SUGGESTED ORIGIN OF THE SUN AND BROWN BODY
and
BINARY STARS (and
EXTRASOLAR PLANETS - F.L,2008)
By T. Frank Lee, Geologist. 4/2/2009.
The
solar system pages in this web site have no particular order. The topics have appeared mostly in the order
they were intensively studied and solved, and only occasional reference has
been made to the major steps of evolution as given by the Single Body Breakup
Hypothesis of Solar System Origin. It
was only in late 2008 that a poster for presentation at geological conferences
in Perth, W.A. and Olso, Norway was produced giving an accurate, detailed
sequence of the breakup as well as over thirty of the planetary relationships
deduced from this. The poster appears
in this web site.
But
of course the poster could not give the full reasons for the breakup. There are three topics, so far only vaguely
mentioned in the papers, which should be presented to help the understanding of
the hypothesis: (i) the transition from molecular cloud ® protostar ® Sun and brown body; (ii) where did Earth’s water come from; and (iii)
an explanation of planetary distribution of the noble gases. This paper is the first of these topics.
The
“Suggested Origin of the Sun and Brown Body” and “Binary Stars” were written in
1993 and formed part of my book “The Origin and Development of the Sun and
Planets”, (1994) ISBN 0646 19165 9, Legal deposit receipt No. LD94/4703. Not seeing any need to “update” the writing,
I give here the extracts verbatim (pp.112-131).
One
thing I would like to “crow” about is the reasoning based on the hypothesis in
“Binary Stars”, which predicted the inevitable presence of extrasolar planetary
bodies of a wide range of types. The
hypothesis predicates these various types of bodies, bodies that were only
detected years later. This is further
support for the hypothesis.
SUGGESTED
ORIGIN OF THE SUN AND BROWN BODY
We
now move back one final step to offer an origin for the brown body and to
discuss a possible mechanism for the formation of the Sun and of the postulated
brown body considered in the previous pages.
A
number of theoretical research workers have studied the development of a star
from the collapse of dense cores within molecular clouds. Stahler (1991) points out that, “In the
1960s theorists had used computer simulations to determine how clouds in
unstable states collapse.
“Although the simulations assumed widely
varying initial conditions, each showed that clouds that are not violently unstable collapse in an
inside-out manner. That is, material at
the center first enters into a true free-fall collapse while the outlying gas
remains static. Gadually the region of
collapse spreads outwards through the rest of the cloud.” A number of suggested collapse sequences
have been proposed from these computer simulations, e.g. Winkler and Newman
(1980) – “A region of supersonic free-fall flow lies outside a central region
in hydrostatic equilibrium, separated by one or more shock fronts, ---.”
In
1980, Stahler, Shu, and Taam suggested that by the time a protostar within a
dense core of 1 MSun had reached a mass of about 0.01 to 0.02 MSun
it developed about itself a thin dust envelope, outside which lay the main bulk
of the gas/dust body. The inside of the
dust envelope is said to be delineated by volatilisation of the dust (“dust
destruction front”) abutting an opacity gap.
The whole structure which is claimed to form is given in a general paper
on the subject by Stahler (1991) in the Scientific American. The development by Stahler , Shu, and Taam
differs from others by incorporating the dust envelope. I have been unable to see why this envelope
is assumed to develop. Perhaps it is
because the resulting protostar to pre-main-sequence star to main-sequence star
is considered to require a disc of dust to finally encircle it; the
understanding being that planets form from an “accretionary disc” about a
young, forming star.
Most
computer models are based on collapse of a non-rotating molecular dense
core. Stahler (1991) argues that as the
region of collapse expands outwards, the higher angular momentum outer material
of a rotating mss, as it falls inwards, misses the small protostar and collects
as a disc about the central body. But
would this account for all the angular momentum? Nowhere does he say momentum is conserved. In fact Hartmann et al (1986) make it clear
(p.290) that collapse theories ignore angular momentum transport.
If
my idea of the central core of a shelled protostar separating from its parent
is correct then somewhat different conditions of core collapse are
required. While it is possible to argue
a simple development of the protostar if collapse is somewhat like a deflating
balloon, the mechanism of “inside-out” collapse gives an even better. However, three basic conditions are
required. First, angular momentum is
conserved by the acceleration of the whole collapsed mass to offset the
momentum lost by inward movement of material to the protostar; secondly, no
dust barrier forms; and three, the cloud consists of gas (dominant), dust
(minor component), and material larger in size than dust – though possibly not
greatly – (very minor).
Again,
I interpret the collapse as being Newtonian.
By this I mean that the individual grains (at least approximately) obey
the law F = GMm/d2. With
this so, of any two grains of equal volume but one having a greater density
than the other then that of greater mass will move more rapidly to the
centre. This condition modifies the
outer face of the region of collapse; it becomes a diffuse zone. It also means that there will be a
differential movement of material towards the forming protostar. While gas (dominantly hydrogen) is moving
inwards and accreting to the protostar, dust and grains of silicate, sulphides,
and carbonides, etc, and even heavier gases, move even more rapidly to the
centre. The result is that the core
will become enriched in these heavier elements and molecules, probably to the
point where they dominate over hydrogen.
Previous
simulations of cloud collapse using non-rotating masses assume that the loss of
potential energy results in acceleration of the individual collapsing particles
all radially to a central point, to supersonic speeds, and final loss of this
energy by heat at the shock front at/near the protostar where accretion takes
place. However, if there is angular
momentum and the angular momentum of the cloud is conserved, the momentum lost
by the infalling material will result in angular acceleration of the infalling
material. Acceleration would probably
be something like a watch spring, being somewhat faster in the inner part of
the cloud and somewhat slower in the outer part and the material would tend to
“wind around” the central point rather than fall to the point. (A multi-armed spiral could possibly tend to
develop in the interior during the early part of the collapse but would
dissolve during the latter part.) The
moment of inertia factor, originally somewhere near 0.4, would also slowly
decrease as the protostar developed.
Much of the potential energy lost by the infalling material would be
converted to kinetic energy.
It
can therefore be argued that the protostar would not attain temperatures
of evaporation of the silicates as Stahler et al claim. The temperature would slowly increase but
not at nearly the rate previous simulations assumed nor to the very high
temperatures calculated. Under such a
condition the infalling material would certainly continue to accelerate inwards
but perhaps not to as great a speed as previously believed. This means that the “shock zone” would be
much thicker than previously argued, with the material entering the zone at an
acute angle to the zone’s face.
Further, as Winkler and Newman (1980) point out (p.210), “If specific
angular momentum is conserved in the early stages of the collapse, any small amount
of rotation inherent in the initial cloud could be greatly magnified, slowing
down the accretion phase.” A longer
period of accretion, say nearer 106 years as sometimes suggested
instead of 105 years, would allow more efficient cooling in the
early and middle phases of the collapse.
It
is accepted that the protostar would not remain spherical as collapse
proceeded. It might or might not form a
sphere-cum-disc but recently reported observations of planetary nebula suggest
to me rather that a prolate ellipsoidally formed body develops (see later in
this discussion). Another difference to
Stahler’s model is that there would be no sharp physical divisions. Instead, the evolving body would be
continuous but with a gradually increasing density towards the centre (rapidly
increasing just outside the protostar) and with a gradual depletion outwards of
the heavier molecules/masses making up the cloud.
For
a 1 MSun cloud such a development would result in a much larger
protostar forming than that argued by Stahler before opacity near the protostar
could cause a rapid rise in temperature.
In fact it can be argued that movement of the heavier molecules to the
centre would be so rapid that the density of 10-13 g.cm-3
of a gas/dust mixture is not reached except at a thin zone at the surface of
the protostar so that little heat is generated and little impediment is given
to heat radiation.
Another
important result of “inside-out” collapse of the dense core is that initially,
transfer of angular momentum is very small but increases, at first gradually
and then at an accelerating rate, as collapse continues. This is the reverse of “balloon deflation”
collapse. Thus the rotation both of the
cloud and of the protostar would be small for a quite appreciable part of the
time of accretion. Only during the
latter part of accretion would rotation become high as progressively larger
amounts of momentum were incorporated in the collapse. This phenomenon is important for two
reasons, now given.
I
suggest that, in the case of our Sun, collapse and accretion continued until a
mass equal to that of the present
Solar System, together with the present (conserved) angular momentum was
reached. Further, that this mass was
finally attained when the breakup speed of a star of MSun,
given as about 200 km.sec-1
by Hartmann at al (1986), was reached by our protoSun. This speed would be reached by the protostar
edge at a radius of about 8 RSun if the cloud collapsed as a
deflating balloon. But Hartmann et al
(1986) on p.290 state, on the basis of “inside-out collapse, that “--, the
surface velocities of stars which have just ended their protostellar accretion
phase should be comparable to breakup speed.
For a 1 MSun with a protostellar radius of 5 RSun
(Stahler, Shu, and Taam 1980), breakup speed is 200km.s-1.” In my original approximate calculations for
core ejection I determined a value of 6RSun and so well within limits.
It
requires simple mathematics to see that at a distance of 5RSun centripetal acceleration would greatly exceed
acceleration due to gravity. However,
the speed of a once distant particle of the cloud, by the time it had neared
the protostar, would be very great.
While the model being suggested here would probably see a lesser speed
than that of the “supersonic freefall flow” (Winkler and Newman, 1980) quoted
in other models, the speed would almost certainly be supersonic. Thus, although the falling matter would
undergo outward acceleration as it drew near the protostar surface (rough
calculations suggest from about 1AU) it would still move inwards – but at a
decreasing velocity. This means that
after a long period of gradual increase in impact velocity, as the protostar
approached its final size, impact velocity would begin to decrease, so
preventing rapid heating by impact.
Remember also, the final material would be depleted in the heavier
elements.
A
stage would be reached, though, when matter would still be collapsing inwards
but its velocity would first become zero at the protostar surface, then
increasingly negative, i.e. outwards, before reaching the surface. Thus there would be for a period of time
outwardly flowing gas and dust, apparently from the protostar but in actual
fact originally infalling material which was unable to reach the protostar due
to velocity reversal. It is obvious
that for a while there would be two-way movement of material in the region of
collapse but because the outward moving gas would both impede the inward moving
gas, and because the outward moving gas would also finally impinge on the
surrounding collapsing cloud, so preventing it from collapsing and even driving
it outwards, growth of the protostar would cease and the surrounding cloud be
removed. What has been described here
is, of course, the T-Tauri stage of star formation. (It will be shown below that the commencement of outflow required
by the madel agrees with the known time.)
It is possible that a weak disc of material could encircle the
protostar, resulting from inward and outward material collision causing suitable
radial velocities for particles at the proper distances from the
protostar. The material should be
mostly dust.
The
suggested model therefore gives a poorly internally shelled protostar with a
radius (if spherical) of about 5RSun and average density about 1/125
that of the Sun, rotating at just on breakup speed. (Below it will be shown that the figure was more likely that of a
prolate ellipsoid, in which case breakup speed would first be attained at
pericentre.) Essentially it has no disc
and although hot it is at a temperature below the vaporisation temperature of
the silicates. There is some
observational support for this model.
Hartmann
et al (1986) studied the rotational and radial velocities of 50 T-Tauri
stars. Observations and comments made
in the paper, and which I believe to be pertinent to my argument, are commented
on as follows. On p.285 Hartmann et al
observe, “Recent studies of open clusters shows (sic) that many solar-type
stars are rapidly rotating when they first arrive on the main sequence
--.” This is support for my model of
increasing rotation as the protostar forms and develops into a
pre-main-sequence star. “The observed
rotational velocities of (p.288) T-Tauri stars in the same mass range [of
youngest open cluster, Per] (Fig.11b) imply that at least some T-Tauri stars spin
up during contraction to the main sequence. (My emphasis.) These data
can be used to constrain the amount of angular momentum lost during the
pre-main-sequence evolution of low-mass stars.” On p.288 they comment, “Low-mass stars rotate at rates far below
breakup velocity at an early age.” [on the main sequence line]. “----
The important thing is that in either [their emphasis] (p.290)
picture, the surface velocity of stars which have just ended their protostellar
accretion phase should be comparable to breakup speed ----. Since almost all [of the stars studied]
rotate at 20 km.s-1 or less, they have only 10% of the angular
momentum predicted by theories which ignore angular momentum transport during
collapse. These stars must therefore undergo
substantial braking during or immediately following (my emphasis) the
protostellar accretion phase; the pre-main-sequence braking cannot last longer
than 105 yr. --- What is needed is rapid angular momentum transport,
i.e. braking,
in the protostellar or early pre-main-sequence phase.” (My emphasis.) But such braking at this time is a requirement of the model I am
about to present. At a later stage in
their discussion, Hartmann et al propose braking to be due to “stellar wind
spindown”. They did not consider sudden
loss of a possible core of the protostar.
They
also comment on p.290, “In the mass range 0.7-1.5 MSun, our stars
have angular momenta per unit mass J between 2.8x1016 and 1.7x1017
cm2s-1 (using a factor of π/4 to correct for
the average sin i). The same quantity for
the Sun is 8x1014 cm2s-1, assuming
solid body rotation (see Brandt, Wolff, and Cassinelli 1969). Thus solar-type T-Tauri stars have angular
momenta 30-200 times that of the present Sun, ----. It is striking that most of these stars have angular momenta well
below the value of 1.6x1017 cm2s-1 for the present-day
solar system (Kraft 1970). Our
results suggest that the Sun never had most of the angular
momentum of the solar system.”
(My emphases.) I emphasise the
figures to show that the Solar System angular momentum is of the order of that
of the T-Tauri stars studied but that the angular momentum of the Sun, alone,
is very much smaller. As will be seen
below this is the result that my model requires. It also requires that the Sun never had most of the angular
momentum of the Solar System (but the protostar did have), which is the
conclusion arrived at, but without explaining it, by Hartmann et al.
The
results of Hartmann et al therefore can be fully explained by – are a
requirement of – the model of evolution of the Sun about to be presented
here. It is now necessary to show that
a “trigger” exists for the suggested mechanism which resulted in Sun and brown
body.
At
the beginning of this paper it was shown that if the small inner core of such a
body was to be displaced, then a mechanism depending on the moment of inertia
factor comes into play and the core must be ejected from the protostar. An astrophysicist who was approached with
this mechanism would not accept it as he could see no way the mechanism could
be triggered. It may not be necessary
to devise or suggest a “trigger”; it may result as a natural consequence of
evolution of the star. I make this
comment for the following reasons.
Hernquist (1991), in an introductory article to one by Boss on
pp.298-300 of the same publication, made the following comment, “It is
certainly the case that the cores of molecular clouds are not smoothly distributed6. In general, one might expect the density to
vary over a number of angular scales ---.”
I suggest that if the cores are irregularly distributed then the gas
forming any one core will have unequal forces exerted on it, due to the
surrounding cores. This being so, I
suggest that the body, particularly in its medium to late stages, will tend to
form a body of triaxial ellipsoidal form.
Again,
Boss (1985) concludes from numerical analysis that a slowly rotating cloud
collapses, at first slowly then rapidly and asymmetrically, to form an asymmetric
core. In his conclusions section, in
respect to the middle stages of collapse, it is stated, “A very slowly rotating
cloud will dynamically collapse into the nonisothermal regime without
undergoing fragmentation. However, the
pressure supported central condensation that results will be surrounded by a
strong nonaxisymmetric region of infalling and quasistatic gas.” It is therefore not at all a ridiculous
statement for me to make in proposing that the collapse was asymmetric in the
Solar System case.
In
my view the development of a collapsing core would be very similar to that of
compact planetary nebulae. Thus I
present selected parts of the paper by Aaquist and Kwok (1991) to show that the
development of a compact planetary nebula gives an identical structure to that
argued in my hypothesis for a developing planetary system.
In
the paper by Aaquist and Kwok are brief descriptions of some of the individual
bodies studied by the writers. Of
interest here is that almost all the objects are noted as being
non-spherical. For example, for object
M2-43: “ The 15 GHz map in Figure 1
shows a classical planetary nebula shape, which can be explained most simply as
a
prolate ellipsoidal shell.”;
for object K3-6: “--- The
spherical shell-like geometry shown in Figure 1 leaves little doubt that this
is a planetary nebula, ---- “This is the only object in our sample that shows a
perfect circular shell geometry --“;
for object K3-29: “-- At 15 GHz K3-29 takes on the apperance (sic)
of a
very regular prolate ellipsoidal shell --. A closer examination of the 5 and 15 GHz images indicate that the
major
axis of the outer halo is slightly different from the major axis of the inner
structure seen in the 15 GHz map.”;
for object M3-35: “-- The deep central emission minimum and the
‘concave’ contours surrounding the peaks imply that this is a
prolate ellipsoidal shell structure with the major axis rotated towards
us by about 60o. There is a
significant brightness enhancement at the SW end of the nebula, indicating
greater densities in that direction.”
(My emphases.)
The
above descriptions make it clear that planetary nebulae tend more readily to
form prolate ellipsoids than spheres or oblate ellipsoids. In particular, object M3-35 shows that the
denser inner material concentrates, or in some cases concentrates,
“off-centre”. It probably concentrates
about a focal point of the ellipsoid.
It is this combination, formation of an ellipsoid and offsetting of the
centre of mass, which is so interesting for the hypothesis presented in the
present paper; for they are the developments required in a molecular core and
its resultant collapsed centre to begin the whole process of planetary
formation. And the comments of Hernquist
and of Boss referred to above indicate that such asymmetry and concentration
does or could take place in a protostar and its precursor. From this, also, it is seen that abrupt
physical movement of the core of the proposed protostar is not necessary to trigger
the ejection mechanism, for such a body as it evolves becomes both
symmetrically and dynamically unstable.
[Years
later I became aware that there are ellipsoidal galaxies and that they follow
the same dynamics as I give in this article, except that some merely detach an
end. See a paper on this in this
website.]
With
such a development, as the nebula, the protostar and envelope, increases its
rotation during protostar growth, the surface velocity of the protostar will
increase. (As pointed out above, p.116,
because buildup is by “inside-out” collapse rotation increase will be slow for
quite some time but will be rapid in its later stages – proportional to d3?) For a sphere the surface speed will be equal
around its equator. For a prolate
ellipsoid the surface speed will vary, being least at apcentre and greatest at
pericentre. That is to say, the maximum
velocity would be nearest to the core of an object that has developed into a
form such as M3-35. It would be here
that breakup velocity would be first reached, leading to a decrease in pressure
along the pericentral axis of the core.
The core would be expected to move outwards along the pericentral radius
of the object.
It
is clear that the descriptions of the planetary nebulae given by Aaquist and
Kwok, when assumed to be applicable to (at least some) protostars, as the
comments of Herquist and Boss suggest
is possible, point to a simple, at first gradual, and inevitable trigger for
the ejection mechanism of the postulated (by me) shelled core of a
protostar. The conclusion then is that
planetary systems, or in some cases probably binary stars, form as a natural
consequence of star development for a certain mass range. The nebula, and almost certainly the
protostar, develop as bodies with off centre mass and increasing rotation. A stage must be reached in such a case when
the protostar core is ejected. This
ejection would take place at about the same time as outflowing of gas from the
object, i.e. the T-Tauri stage of the protostar as it passed through the
pre-main-sequence phase of star development to main-sequence stage.
The
above arguments attempt to show, amongst other things, that a mechanism is
inherent in prolate ellipsoidal protostars requiring the core to be displaced
along the pericentral axis when the pericentral speed reaches escape
velocity. It is now intended to
consider the effect displacement of the core would have on the system. The investigation cannot be mathematically
rigid as the physical values to be entered in the formulae used can only be
approximated. Still, it is intended to
show that even so, the results support the hypothesis of this paper.
The
moment of inertia factor of the present Sun is given as 0.05 (Cole, 1990). (Cole (1979) gives a value of 0.06 but as
this figure was seen after calculations were made the value of 0.05 will be
used.) The protostar from which the Sun
and its satellites originated must therefore have had a much smaller moment of
inertia factor. This because not only
was its radius at breakup equal to 5RSun but it also carried at its
centre the heavier elements which ultimately made up the planets and so had an
even relatively smaller, dense core than the present Sun. For computation purposes I shall assume a
moment of inertia factor of 0.05/52 = 0.002.
It
is assumed that instability of the protostar resulted in displacement of its
core outwards along the pericentral axis.
Should such a movement occur the protostar would, in effect, become a
system of two interacting bodies rather than a single body and the moment of
inertia factor of the system would rise.
The factor for the brown body as a sphere has been calculated as about
0.357 (p.85); for the remaining bulk of the gases, that which ultimately became
the Sun, would abruptly rise towards 0.4.
I shall use a conservative figure of 0.25, making it slightly less than
present Jupiter, although it would probably have been greater.
Using
the above figures, the masses of the present bodies in the Solar System, and
the radius of the Sun – considering the body as a sphere – we can say that
before the core moved from the centre of the protostar (When the calculations were made in the early 1990s my belief was that
the mass of Jupiter was 1999x1024 kg. This figure was taken from
“Early Physical Conditions of the Planets and Satellites” by G.H.A.Cole, Table
2, p.5 [“Geophysics. 1990]. It was not until some considerable time later that
I discovered the value to be 1899x1024 kg. I have not recalculated the answers to the following equations as
the differences in values is small, a maximum of 5%, and with the assumptions
being used the difference cannot change the argument.):
Angular
momentum of the protostar = IШ = aMr2Ш =
0.002x1.99276x1030x52x 6.992x1010xШ
= 312.6x1035 kg.km2.sec-1 (from Cole, 1990, p.11)
And therefore Ш = 6.4211x10-4 rad sec-1
Thus
the kinetic energy of the body would have been = 1/2.aMr2Ш
= 0.5x0,002x1.99276x1030x52x6.992x1010x6.42112x10-8
=10.036x1033 kg.km2.sec-2
After
the core had moved from the centre:
Approximate angular velocity of the main mass of the protostar (the
protoSun) plus displaced core (on the assumption the core is still within the
main mass)
= Ш = 312.6x1035/(0.25x1.99276x1030x52x6.992x1010)
= 5.1369x10-6 rad.sec-1
That
is, there would have been a large decrease in angular velocity.
The
body would then have had kinetic energy approximately
= 0.05x0.25x1.99276x1030x52x6.992x1010x5.13692x10-12
= 8.029x1031 kg.km2.sec-2
Thus
there was a great change of energy, also.
It
was calculated on p.17 that the energy in the present Solar System (Saturn
omitted) about the Sun is approximately = 183x1033 kg. m2.sec-2 or 183x1027 kg.km2.sec-2
For
Saturn, K.E. = 0.5x568.6x1024x9.62 = 26.2x1027
kg.km2.sec-2
P.E.
= -6.67x1.99x1030x568.6x1024/(1011x214.9x6.99x108x9.54x106)
= -52.7x1027 kg.km2.sec-2
Therefore
total energy at present in the Solar System of planets approximately =
(183+26.2)x1027 = 209.2x1027
Say
210x1027 kg.km2.sec-2.
This
is the energy which would have been required to drive the planets from the
protoSun centre assuming no energy loss.
But the difference between the energy before and after ejection of the
brown body core was approximately 10.0x1033 – 8,0x1031 =
9.92x1033 kg.km2.sec-2
That
is, some 4.7x104 times more energy was released than was required
for ejection. Allowing for energy loss
of 50% from heat radiation, gas flow, etc. there would have been 100 times the
energy required. The energy required was
very little.
But
assume that the excess energy of 9.992x1033 kg.km2.sec-2
was required to separate the two bodies – the Sun mass and the brown body mass
– at escape velocity (as the brown body did not escape this case is worse than
actual.), then the two body centres would have had to have been separated by
the following fraction of the present Sun radius: s = 9.92x1033x1011x6.99x105x106/(6.67x1.99x1030x2.76x1027)
= 0.0189 (Note: 1 - 0.0189 = 0.9811,
that is 0.98.)
That
is, the core would need only to have been displaced by about 13230 km from the
centre of the original body, which at 3495000 km radius if taken as a sphere
was very little indeed. If it is
accepted that the brown body began to move independently, due to the escape
velocity being reached at the pericentre of the protostar, the distance between
the body centres would already exceed the distance required; unless ellepticity
was small.
Thus
the calculations show that even if worse case scenarios are used there was
ample energy to eject the brown body core to 5.3 AU from the remainder of the
protostar mass, i.e. from the protoSun.
The
calculations show that as the core was ejected the remainder, the protoSun,
slowed down. This must have occurred
fairly rapidly. Oldshaw (1968) on p.8
states, “--, the predominant forces controlling everything [of a collapsing cloud]
are the gravitational forces and the forces due to the rotation.” Rotational forces oppose the gravity
forces. The rapid and large reduction
of the velocity of the protoSun would mean the outward force due to rotation
would be greatly reduced and the effect of gravity greatly enhanced over that
short time. This would result in a
rapid decrease of the protoSun’s radius as the dominantly gas cloud collapsed
inwards – in this case from 5RSun to 1RSun. There would be a concomitant decrease in the
moment of inertia factor with decrease in the radius – to say its present-day
value of 0.05 – as the pressure towards the centre of the body would rapidly
increase to beyond the atomic bond collapse pressure and so cause a density
concentration at the centre. It is
assumed that because of the collapse the pressure at the centre reached fusion
pressure for hydrogen and the Sun came into being.
With
the figures used above, at the end of the collapse the new body, the Sun, would
have an angular momentum = 0.05x1.99x1030x6.992x1010x5.1369x10-6 =
2.4973x1035 kg,km2.sec-1
This
is of the order of the present Sun value of 1.6x1035. Note that this value for the Sun is
2.5/312.6 = 1/125th of the value before slowing down. This accords with the values determined from
a study of present protostars by Hartmann et al (1986, p.290) and quoted on
p.118 of this paper, viz. J per unit mass between 2.8x1016 and
1.7x1017 cm2.sec-1 compared with the Sun of
8x1014 or a ratio of approximately 1.7x1017/(8x1014)
= 125! It also shows why the
difference. The momentum has been
transferred to an ejected body which ultimately broke up to form the present
planets. The sum of momenta for Sun and
planets approximates determined protostar values.
Actually,
when the idea of this mechanism first originated, and before reading other
workers’ papers, the writer roughly deduced an unstable radius of approximately
6RSun and used an original moment of inertia factor of 0.0015. These values give an angular momentum for
the Sun of 1.73x1035 kg.km2.sec-1.
(The
rotation of the Sun calculated here is 5.1369x10-6 rad.sec-1
compared to a value somewhere near 2π/(26.5x3600) or 6.5x10-5
rad.sec-1. However, the
exact rotation cannot be estimated as the body does not act as a perfect solid
and rotation varies with latitude and possibly with depth. When the Sun’s known angular momentum of
1.6x1035 kg.km2.sec-1 is inserted in the
equation to find angular velocity a value of 3.2911x10-6 is
obtained.)
The
actual factors used do not really matter; they show, no matter which reasonable
figures are selected, that the angular momentum of the Sun in a closed system
Sun and satellites, and derived from a protostar due to ejection of its core,
MUST be less than that of the remainder of the system. The hypothesis thus accounts for the so-far
unsatisfactorily explained phenomenon of the main member of the Solar System
containing less angular momentum than its system of satellites. In fact, it is a requirement of the
mechanism.
The new kinetic energy of the
main body, the Sun, would = 0.5x0.05x1.99x1030x6.992x1010x5.13692x10-12 = 6.4x1029 kg.km2.sec-2,
i.e. a further loss of energy, though much less than before.
As
angular momentum is conserved in a closed system, and there were only two members
in the protostar system considered, then the loss of momentum by the protoSun
to give the Sun could only have been taken up by the ejected core, i.e. the
brown body. (Note that it is assumed
that the direction of the force of ejection of the brown body was through the
centre of the protostar.) It has been
shown above that there was ample energy generated to give the brown body the
speed required to absorb the momentum change,
Before
the collapse of the protoSun, immediately prior to the brown body core moving
from its “central” point, the brown body would have been rotating approximately
in harmony with the protoSun mass. At
5RSun this rotation was 6.4211x10-4 rad.sec-1
(previously calculated, p.122) assuming a rigid body. Because of its much greater density compared with the protostar
remainder, its much greater rigidity , its greater moment of inertia factor,
and its comparatively small surface area, the rotation of the brown body would
decrease only slightly as it passed out of the protoSun. (It would be expected that its path outwards
would be a spiral as both a tangential/circumferential force and an axial force
would be acting on it. However,
ejection would have been completed after about one revolution.) Say that at final ejection the brown body
axial rotation had slowed 10% due to the friction of the gases through which
the body passed. This suggests the
brown body had a rotation of 6.4211x10-4x0.9 = 5.7790x10-4
rad.sec-1. The brown body
radius as a sphere has been determined as 85332 km. Therefore the surface speed of the body = 85332x5.7790x10-4
= 49.3 km.sec-1. This
compares with a previously calculated particle speed (p.101) of 46.5 km.sec-1. The near equality of the values may be pure
coincidence; then again, they may indicate that the hypothesis has some
substance.
It
is probably worth pointing out here that the brown body when at the centre of
the protostar reflected the structure of the protostar. That is to say, it was probably prolate
ellipsoidal in shape. This would have
greatly facilitated its elongation after leaving the protoSun; it was already
partway to it.
Another
comment that can be made is that if the total energy of the planets estimated
on p123 – 210x1027 kg.km2.sec-2 - is equated
with 1/2mv2 then v = 12.3 km.sec-1 for
the initial radial movement of the brown body.
Also
as shown in Test 10 the orbital speed of a body in our Solar System at 5RSun
from the centre of the Sun would be 195.4 km.sec-1, which, being
closely the breakup speed of 200 km.sec-1 at that distance as
calculated by Stahler et al (1980) for a protostar of 1MSun,
suggests that this was the orbital speed of the brown body about the protoSun
as it passed out of that body. It can
be argued, then, that the orbital (tangential) speed of the brown body as it
left the protoSun was considerably greater than the radial speed from the
protoSun centre. Thus the requirement
as enunciated by Motz and Duveen (1966) that such a condition must exist for a
near-circular orbit of a satellite about the Sun is met in this case.
BINARY STARS
As
a digression I would point out that the mechanism of breakup of a 1MSun
protostar as given here allows some light on binary star systems where the
companions are relatively near to one another.
The mechanism that follows is not considered the only mechanism that
produces binary stars. It may, however,
be the only one to produce planets.
Cole
(1990) has pointed out that there appears to be a distribution gap between the
smallest of brown bodies and the largest of planetary bodies. For some reason or other intermediate-sized
bodies appear to be lacking. He writes
(p.44), “It is strange that the greatest planetary mass (Jupiter) is very close
to the maximum possible (see Section 4.5),
This is the more interesting because no known dwarfs have yet been
unambiguously identified .but theory (Section 4.4) predicts a continuity of
mass at least to the stellar region.
Barnard’s star (see Section 8.2) provides a case where the situation is
far from clear, but otherwise the known small companions of stars have masses
in excess of 1029 kg, which is in the stellar range. The surface temperature of a brown dwarf is
likely to be in the range 1000-2500 K and this is low for present detection
methods, and this could be the explanation of why they have not been detected.”
From
these points, and interpreting the effects of variation in mass and angular
momentum by use of the hypothesis, we can suggest the following. I shall use the term dense clouds for the
usual term dense cores in molecular clouds.
This is to avoid confusion with the dense core of the protostar.
For
increasingly smaller dense clouds, even if the body became ellipsoidal, a size
would be reached where, assuming the escape velocity could be reached, ejection
would either approximate the breakup of the brown body discussed above or no
core breakup would take place. The
outer gas would simply pass off. This
latter case would occur where temperature was relatively low and ellipticity
(perhaps in part as a result) was not great so that movement of the core would
be insufficient to trigger the moment of inertia mechanism. It is interesting to note that the
postulated brown body before breakup into the planets lay just within and at
the bottom of the hypothetical brown body field; while Jupiter, its remnant,
lies just below.
In
the case of a large dense cloud mass, its initial angular momentum would have
to be large, indeed, for the derived protostar to be sufficiently prolate for
escape velocity to be reached before core fusion.
Thus
it would be expected that binary star systems would tend to be commonest in a
restricted, “medium-sized” mass range for stars.
For a 1MSun ellipsoidal
protostar, it would be expected that (i) a single star would form: (ii) a star
with a planetary system would form; (iii) a star with a brown body companion
would form; or (iv) a star companion would form depending on whether the
initial angular momentum of the dense cloud was low, moderate, somewhat higher
than moderate or high. (2008 emphasis. F.L)
For
larger masses with slightly lower angular momentum the ejected core (with
captured gas) could be sufficiently large to develop a star, so that a binary
system would form. For yet larger
masses the core could be ejected to form a star but the protostar collapse only
slowly to allow a repetition of the trigger mechanism, so forming a ternary
system. However, a mass would be
reached where even for high angular momentum breakup would be unlikely to
occur. (It is also possible in large
mass cases that Jean’s mechanism would come into play in some cases, resulting
in peaceful separation of the protostar.)
[See the paper in my website on my
much later brief study of ellipsoidal galaxies with accompanying qasars. This latter description fits their breakup
very well. FL, 2008.)
It
can also be said from the above that a large force of ejection would propel the
core out to a larger distance from the originating protostar than a small force
of ejection. This varying radial
velocity has important consequences. As
has already been mentioned, Motz and Duveen (1966) – see 523 M923 – p.147 point
out that if any particle is thrown into space in the neighbourhood of the Sun
“it can move in a circular orbit only if at the moment it is thrown, it is
moving exactly at right angles to the line from it to the sun and exactly at
the speed given by the previous equation [V2circle = G(MSun+m/a].” Any deviation from these conditions results
in a noncircular orbit. Thus the larger
radial ejection force the more elliptical will be the orbit of the ejected
body; for the escape velocity ill almost certainly increase only slightly for
increasing body size. (Note for the
brown body previously discussed the escape velocity for the body which is one
thousandth the size of the protostar has been computed as 133.8 km.sec-1
compared with 200 km.sec-1.
How
does all the above fit with field data?
In
reporting on AU Colloquium 135 of April, 1992 Clarke (Cathie Clarke, 1992,
“Stars arriving two by two”, Nature 357 pp.197 and 8) records,” The picture that
is starting to emerge from this wealth of data is that binaries are very common
among main sequence stars, current estimates suggesting that the fractions of
binaries among solar type stars is at least 65% (M. Mayor, Geneva Observatory). These binaries span six orders of magnitude
in separation, from systems that are almost in contact to those that are so
wide that they are destined to be dissolved eventually by encounters with field
stars.”
Clarke
goes on, “The orbits are generally highly eccentric, implying that binary
companions must form in a fundamentally different way to planets, whose orbits
are pretty much circular. One notable
result is an apparent dearth of the highest eccentricities among systems with closer
separations, implying that the binary formation process cannot be entirely
independent of scale. ---- but the lack of high-eccentricity orbits in somewhat
wider systems [than for orbital circularization, FL] remains a puzzle.”
The
twenty pre-main sequence binaries reported by Clarke (op sit) are
explainable. In fact, as the model
requires the core of the protostar to be ejected immediately prior to the
development of the ensuing star it is a necessary part of the model that there
be “binaries in their stellar youth”.
“Such stars are probably less than a million years old, so that binaries
can evidently form very early in the stellar evolution process (R. Mathieu,
University of Wisconsin).” (Clarke op
sit.) The suggested model requires
exactly this. “—It is among these
systems [pre-main sequence binaries in regions of star formation, FL], with
separations in the range ten to several hundred astronomical units (----), that
there would appear to be an excess of binaries compared with main-sequence
counts ---- it poses a number of questions: either a large number of systems
are being missed in mid-sequence surveys or else many binaries in this
separation range are somehow destroyed during their main-sequence
lifetimes.” The model requires at least
some of the ejected star companions to break up into planets and so there
should be an excess of binaries pre-main sequence compared with main sequence.
Finally,
the hypothesis can also explain why some stars have a gas/dust disc surrounding
them; without the disc necessarily being the precursor of planets. It can be envisaged that in some cases a
protostar may form as a sphere or an ellipsoid of small ellipticity so that the
escape velocity would be reached simultaneously along a long length of its
circumference, and probably over a reasonably broad band of low latitude. Also, the core, even if it was not at the
centre of the protostar, would be displaced only a relatively small amount so
that insufficient energy would be available to eject the core. The result would be for only surface
material to be repelled outwards and some of this material would move outwards
sufficiently slowly to be captured by the forming star. Thus the star would become surrounded by a
disc. Obviously, occasional stars must
form which bridge the difference between both types and so exhibit both disc
and partner(s). (The Sun, itself, has a
weak disc and can be explained by similar reasoning.)
In
the writer’s opinion the model derived from the hypothesis gives an adequate
explanation of binary star frequency and distribution and of discs and so can
be used as a basis for their study.
However, this is not germane to the topic of the paper and so will not
be developed further.