Some remarks on the jeans rule

CHINESE ASTRONOM‘k AND ASTROPHYSICS Chinese Astronomy

PERGAMON aa99a.ctx

and Astrophysics

27 (2003) 167-175

Some Remarks on the Jeans Rule+ * GAO Chong-yi College of Physical Science and Technology, Langzhou University,

Langzhou 730000

Abstract From the basic laws of physics one can deduce the Jeans empirical rule, which should preferably be called the Jeans rule. Its representations in velocity, height and energy are completely equivalent and may be arbitrarily chosen. This rule is most easily interpreted in terms of energy. The Jeans rule merely provides a necessary condition for the particles to be the main atmospheric constituent of a planet or satellite with a dense atmosphere. Its range of applicability can be expressed by equations or figures, or by a nomogram. It is applicable

to planets,

satellites,

asteroids

and meteoric

bodies

tem, as well as to some objects in the outer solar system, the Kuiper belt and the Centaurs. This rule still possesses significance even at present day. Key words:

planets

and satellites:

general-

in the solar sys-

such as the objects in a general and realistic

asteroids-Kuiper

belt

1. INTRODUCTION In planetary atmospheres.

physics,

the Jeans

The rule ill states,

empirical

rule is frequently

if the escape velocity

used in the study

on the surface

(some people use 5 L21)or more times the root-mean-square in the atmosphere, i.e., if

velocity

of a planet,

of planetary veSc,o, is 4

(A&?) of a given species

of particles

then this species of particles will escape from the atmosphere only exceedingly slowly. practice, this particular constituent will “eternally” exist on the surface of the planet. t Supported by Ministry of Education and Langzhou University Received 2001-10-08; revised version 2002-01-04 ?? A translation of Acta Astron. Sin. Vol. 43, No. 4, pp. 366-374,

2002

0275-1062/03/$-see front matter 0 2003 Elsevier Science B. V. All rights reserved. DOI: lO.l016/SO275-1062(03)00038-9

In

168

GAO Chong-yi / Chinese

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27 (2003)

167-l 75

For a long time the Jeans rule has been regarded as a purely empirical rule. Its form of representation is simple, the factors in its expressions are various and it has not been justified by a theoretical derivation. Hence its reliability is uncertain and the range of applicability is unclear. In practice, it has been applied only to a few known major planets and satellites. We have now studied some aspects of the Jeans rule according to the basic laws of physics, aspects such as its theoretical derivation, forms of representation, physical interpretation, range of applicability and realistic significance. We have theoretically derived the Jeans rule 131,determined

the factor in its velocity

representation

to be 5 I31 and obtained

its

representations in height and energy [4,51. By means of the statistical theory, we proposed a physical interpretation ~1, investigated whether it is a necessary and sufficient condition for the particles to be a constituent of the atmosphere of the planet or satellite 161. We have determined

its range of applicability

f71. Moreover,

we have clarified

its realistic

significance

(731.

2. THEORETICAL

DERIVATION

The planetary atmosphere is assumed to be an isothermal and ideal gas. Then from the hydrostatic equation, the equation of state for ideal gas, the gravitational law and the law of Maxwellian

distribution,

one may derive the following

ln{Gpr2K2(Rc/&)2[(R,/Ra)

expression

+ (3/2)K2]“}

(2)

= 3K2,

where G is the gravitational constant, p is the average density of the planet, r is the escape time, i.e. the time for the number of particles of the given kind to be depleted e-fold through escape, Ro is the radius of the planet, R, is the distance between the planet’s center and the critical height of the base of the outer atmospheric layer, and K is the ratio of the escape velocity at the planet’s surface to the rms speed of the particles considered, K = ~,,,,~/~. Eq.(2)

shows,

when

the value of K is determined

r is given,

and the value of R, JRo, which

can be found

in various

by the mean

ways from the magnitude

density

p

of the

atmospheric pressure at the surface 131. Starting from known values of R,/Ro and T of the planets/satellites, and taking r = 45.5 x 10’ yr (the age of the solar system), we obtain the values of K given in Table 1. Table

Although of magnitude

1

The

values

of K for various

planets

and

satellites

Planet/satellite K

Mercury 4.99

Venus 5.00

Earth 4.99

Mars 4.99

Jupiter 4.94

Saturn 4.92

Planet/satellite K

Uranus 4.94

Neptune 4.95

Pluto 5.00

Moon 4.97

IO 4.98

Titan 4.99

the values of p of the different planets and satellites differ by about and their R,/& values are also different, yet all the calculated

are close to the value isothermal atmosphere. not strictly isothermal,

one order K values

5: K = we.,,0/@ x 5.0. This is a result given by the model of In practice, although the atmospheres of planets and satellites are their vertical temperature distributions are rather similar and they

GAO Chong-yi

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Astronomy

and Astrophysics

27 (2003)

167-l 75

169

have common characteristics. The low layer close to the surface is analogous to the earth’s troposphere, in which the temperature decreases with increasing height: it is close to an adiabatic atmosphere. The high layer is analogous to the heated layer of the earth, i.e., its temperature increases with increasing height. For the atmosphere as a whole, the vertical variation of temperature is rather small. Therefore, on the whole, the atmospheres of the planets and satellites are quite close to being isothermal atmospheres. Because in Eq.(2) the computation involves logarithm, the approximations adopted in its derivation a large influence on the calculated value of K. As demonstrated by the above results, if the specific particles in a planetary

do not have atmosphere

satisfy Eq.(l), then their escape time will not be less than the age of the solar system, i.e., their present total number will not be less than l/e of the primitive value: the order of magnitude constituent

will not have changed. In practice, we may then regard that the atmospheric composed of such particles will “eternally” exist on the planet’s surface. In this

way we have in theory approximately that this rule should simply be called

3. FORMS

derived the Jeans empirical rule. Therefore, it seems “the Jeans rule” without the word “empirical”.

OF REPRESENTATION

Eq.(l) may be regarded as the representation of the Jeans rule in terms theoretical physics, we can also express it in terms of height or energy. The representation

of the Jeans

rule in height

of velocity.

From

is

Rs >_ 37.58, where

(3)

H is the scale height of the specific particles in an isothermal The representation of the Jeans rule in energy is

atmosphere.

(4)

I &p 12 25Z’, where cp and ~1 are, respectively, the gravitational potential kinetic energy of the specific particles in the atmosphere.

energy

and mean translational

equivalent, and they may be used as one pleases Eqs.(l), (3) and (4) are completely according to circumstances. They are all the same criterion for the “eternal” existence of specific particles as an atmospheric constituent on the planet’s surface. The Jeans rule can also be called “the Jeans escape criterion”. It is better not to further abbreviate the last to “the Jeans criterion”, because the term “the Jeans criterion” has customarily been used to mean “the Jeans instability criterion”. For some planets, the values of Rc and H can be directly read off in handbooksl’Ol. Then, it is especially convenient to use the representation in height (3).

4. PHYSICAL

INTERPRETATION

Although one may understand the Jeans rule from the theoretical derivation of Eq.(l), yet Eq.(4) offers a more explicit physical picture. On the basis of the statistical theory, it is easier to understand the physical nature of the Jeans rule in terms of energy 151.

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/ Chinese

Astronomy

and Astrophysics

When the JQ,J of some given specific particles their mean kinetic energy &k, i.e., when

it is possible

for the particles

to break

27 (2003)

in the planetary

free of the shackles

167-175

atmosphere

of the planet’s

is lower than

gravitation.

Let’s

assume that the particles in the outer layer of the atmosphere still obey the law of Maxwellian distribution, then, of their total number the percentage of those with velocities greater than a given value 21is [11,121 AN/N

= 1 + (2/fi)zexp(-s2)

- erf(z) ,

(6)

where x = v/up, up being the most probable speed, and erf(z) is the error function. Hence, from Eq.(6) that, of the total number of particles, the percentage of those with speeds v 2 0 (i.e., particles with average kinetic energies ek > ck) is as high as 39.16%. Hence if we have 1~~1 = &, the particles will rapidly escape in large quantities. Therefore, it is only when (~~1 is much higher Maxwellian speed distribution

than ~1, that the particles located in the tail of the curve of with rather high mean kinetic energies can just barely break

away. From Eqs.(4)

and (5) we have &k 1 25%.

As shown by this formula,

when the Jeans

planetary gravitation and break away must hence must account for a very small fraction species. In practice, when Eq.(7) holds, the to 3.623 x 10-16. Thus, when Eq.(4) or (7) translational kinetic energies the total number. Therefore, escape of such particles

cannot

(7)

rule holds, particles

that

can overcome

the

possess very high translational kinetic energies, of the total number of the particles of the given AN/N found by Eq.(6) is approximately equal is valid, the particles that have sufficiently high

to escape only account for an exceedingly small percentage of even in a period as long as the age of the solar system, the change

the order of magnitude

of their

original

number.

In

practice, such particles can then be regarded as constituting an “eternal” component of the atmosphere. The foregoing situation implies that the factor in the representation in energy should be far greater than unity. However, this factor, 25, must not be too great. If, for example, it is 50, i.e., if Eq.(4) is rewritten as

I~pl25OZ,

(8)

then the AN/N given by Eq.(6) decreases to about 2.635 x 10-32. True, in this case the specific particles which satisfy Eq.(8) also satisfy Eq.(4) an so are “eternal”. However, it could be that some other particles, which do not satisfy Es.(B), may yet also be”eterna1”. For instance, in the dense atmosphere of Titan CH4 accounts for 1 - 6%. By using the data in Ref.[lO], we may find that for the CHd in Titan’s atmosphere, we have 1 En Ix 47.7%.

(9)

GAO

This expression

Chong-yi

/ Chinese

demonstrates

Astronomy

that

and Astrophysics

27 (2003)

for CH4, which is a main

171

167-175

constituent

of Titan’s

atmo-

sphere, Eq.(B) does not hold, so it cannot be substituted for Eq.(4) and cannot be used as the criterion of escape. However, the factor in the representation of energy cannot be much smaller than 25, either. Eq.(4) is rewritten as

For example,

suppose

this factor

is taken

to be 23, i.e., suppose

00) then the following

case shows that it cannot be taken as a criterion for “eternal” existence. we have Using the data in Ref.[lO] one may find that for the helium in Venus’ atmosphere the following relation: 1 sP 1% 23.6G.

(11)

Thus, in this case, Eq. (10) is satisfied, yet we know that helium escaped in large amounts from the primitive atmosphere of Venus and only a tiny amount now remains. Thus, Eq.( 10) cannot be substituted for Eq.(4) an d used as the criterion of escape. It thus appears that in the representation in energy the factor 25 is appropriate. should not be replaced with either a much larger or a much smaller value.

5. DISCUSSION Taken as the escape “eternal”

existence

OF criterion, of specific

NECESSARY the Jeans particles

AND

SUFFICIENT

rule is a necessary on the planetary

CONDITION

and sufficient surface,

It

condition

for the

but it is not a necessary

and sufficient condition for the specific particles to become a constituent atmosphere. Evidently, the Jeans rule is not sufficient for the specific particles

of the planetary to become

a con-

stituent of the atmosphere, because the atmosphere of a planet do not necessarily contain all the particles that satisfy the Jeans rule. Moreover, in practice, the Jeans rule is only a necessary condition for specific particles to become a chief atmospheric constituent of planets or satellites that possess a dense atmosphere. By using Eq.(l), one may find that the molar mass of the lightest gas that can “eternally” exist in the atmospheres above the planetary surfaces is: Pmin

= 37.5(NAh)T/(9OfiO)

J

(12)

where go is the gravitational constant at the planetary surface, T is the thermodynamic temperature, NA is the Avogadro constant, Ic is the Boltzmann constant, and NAk is the universal gas constant. The molar mass p of the lightest major gas in the atmospheres of selected planets and satellites as well as their pmin values calculated with Eq.(12) are listed in Table 2.

172

GAO Chong-yi

Table

2

/ Chinese

Values

Planet/satellite Lightest major gas P/k Pmin/(g

molel)

mole’1

Planet/satellite Lightest major gas d(g mol-‘) . mol-‘) p,i,/(g

Astronomy

and Astrophysics

of p and pmin of planets

Mercury H 1.01 14.2

Venus

Earth

28.0 4.23

18.0 1.44*

Uranus Hz 2.02 0.0748**

Neptune

Pluto CH4 16.0 17.3

N2

Hz0

Hz

2.02 0.0514”

27 (2003)

167-175

and satellites

Mars

Jupiter

N2

Hz

H2

28.0 5.36

2.02 0.0242**

2.02 0.0572”

Moon He 4.00 30.3

10 SO2 64.1 10.6

Titan CH4 16.0 8.43

Saturn

* Calculated with the surface temperature of the earth set equal to 288K. height 384 km is 955 K, then pmi,, = 4.97g.mol-‘.

If the temperature

at

** Calculated with the effective temperature As shown by Table 2, for objects with dense atmospheres, Venus, Earth, Mars and Titan, the inequality

such as the Jovian

planets,

(13)

P > Pmin

is valid. For objects but some others

possessing

only thin atmospheres,

(e.g., Mercury

and the moon)

some of them (e.g., 10) satisfy

satisfy

P < Pmin

Eq.(13),

the inequality (14)

.

If we compare the molar masses of the secondary gases in the atmospheres of the planets and satellites with their prnin values[l~lOl, we shall easily see that sometimes the former is the greater (e.g., the CO in the atmospheres of Venus, earth, Mars, Jupiter and Titan), and sometimes the latter is the greater (e.g., the Hz in the atmosphere of Titan). Therefore, the fact that specific particles satisfy the Jeans rule Eq.(l), Eq.(3) or Eq.(4), is only a necessary condition for that species to be the chief constituent of the dense atmosphere of a planet or satellite; it is not a sufficient condition. Moreover, it is neither a necessary nor a sufficient condition for the species to be the chief atmospheric constituent (if the atmosphere is thin) or a secondary atmospheric constituent. It is very important to understand these points for a reasonable application of the Jeans rule in the study of the atmospheres of planets and satellites.

6. THE RANGE

OF APPLICABILITY

As pointed out above, when R, / & and r are given, the value of K = vesc,o/x@ is determined by the p of the planet/satellite. If the value of K calculated by Eq.(2) satisfies the inequality

4.85 5 K 5 5.15, then the Jeans rule is applicable. in a certain range, namely,

In this case, according

(15)

to Eq.(15)

the value of p should

be

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173

pmin = exp[3 X (4.85)2]/{(4.85)2G~2(Rc/Ro)2[(RC/Ro)

+ (3/2)

X (4.85)2]“},

(17)

P max = exp[3 x (5.15)2]/{(5.15)2G~“(R,/R~)2[(R,/R,-,)

+ (3/2)

x (5.15)“]“}.

(18)

These three expressions specify the range of applicability of the Jeans rule. Still taking T = 45.5 x 108a and using Eqs.(l7) and (18), we determined the values R,/R for to a series of (actually 36) values. Then on one and the same 0 of pmin and pmax diagram we drew curves lg pmin-lg(Rc/Ro) and lg pmax -lg( R,/ Ro) . The area lying between these curves marks the range of applicability of the Jeans rule L71(see Fig.1).

0.2

G.l

0.0

0.4

0.3 k(R,

Fig. 1

0.5

0.7O

0.6

lRo)

The range of applicability

of the Jeans rule

In Fig.1, both curves are very close to being straight lines, so they can be replaced by linear regression lines. For a number of degrees of freedom of 34, the correlation coefficients are equal to -0.99993 and -0.99994, indicating a high level of confidence. The two regression lines are:

lgp,i,

lgp,,,

= lg 106.4 - 2.1261g(R,/Ro),

= lg(6.036

x 105) - 2.1121g(R,/&)

Using Eqs.( 16), (19) and (20), the range of applicability as 106.4/(R,/&)2.‘26

(19)

< p/(kg . me3)

.

(20)

of the Jeans rule can be summarized

5 (6.036 x 105)/(RC/&)2.112

We can also construct a nomogram according to Eq.(21), whether the Jeans rule is applicable or not (see Fig. 2).

which will enable

(21) us to judge

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10.

In the nomogram shown in Fig.2, the left ordinate marks R,/Ro and the right ordinate 7marks p. The line GC represents the range of < IO5 6applicability. To use the nomogram, locate the sG points on the two ordinate axes with the values 7 lo4 Go4 ? of R,/& and p of the given planet/satellite, and connect the two points with a straight ;3710’ g : 9 line. If the line crosses the line GC, then ? 102 the Jeans rule applies; otherwise, it does not. For instance, Triton has R,/Re = 1.44 and IO’ p = 2.05 x 103kg rne3, and we find the straight line connecting the two points crosses GC. So we conclude that the Jeans rule is applicable Fig. 2 Nomogram for judging whether the to Triton. Jeans rule is applicable or not Thus, the range of applicability of the Jeans rule can be either expressed by equations and their corresponding figures, or by a very simple nomogram. The RcfR,-,'s in Fig. 2 and Eq.(21) need to be determined with different formulae according to the magnitude of the atmospheric pressure on the planetary surface 131. If p. is 98-

3 106

= 1. Only when pe is comparatively high, do we have R,/Ro> 1. Among very low, R,/R.o the planets and satellites in the solar system, not many objects have high pa. Calculations show that their R,/&'sare less than 2. Even in extreme cases, the RJR0 cannot exceed 4. All the asteroids and meteoric bodies in the solar system have rather small masses, so even if they have atmospheres, the atmospheric pressures on their surfaces are certainly quite low. Hence for them R,/Ro= 1. From this it can be inferred that for planets, satellites, asteroids and meteoric bodies (if possessing atmospheres) the values of their R,/Ro's should lie in the interval 1 - 4. On the other hand, their p’s are in the range 1 x lo2 - 3 x 104kg.m.-3, and are located rather Eq.(21) that the Jeans

far from the two end values. Therefore, it follows from Fig. 2 and rule has a wide range of applicability. It is usable not only for the

planets and satellites but also for the asteroids and meteoric bodies (if indeed they possess an atmosphere), and even for bodies outside the traditional solar system, such as the Kuiper belt objects and Centaurs 171.

7. THE REALISTIC

SIGNIFICANCE

In t.he last decade, with the discovery of 1992QBl li51 the prediction made 40 years ago, of the existence of the Kuiper belt outside the orbit of Neptune, was at last fulfilled. In recent years the exploration and research of objects in the outer solar system has been active [16-301. Up to August 2001 13’1, there have been discovered 418 classical Kuiper belt objects (among them 303 have diameters from 100 km to 1000 km) and 48 objects with scattering disks. During the same time, 29 Centaurs, as well as (2060) Chiron, were discovered. Some of the discovered objects are quite large. For example, the diameter of (20000) Varuna (2000 WR 106) is 900 km 1281,and its absolute magnitude is 3.7. The absolute magnitude of 2001 KX 76 is 3.2. Therefore, outside the orbit of Neptune there are bodies as large as Pluto, maybe more than one, may even be several 1271. It is quite possible that these objects possess atmospheres. We now know that, in the outer solar system, 10 km-size objects may number

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[27]. Their

several billion, and 100 km-size ones, 100 thousands are manifold and abundant. Yet it may be inferred

175

167-175

characteristics,

from the analysis

of existing

of course, material

that they are not entirely different from the objects we know [7-10~14-281. Therefore, in the present era of exploration of planets of the solar system, especially of the outer solar system, the Jeans rule, which has been applied for a long time, still possess a general and realistic significance. References 1

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