A 3,
the values of AP/2z are especially large for the
This makes us realise that it is not appropriate to
define the boundary of a planetary region by the geometrlcalmean of the distances of the planet and its nelghbour.
Since AP increases with increasing m, the boundary of a planetary
region must also be related to m.
Suppose mn+ I > mn, then the llne of demarcation between
the (n+1)-th and the n-th regions m u s t be closer to the n-th planet.
We shall take the
demarcation line to be where the tidal forces of the two nelghbours are equal, that is, we divide the l e n g t h ~ + 2 - 2 , n into two parts in the ratlomn]/3 a more reasonable boondarybetween
the two neighbouring regions.
: mn+2 I/3 and thereby fix For the four planets,
Mercury, Pluto, Mars and Jupiter, we still suppose the inner and outer two halves of each region to have the same width.
The planetary regions so calculated are given in Column 2 of
18 8
Titius-Bode Law
TABLE 4.
Using the new boundaries, the r e v i s e d
values of AP/2x for the planets are given
in Column 3, the dispersion is much smaller than before.
For the regular satellites of
Jupiter and Saturn and the satellites of Uranus, the revised values of Ar/2.z calculated by the same method are given in the last column of TABLE 5; here, too, the dispersions are much less than before. Tab,le 4
Limits ofPl,anetayy Regions_..and Revised Values of Ar/2x Limits of Planetary Region (AU) ....
Planet Mercury Venus Earth Mars Jupiter
0.287 0.486 0.857 1.355 2.60 -
Saturn Uranus Neptune Pluto
Table 5
7.80 15.82 24.47 37.97
Values of ~ m
Satellite
(g)
-
0.486 0.857 1.355 1.693 7.80
15.82 24.47 37.97 40.91
Revised Value of ~" / 2 z 66.0
27.5 24.9 23.3 7.31
9.20 9.22 8.69 7.76
pj AP/2x of Regular Satellites ff
(10 '° cm)
=r/2x
3.42X10 't
1.81
7.89X10" 4.764X10" 1.535X102~ 9.113X10"
4.22 6.71 10.70 18.83
1.6X10 '~ 3.75X 10'2
1.59 1.86
67.2 36.9
34.6 36.4
S-II S-Ill
8.42X10'* 6.228X 10'3
2.38 2.95
S-IV S-V
1.160XI0 u 1.SXI0 u 1.4003X 10" 1.1 X 10~3
3.77
31.3 16.2 16.9
28.0 17.6 17.3
5.27 12.22 14.83
33.2 5.12 23.0
19.8 7.55 3.71
1.30 1.92
31.1 10.4
19.9
2.67 4.38
17.2 7.36
5.86
6.35
J-V J-I J-ll J-Ill J-IV S-X S-I
S-VI S-VII U-V U-I U-II U-Ill U-IV
8.7X10" 1.31XI0 u 5.25X 10'3 4.37 X 10" 2.54X 10*4
625 12.6 11.6
=r/2s revised
8.92
9.77
53.6 18.1 10.1 10.6 7.83
13.3 13.1 8.72 5.39
The calculated results show that the distance of a planet (or regular satellite) from its central body is related to its mass, that the width of a planetary (regular satellite) region is roughly proportional to its gravitational region, and since the latter is proportional to m 1/3, the width of the planetary region increases with the planetary mass. The size of the gravitational region is also proportional to the distance of the planet (satellite) from its central body, but Form (2) of Titius-Bode Law a ~ L ~ p ~ e ~ e ~ only one aspect that the width of a planetary region increases with increasing distance, it does not
Titius-Bode Law
express the relation between the width and the mass.
189
There is another result of the calculation
that should be noted: the further out the planet (satellite), the smaller is its A1-/2x. (In) To explain Titius-Bode Law, we must answer the following three questions: (I) Why are the widths of the planetary regions proportional to the diameters of the gravitational regions ? What problem does this elucidate ? (2) Why is it that the regular satellites possess the same type of law of distances as the planets do ? (3) Why are the Ar/2x values smaller at larger distances from the central body ? There have already been over twenty nebular theories of the origin of the solar system that maintain that both the planets and the satellites were formed from disk surrounding the Sun.
matter in a nebular
As regards how the planets were formed out of this matter, there
have been two quite different views.
According to one, because of gravitational instability,
the nebular matter first formed a number of very large proto-planets.
Inside the proto-planets,
the gas condensed into the solid particles, which sank to the centre and there formed a solid core, the gas in the outer part then left thp proto-planet through evaporation and through being driven away by the various solar radiations, so that only the solid core remianed behind (for Jupiter and Saturn, with part of the gas in the inner part added on), and so the proto-planets were transformed into planets.
The other view is that, because of
their relative movements, the solid particles inside the nebular disk underwent mutual collisions at which the smaller particle was often swallowed up by the large one, the two joining together to become one. called planetesimals.
In this way,.small particles grew into large solid peices
When a planetesimal became sufficiently massive, it began to "accrete"
other approaching planetesimals without the latter colliding with it.
Some of the larger
planetesimals thus became planetary embryos, and when the embryos grew to certain sizes, they would begin to accrete gas, if there was gas about.
Jupiter and Saturn were probably
formed in this manner, for their solid cores are only a small part of their total masses, most of which is gaseous or liquid.
The embryos grew gradually into planets.
We shall now answer the first question.
The fact that the widths of the planetary regions
are directly proportional to the masses shows that the second view is probably the correct view.
If the planets were fromed from proto-planets,
then this would not demand that the
widths be proportional to the masses, whereas if the planets were formed through the accumulation of material particles, then this feature would be demanded.
The gravitational
region of an embryo grows with the embryo, although even at the end of the formation process, the dlamter of the gravitational region is only a fraction of the width of the planetary region, the relative movement among the planetesimals means that the accretion region of an embryo is much greater than its gravitational region.
Because of their mutual perturbations
the orbital eccentricities and inclinations of the planeteslmals were constantly changing, thus producing their relative movements. the gravitational region of an embryo.
Planetesimals would thus be constantly entering If the velocity of entry is less than the velocity of
escape a t t h e s u r f a c e of t h e g r a v i t a t i o n a l a c c r e t e d by t h e embryo.
This v e l o c i t y
r e g i o n (2GM/~) ½ , t h e n t h e p l a n e t e s i m a l w i l l be
f o r the i n c i p i e n t Mercury, E a r t h , J u p i t e r and Neptune
is respectively equal to 0.45, 0.73, 2.18 and 0.34 km/sec.
If the planetesimal enters the
19 0
Titlus-Bode Law
gravitational sphere with a greater velocity, then so long as its impact parameter * is less than x, or if it is going to directly hit the embryo, then the planetesimal will still be accreted.
During the period of formation of the planets, planeteslmals in general have
small velocities, so that as soon as one enters a gravitational sphere, it will generally be accreted.
In this manner, the larger the gravitational region, the greater is the
possibility of accreting planeteslmels, therefore the observed fact that the width of the p l a n e t a r y
region
is roughly proportional to the diameter of the gravitational
region shows that the planets were formed through the accumulation of planetesimals and not from proto-planets. The second question was: "Why do the systems of regular satellites have the same type of distance laws as the system of planets has ?". were formed in the same way as the planets.
The answer is that the regular satellites
In the Jupiter and Saturn region of the
nebular disk, the temperature was lower than in the terrestrial region and so a large part of the volatile gases (mainly hydrogen and helium) remined there.
Furthermore, the gas
t h e r e d i d n o t e s c a p e a s d i d t h e g a s i n t h e r e g i o n o f U r a n u s , Neptune and P l u t o , where the velocity
o f e s c a p e was s m a l l b e c a u s e o f t h e weaker s o l a r g r a v i t a t i o n .
formation process of Jupiter planets.
core which, after first
reaching a certain
formed a l a r g e f l a t
the shell,
their
eccentricities orbits portion
and S a t u r n was n o t q u i t e
Each b e g a n w i t h s m a l l s o l i d p a r t i c l e s
velocity
shell
t h e same a s t h a t o f t h e o t h e r
and i c y p i e c e s a c c u m u l a t l n g i n t o a s o l i d
mass, began to accrete
around the solid core.
gas gravitationally.
the planets
to the p r o t o - p l a n e t s
p o i n t i s f a v o u r a b l e to the a c c r e t i o n
According to Kuiper's
500 t i m a s t h e p r e s e n t E a r t h ' s
present value, sufficient
o r 68 k m / s e c .
in their
theory of the proto-planets,
m a s s , so t h a t
distances
proto-planets,
i n e x p l a i n i n g how t h e e x c e s s m a s s e s l e f t
o f t h e d e n s i t y now, t h e n t h e v e l o c i t y
was lower s t i l l , proto-planets.
t h e m a s s o f P r o t o - E a r t h was
o f e s c a p e a t t h e s u r f a c e would be 6 . 0 7 t i m e s t h e
The t e m p e r a t u r e o f t h e g a s on t h e P r o t o - E a r t h would n o t be
and i t would be more d l f f l c u l t If the sate11ites
still
For t h e J o v i a n p l a n e t s ,
would f a l l the temperature
f o r the e x c e s s mass to leave the
were a l s o t r a n s f o r m e d f r o m h u g e p r o t o - s a t e l l i t e s ,
i t would be e v e n more d i f f l c u l t
t o e x p l a i n how t h e e x c e s s m a s s was g o t r i d o f ,
central
systems - the planets
bodies of the satellite
t h e n we
i f we s u p p o s e t h e d e n s i t y t h e n t o be ] / S t h a t
t o d r i v e away 4 9 9 / 5 0 0 o f t h e m a s s .
l l k e t h e Sun d i d .
as
the proto-
f o r t h e g a s t o e v a p o r a t e away and a l s o t h e v a r i o u s s o l a r r a d i a t i o n s
by a l a r g e f a c t o r
A large
h y p o t h e s i s and u n f a v o u r a b l e
I f t h e p l a n e t s were o r i g l n a l l y
dlfficulty
sate11ites.
became t h e m i d d l e and o u t e r l a y e r s o f t h e p l a n e t .
h a v e t h e same t y p e o f r e g u l a r i t y
hypothesis.
would h a v e t h e g r e a t e s t planets.
later
satellites
do, and t h i s
orbltal
would a l s o be r e d u c e d so a s t o r e v o l v e i n r o u g h l y c l r c u l a r
of the gas of the shell the regular
entered
o f t h e g a s , and t h e i r
a r o u n d t h e s o l l d c o r e and s u b s e q u e n t l y t o g a t h e r i n t o r e g u l a r
Therefore,
The g a s
When t h e p l a n e t e s i m a l s
would be r e d u c e d by t h e r e s i s t a n c e
and i n c l l n a t i o n s
Hence t h e
then
since the
- d i d n o t e m i t any s t r o n g r a d i a t i o n s
Recently, it has been discovered that,
slmilar
t o what i s on t h e
s u r f a c e o f t h e Noon, t h e r e a r e n u m e r o u s c r a t e r s
o f a l l s i z e s on N e r u c r y which h a s p r a c t i c a l l y
~rhe formula for the impact parameter is,
b ° = b [I
denote respectlvely p l a n e t e s l m a l when i t
[6],
+ 2 ( ~ / ~ v 2 ] ~ , where m and b
t h e m a s s and r a d i u s o f t h e embryo, and v i s t h e v e l o c i t y i s f a r f r o m t h e embryo.
of the
791
Titius-Bode Law
no atmosphere and on Mars which has only a very thin atmosphere, which shows that there had been planetesimals of all sizes falling on these bodies.
The irregular satellites are
probably objects captured by the planets, though they were also formed by the accumulation of solid particles, they did not enter the extended gaseous shells surrounding the planets, or the planets concerned did not have such shells, hence they have in general large orbital eccentricities and inclinations. The third question was "Why were the values of Ar/2x smaller at larger distances from the central body ?".
The width of a planetary region is not only related to the size of the
gravitational region, but also to the relative velocities of the planetesimals.
We have
already mentioned that the mutual perturbations of the planetesimals will change their initially circular orbits into ellipses of various values of e and i, giving rise to mutual relative velocities.
Suppose at a distance P from the Sun, not only does the
circular orbit of a planetary embryo pass, but also pass the perihelion point of a planetesimal orbit of eccentricity e I and the aphelion point of another orbit of eccentricity e 2. Suppose further that the two planetesimal orbits have zero inclinations and semi-major axes u I and u~ respectively.
We then have s,(1
-
e,)
, -
.,O
+
e,) ~
(13)
,.
Using the vis viva integral, it is easily shown that (14)
~ v 2 - - vo - - v , ffi
(1 -- V~
-- e,);
(is)
in these, V° = (GM/r) ~ is the velocity of the embryo in its circular orbit. V O % 30 km/sec, and if AV ffi I km/sec, then e I = 0.068 and e2 = 0.066. maximum
For the Earth,
Here AV is the
relative velocity of a planetesimal falling into the Earth embryo at the final
stage of the Earth's formation.
At this time, the width of the accreting zone is roughly
equal to
[at(1 + et) - - r] + [r - - s , ( 1 - - e2)]. Substituting from (13), we have
2(e1+ e~)
or 0.268r.
(16)
This is 13.4 times the diameter
of the gravitational region, and is 54% of the Earth region of 0.498r.
A slight increase
in the value of the maximum relative velocity of the planetesimal will make the Earth's accreting region at the final stage of the Earth's formation as wide as the Earth region. The nebular disk surrounding the Sun is mainly gas, and because the z component of the solar gravitation does not fall with increasing distance as fast as does the force of presuure gradient, the disk is thin in the inner part and thick in the outer.
The small
solid particles and icy pieces, in virtue of the z component of the solar gravitation, slowly overcome the viscous force of the gas and settle to the neighbourhood of the plane of symmetry of the disk, and ther form a thin absorbing layer.
This absorbing layer, also
because of the outward decrease in the z component, is also thin in the inner part and thick in the outer, which means that the large the seml-major axis, the greater is the average inclination.
The larger the inclination, the smaller is the eccentricity projected
on the plane of symmetry, and hence the smaller is the projected relative velocity so that the width of the accreting zone, this is, the width of the planetary region AI• will
19 2
Titius-Bode
be correspondingly we h a v e e x p l a i n e d ~/2m.
smaller. why i t
In our calculation
masses, oreder
A similar
is that,
of the m-values
but a more reasonable of magnitude
This explains
interpretation
procedure
down on t h e i r
why o u r v a l u e s
We h a v e now a n s w e r e d a l l
situation
the further
entire
Law
holds
of Jupiter
of the Titius-Bode
the regular
and Saturn,
satellites.
body,
bodies),
questions
the m-values
we u s e d t h e i r
and Saturn
should
Thus
the smaller
w o u l d be t o u s e t h e m a s s e s o f t h e i r
of Ar/2x for Jupiter three
for
away f r o m t h e c e n t r a l
solid
is
total cores
then be smaller.
are somewhat too small.
and have thus given a rather
satisfactory
Law.
REFERENCES
[1] [2] [3]
Fessenkov, V.G., Astr. gh. 28 (1951), 492-517. Kuiper, G.P., Astz~hZ~S. J. 109(1949), 308-313. Kulper, G.P., "Origin of the Solar System", in "Astrophysics" edited by J.A.Hynek, Chapter 8, pp.357-424, partlcularly p.380 (1951).
[4] [5]
Lyttleton, R.A., t4on.No"c.R,astz,on.,F,oe.15~(1972) 463-484. Houlton, F.R., "Celestial Hechanics" (1914). H a r t m a n n , W . K . , Astz,ophys. J. ] 5 2 ( 1 9 6 8 ) 3 3 9 - 3 4 2 .
[6]
(one