Download Understanding Solar System's Angular Momentum: Torques, Eddy Transfer, and Nebula Models and more Schemes and Mind Maps Geometry in PDF only on Docsity!
ANGULAR MOMENTUM FLUX IN THE FORMATION OF THE SOLAR SYSTEM. by NORMAN EUGENE GAUT A.B., University of California at Los Angeles (1959)
SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY January, 1964
Signature of Author (^)... .,. --.---. * - v-w-...... .. Department of Meteorology, January 20, 1964
Certified by. ... .......... (^) Thesis W L.- (^) Supervisor -v.. ......
Accepted by (^) Chairman, Department.......... (^). (^) Committee.... (^). (^) on. (^) Graduate.... (^). (^) Students .......
.WStES
ANGULAR MOMENTUM FLUX IN THE
FORMATION OF THE SOLAR SYSTEM.
by Norman Eugene Gaut
Submitted to the (^) Department of Meteorology on January 20, 1964 in (^) forpartial the degreefulfillment of Master of theof requirementScience.
ABSTRACT
introduction.The^ paperThe secondis^ divided presents^ into a threecomprehensive^ sections. historicalThe^ first reviewis an of theories explaining the origin of the Solar System. and eddy transferPart^ three of^ introducesangular momentum^ the^ concept in a hypothetical modelof^ gravitational^ torquesof a solar nebula. It is found that if a spiral asymmetry to the mass andistribution imaginary^ isradial^ present, wall,^ theon^ gravitationalthose inside,^ torquestends to^ ofdraw^ particles angular^ outsidemo- mentumthe same radially distribution outward. is (^) inward, butThe eddy transport seems to ofbe angularless, momentum for an order of magnitude of the gravitational torques. however, within
Thesis Supervisor (^) : Victor P. Starr Title : Professor of Meteorology
TABLE OF CONTENTS
Page Chapter 1. INTRODUCTION Chapter 2. REVIEW - Descartes^ OF^ IMPORTANT^ SOLAR^ SYSTEM^ COSMOGONIES.
-^ - KantSwendenborg - - LaplaceChamberlin and Moulton - - JeansRussell - - (^) AlfvdnLyttleton - - WeizsackerKuiper - Hoyle Chapter 3. ANGULAR NEBULA. MOMENTUM CONSIDERATIONS OF^ THE^ SOLAR
Appendix I. CONSTANTS AND CONVERSIONS Appendix II.TABULATION OF ADVECTIVE PARAMETERS REFERENCES
42^36
A-
A-
LIST OF FIGURES
Chapter 2. Figure 1. Production (^) of spiral filaments by dynamic encounter Figure 2. Orbits resulting from filaments drawn out by a passing star. Figure 3 and 4. Reduction of ellipticity by accretion of matter. Figure 5. (^) matter.Forward rotation resulting from the accretion of Figure 6. Production of planets by.dynamic encounter. Figure 7. Figures of equilibrium for a rotating body. Figure 8. Rotationalof a instability and consequent break up liquid (^) body. Figure 9. Equipotential surface for M'/M = 2. Figure 10. Equipotential surface for M'/M = a Figure (^) 11. Possible orbits of one body about another. Figure 12. Ionisation versus temperature for selected elements Figure 13. Geometry (^) of the magnetic field and coordinate system Figure 14. Relationship2) magnetic fieldbetween line, (^) : (^) 1)3) ionization minimum magnetic surface, line, 4), 5), 6) are separation surfaces. field Figure 15. Mass (^) distribution curves for reduced equatorial distance x. Figure 16. Ellipticalfor angular (^) andmomentum. circular orbits with similar values
Figure 17. (^) by The (^) Weizsackerconfiguration of turbulent eddies envisioned
Figure 18. Magnetic (^) lines of force connecting sun and disc as seen in the equatorial plane. Figure 19. Polartheory view of theof (^) originthe magnetic of solar field nebula. in Hoyle's
Page 15 15 16 18 19 24 29a 30a 30a 39 47 58
59 61 63 70 94
1. INTRODUCTION
The origin of^ the^ Solar^ System^ is a^ natural^ extension^ of^ the history of man and his environment. As^ such,^ it^ has^ been^ of^ interest to natural scientists since its recognition as a semi-closed system. The details of its birth, however, are^ obscured^ by^ time.^ Attempts^ to relate its beginning to^ similar^ phenomena^ occurring^ now^ in^ our^ galaxy have been nullified by distance. Speculation based^ upon^ physical insight, probability,^ and^ mathematical^ analysis^ have^ been^ and^ remain the chief means of unraveling the mystery of its origin. The purpose of this paper is two fold. Firstly,^ it^ is^ to^ re- view the most important ideas published on planetary^ evolution^ since Copernicus. Particular emphasis will be^ placed^ upon^ their^ explanations of the present distribution of angular momentum within the system. Not all theories will be^ presented^ in^ the^ same^ detail.^ A^ particular^ school of thought will be explored by presenting^ rather^ completely^ the^ ideas^ of its foremost proponent. Only significant modifications by other members of the same school will^ then^ be^ discussed. Secondly, its purpose is to introduce certain ideas of momentum flux due^ to^ gravitational^ torques^ and^ advective^ processes^ in^ a^ hypotheti- cal spiral nebula.^ It^ is^ hoped^ that^ these^ latter^ processes^ will^ shed light on a possible mechanism^ for^ angular^ momentum^ transfer^ during the early formative years of^ the^ solar^ system.
- REVIEW OF IMPORTANT (^) SOLAR SYSTEM COSMOGONIES.
The (^) acceptance of the heliocentric view of our planetary family, as proposed by Copernicus (^) in 1543, presented the problem of how such a system (^) could evolve. Any theory must explain several readily observable features. For instance, (^) why do all the planetary orbits approach (^) a common plane? Why does the sun, the planets, (^) and most of the planetary satellites rotate with the (^) same directional sense? And why are there nine major planets (^) and not, say, one thousand small ones or one (^) large one? Equally important as (^) the regularities to explain are the pe- culiarities of the (^) Solar System. What is the origin (^) of Saturn's rings? What is the significance of (^) the asteroid belt? And, what does the pre- sent distribution of angular (^) momentum imply about the early history (^) of the system? The last question (^) has been the major stumbling block to (^) the acceptance of several (^) otherwise plausible theories. Straightforward calculations (^) show that the sun carries 98.8% of the (^) entire mass associ- ated with the solar (^) system. Yet, its angular momentum (^) represents only about (^) 2% of the total (^) distributed among (^) these bodies. (^) Jupiter, alone, carries approximately 65% of the total angular momentum and the four major planets together (^) account for close to 98%. Since the 16th century, many ideas (^) have been presented to answer the questions (^) mentioned above and others equally interesting and demanding. (^) The major ideas in chronological order are presented below.
ment of the^ Nebular^ Hypothesis.^ In^ 1734,^ almost^^50 years^ after^ Newton's Principia was published, he^ presented^ his^ explanation^ of^ the^ evolution of the planets. It was not^ scientifically^ rigorous^ but^ incorporated the first use^ of^ dynamic^ instability^ as^ the^ basis^ to^ planetary^ birth. In an unexplained manner, Swedenborg^ envisioned^ a^ chaotic mass in^ the^ universe^ coming^ together^ into^ a^ rotating^ sphere.^ Because of its rotation, this^ sphere^ threw^ off^ a^ ring.^ The^ expanding^ ring eventually burst and the remains^ coalesced^ into^ the^ planets. It is evident that the knowledge of^ Saturn'^ s^ rings^ influenced Swedenborg to propose his theory.^ The^ rings^ were^ first^ seen^ by^ Galileo in 1610 and^ interpreted^ correctly^ by^ Huygens^ in^ 1655.
Immanuel Kant - (1755) Although Immanuel^ Kant's^ fame^ rested^ upon^ his^ writings^ and teachings as^ a^ metaphysician,^ during^ his^ early^ lifetime^ he^ was^ interested in mathematics and^ natural^ science.^ He^ studied^ Newton,^ lectured^ on mathematics, and taught physical^ geography^ for^ many^ years.^ His^ scien- tific pursuits led him to propose^ a^ theory^ of^ solar^ system^ development. Kant saw^ the^ great^ similarities^ which^ mark^ the^ dynamical^ char- acteristics of^ the^ system^ :^ six^ planets^ (in^ his^ day)^ with^ nine^ satellites revolving about the sun in^ the^ same^ direction,^ almost^ in^ the^ equatorial plane of^ the^ sun,^ and^ rotating^ upon^ their^ own^ axes^ in^ the^ same^ direction as the solar star itself.^ The^ relative^ lack^ of^ matter^ outside^ the^ rota- ting bodies was^ also^ perceived^ by^ Kant.^ He^ reasoned^ that^ the^ material once pervading this part of space was now collected into the planets,
their satellites, the sun and comets. The simplest assumptions possible^ were^ explicitly^ made^ by Kant :^ the^ eventual^ material^ of^ the^ solar^ system^ was^ originally^ scattered as atoms throughout the entire volume^ now^ occupied^ by^ the^ system.^ In^ his theory, this diffuse^ nebula^ shrank^ due^ to^ self-gravitation,^ and,^ in^ the process, Kant erroneously^ reasoned^ that^ the^ mass^ acquired^ rotation^ through the irregular collisions of its^ particles.^ Most^ of^ the^ material^ collected in the system's central^ star.^ But^ a^ thin^ disc^ of^ matter^ remained^ with each of its discrete constituents orbiting^ about^ the^ sun.^ He^ reasoned further that the outer parts of the nebula contained (^) the most matter, because of their greater^ volume,^ but^ that^ the^ inner^ parts^ of^ the^ disc were more dense. The^ planets^ and^ their^ satellites^ were^ first^ formed^ by the adhesion of colliding particles, and, as^ they^ grew,^ also^ through gravitational attraction. Forward^ rotation^ for^ all^ the^ bodies^ was deduced because^ the^ particles^ collected^ from^ the^ larger^ orbits^ were moving with greater velocity than^ those^ collected^ from^ smaller^ orbits. (This is not adequately proven^ in^ light^ of^ the^ supposed^ free^ orbiting particles making^ up^ the^ disc). The heat^ developed^ from^ the^ particles^ striking^ the^ growing planets, caused them to liquefy. While^ in^ this^ liquid^ state,^ the earth's moon was^ reduced^ to^ synchronous^ rotation^ by^ tidal^ friction^ in^ the earth's gravitational field.^ Further,^ some^ elemental^ segregation^ in^ the planets, based^ on^ density^ took^ place^ under^ these^ liquid^ conditions. Since, in Kant's lifetime,^ Jupiter^ and^ Saturn^ were^ believed^ to^ still be liquid, this^ segregation^ process^ was^ thought^ to^ be^ still^ active^ in
system, the rings persisted. Of course, the^ major^ defect^ of^ the^ dynamical^ explanation given by Kant was his belief that angular momentum would^ be^ generated when (^) the solar nebula was initially collapsing. (^) This is unexpected, since he asserts that^ angular^ momentum^ throughout^ the^ Universe^ is conserved. Moreover, it is clear that only the^ first^ cycle^ of^ the solar system is^ without^ aprevious^ history^ of^ rotation.^ For^ the^ sun^ is described as rotating rapidly before it expands^ into^ the^ rarefied nebula to start a new cycle. Laplace - (1796) Pierre Simon Laplace has been popularly associated with the origin of the Nebular Hypothesis of^ solar^ system^ evolution.^ The^ expo- sition of his ideas, however, was in a semi-popular book, and the dis- cussion was brief, speculative, and non-quantitative. His ideas were similar to Kant's in many^ essential^ respects,^ although^ he^ did^ not^ make the errorcf creating angular momentum where none^ before^ existed. Like Kant, Laplace was^ impressed^ by^ the^ regularity^ of^ the planetary orbits and their axial revolutions.^ (Even^ though^ Herschel^ in 1787 had discovered^ the^ retrograde^ satellites^ of^ Uranus,^ Laplace^ made no mention of them). He postulated that the only progenitor of such dynamic similarities could be a fluid of immense extent. Because of excessive heat, the sun at one time extended beyond^ the^ present^ limits of the solar system, (perhaps from^ an^ outburst^ as^ a^ true^ nova),^ and constituted this immense fluid.^ As^ this^ primal^ rotating^ nebula^ cooled 'and contracted,^ its^ angular^ momentum^ remained^ constant.^ To
compensate for the reduced moment of inertia, the angular velocity increased. Instability at the equator, from the ever increasing angular speed, caused the shrinking sun to leave rings of matter behind^ at^ regular^ intervals. These rings contracted and^ condensed^ into^ the^ planets.^ Some^ of^ these planets produced satellites by a^ small^ scale^ version^ of^ their^ own^ births. Laplace reasoned^ that^ a^ remnant^ of^ this^ nebula^ still^ existed^ and^ reflected radiation from the sun to the earth as the zodiacal light. Because his^ theory^ lacked^ computational^ and^ observational verification, (^) Laplace placed little confidence (^) in it. Nevertheless, this became^ the^ accepted^ theory^ of^ the^ origin^ of^ the^ solar^ system^ for many years. Chamberlin and Moulton - (1902) The Nebular Hypothesis of Kant and Swedenborg was not the^ only theory which preceded Laplace. The french mathematician had examined and rejected a suggestion made by Buffon in^^1750 that^ the^ planets^ had^ been formed from (^) the collision of the sun and a (^) massive comet. This idea, despite its implausibility today^ must^ be^ considered^ as^ the^ first^ sug- gestion of^ a^ collisional^ origin^ to^ the^ solar^ system.^ Such^ a^ beginning was seriously examined^ by^ two^ Americans^ around^ the^ turn^ of^ the^ 20th century, F.R.^ Moulton,^ and^ T.C.^ Chamberlin.^ From^ their^ studies,^ the Planetesimal Theory^ of^ Solar^ System origin^ evolved. T.C. Chamberlin was primarily a geologist. However,^ his^ studies of ice ages and their causes^ led^ him^ to^ doubt^ the^ gaseomolten^ concept^ of the origin of^ the^ earth^ as^ required^ by^ Laplace.^ Evidence^ was^ lacking^ in geologic history wbich could be interpreted as supporting
cules and atoms. The self-gravitation of^ such^ a^ ring^ would^ be^ very^ small because the matter which made it up would be^ distributed^ over^ the^ present orbit of the earth, a linear distance^ of^ roughly^ twenty nine^ million miles. The^ center^ of^ gravity^ for^ the^ ring^ would^ be^ the^ center^ of^ the sun. The kinetic speed^ of^ the^ constituent^ atoms^ and^ molecules^ at^ a temperature which would keep refractory^ elements^ gaseous^ would^ cause^ all the lighter elements, if not the entire^ ring,^ to^ dissipate^ into^ space. Chamberlin did not^ test^ the^ nebular^ hypothesis^ only^ by^ applying kinetic theory to its^ various^ stages^ of^ evolution.^ The^ dynamic^ incon- sistencies between the solar system as^ it^ now^ exists^ and^ the^ solar^ system as it must have been when the sun had^ initially^ contracted^ to^ its^ present size were^ offered^ as^ more^ evidence^ that^ the^ nebular^ hypothesis,^ at^ least as outlined by Laplace, could not^ be^ true. The basic cause^ of^ the^ separation^ of^ the^ planetary^ discs^ was the increase^ in^ rotational^ speed^ of^ the^ nebula^ as^ it^ contracted.^ The increase in rotational speed^ was^ necessary^ to^ keep^ the^ total^ angular momentum of the nebula constant, a^ dynamic^ requirement^ in^ any^ closed system. When the^ centrifugal^ force^ on^ particles^ in^ the^ equatorial^ plane of the nebula equaled the inward^ gravitational^ pull,^ these^ particles^ were essentially in^ free^ orbital^ motion.^ They^ were^ no^ longer^ subject^ to^ the gravitational contraction^ of^ the^ disc.^ Further^ cooling^ and^ further contraction of^ the^ nebula^ left^ a^ ring^ of^ these^ particles^ to^ circle^ as a separate entity. Chamberlin pointed^ out^ that^ for^ separation^ to^ occur^ at^ the orbital radius of^ Neptune,^ the^ linear^ speed^ at^ the^ equator^ of^ the
contracting cloud must have been three and four-tenth miles^ per second; at the radius of Jupiter, eight miles per^ second;^ at^ the radius of the earth, eighteen and^ a^ half miles^ per^ second;^ and^ at^ the radius of Mercury, twenty nine miles per secnnd. Further contraction of the nebula to the present size of^ the^ sun^ would^ have^ raised^ its equatorial speed to 270 miles per second (435^ kilometers^ per^ second). However, careful measurements of the equatorial speed today reveals that it is only about one and a third miles per second (2 kilometers per second). This meant the sun is rotating at one-half of one per cent as fast as the nebular hypothesis requires in^ the^ absence^ of^ any^ contravening influences. What these contravening influences might^ have^ been,^ if^ any existed, were not obvious to Chamberlin. Darwin had already shown that the tidal forces of the planets on the sun, which might act to slow its rotation, were negligible. Chamberlin^ surmised^ that^ in^ the^ absence^ of some independent agency which could effectively draw angular momentum from the nebular disc, and this^ agency^ operating^ only^ after^ the^ contracting cloud had spawned the.planets, the nebular hypothesis as Laplace had suggested it was fatally weakened. Further considerations cnnfirmed^ Chamberlin's^ doubts.^ It^ had been known that the^ plane^ of^ the^ sun's^ equator^ did^ not^ coincide^ with the orbital plane of the earth. Indeed, the orbital plane^ of^ the^ planets were not even similarly inclined. If one defined an "invariable plane" for all the planets based on^ mass,^ its^ inclination^ to^ the^ solar^ equa- torial plane was five degrees. Even though this^ deviation^ is^ small,^ it represents enormous amounts^ of^ inertia^ from^ the^ coplanar^ configuration
NNW
faster than^ the^ primary^ rotates^ (as^ Phobos^ does^ about Mars)? The overwhelming^ evidence^ against^ the^ Nebular^ Hypothesis^ led Moulton and Chamberlin^ to^ seek^ an^ entirely^ fresh^ approach^ to^ the^ problem of planetary production;^ an^ hypothesis^ which^ could^ account^ for^ the^ slow rotation of the^ sun^ and^ at^ the^ same^ time^ explain^ the^ high^ angular^ mo- mentum of the planets.^ In^ it^ they^ wanted^ to^ have^ the^ ecentricities^ of the planetary^ orbits^ and^ the^ obliqueness^ of^ the^ sun's^ and^ planets!^ rotation- al planes a natural^ result^ of^ the^ evolutionary^ process. The final hypothesis departed^ entirely^ from^ the^ Laplacian viewpoint. The^ essential^ feature^ of^ their^ scheme^ was^ the^ close^ approach of a^ wandering^ star^ to^ our^ own^ sun.^ An^ approach^ so^ close^ that^ giant tides were^ raised^ on^ the^ surface^ of^ the^ sun^ which^ greatly^ agitated^ the prominent disturbances which^ sporadically^ and^ naturally^ occur^ in^ its surface layers.^ During^ the^ time^ the^ star^ was^ closest^ to^ the^ sun,^ the combination of natural activity^ and^ the^ abnormal^ tides^ raised^ by^ the gravitational attraction^ of^ the^ wanderer^ caused^ jets^ of^ material^ to^ be ejected with four results : 1) Some jets were ejected from^ the^ sun^ only^ to^ fall^ back^ again into the^ sun.^ However,^ due^ to^ the^ tangential^ speed^ they^ acquired^ from the movement of the second^ star^ and^ sun^ about^ their^ common^ center^ of gravity, they^ transferred^ angular^ momentum^ to^ the^ sun^ along^ the^ plane^ of interaction, modifying the^ original^ rotation^ of^ the^ sun.
- Some jets of matter^ were^ shot^ out^ so^ far^ that^ the^ tangential speeds they^ acquired^ were^ enough^ to^ allow^ them^ to^ miss^ the^ sun^ entirely when they^ fell^ back.^ They^ went^ into^ highly^ eccentric^ orbits^ about^ the
sun. From these masses the^ planets^ evolved. 3) If a jet was ejected out^ to^ a^ critical^ distance,^ attraction of the sun and^ the^ passing^ star^ were^ balanced^ so^ that^ it^ escaped^ the influence of both.
- If the wandering star^ approached^ closely^ enough,^ and^ was enough larger than the^ sun^ to^ remain^ approximately^ dynamically^ inert,^ it could have produced eruptions of material on^ the^ sun^ which^ were^ drawn into the wandering star. Moulton and Chamberlin used^ the^ first^ possibility^ to^ explain both the slow rotation of the sun and its oblique plane of^ rotation^ to the invariant plane of the planetary orbits. The sun originally rotated opposite to the direction it is now seen to have and at an angle to the invariant plane of the planets. The jets of material drawn out only^ to fall back into the sun overcame the original rotation and succeeded to cause the sun to rotate slowly almost in the plane of^ the^ dynamic^ encounter; only the slight obliqueness of its rotational plane to^ the^ invariant plane of the planets shows that it once rotated with another vectorial value for its angular momentum. The second effect of the solar eruptions caused spiral jets to be produced with enough tangential speed to miss the sun upon falling back and thus remain^ in^ highly^ eccentric,^ elliptical^ orbits^ with^ the^ sun at one focus. An illustration of such spiral ejections is shown^ in Figure 1 below
