Wolfgang Westenberger

Dark matter : Who will save the materia obscura ?

Wer rettet die Dunkle Materie ?

Copyright Wolfgang Westenberger 2011Herstellung und Verlag:Books on Demand GmbH, NorderstedtISBN 978-3-8423-4883-7

CONTENTS

INHALTSVERZEICHNIS

Vorwort an Alle, die sich für Astronomie interessieren 5

Vorwort an aufgeschlossene Studenten und Einsteiger der Astrophysik 6

Preface to open-minded students and young professionals of astrophysics 8

Preface to professionals and standard cosmologists 9

Part 1 : Calculations of local gravitation within spiral galaxies 11

Part 2 : What about Zwicky's dark matter proof? 27

Teil I : Berechnungen der lokalen Gravitation in Spiralgalaxien 31

Teil II : Und was ist mit dem Dunkle-Materie-Beweis von Zwicky? 49

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Vorwort an Alle, die sich für Astronomie interessieren

Seit Jahrzehnten hat sich bei den Astronomen immer mehr die Überzeugung durchgesetzt, dass es neben der uns bekannten Form der Materie, aus der wir Menschen ebenso wie die Erde, die Sonne und die Sterne aufgebaut sind, noch eine zweite Form der Materie geben müsse, die sogenannte „Dunkle Materie“. Von dieser Dunklen Materie soll es mehr als fünfmal so viel wie von der normalen Materie geben. Die Dunkle Materie kann bisher nicht direkt beobachtet werden, nur durch ihre Gravitation soll sie sich bemerkbar machen. Jetzt zeigt es sich, dass sich die Wissenschaft vielleicht doch zu früh auf diese Hypothese festgelegt hat.Ihren Ausgangspunkt hatte die Dunkle-Materie-Hypothese in der Beobachtung von Fritz Zwicky, der 1933 feststelte, dass für die Erklärung der Bewegungen von Galaxien eine Menge Masse fehlte. Dieses Problem der „fehlenden Masse“ ergab sich auch aus den Beobachtungen von Vera Rubin, die in den 1960er Jahren die Bahngeschwindigkeit der Sterne in der uns benachbarten Andromeda-Galaxie maß.Obwohl die Dunkle Materie inzwischen zu einem unverzichtbaren Bestandteil der Standard-Kosmologie geworden ist, kommen auch bei renommierten Wissenschaftlern zeitweise Unbehagen und Zweifel auf. Beispielsweise formuliert der bekannte Astrophysiker und TV-Wissenschaftler Harald Lesch: „Die Dunkle Materie ist eine Katastrophe für die Wissenschaft.“Selbst mit den größten Superrechnern der Welt ist es nicht gelungen, die Beobachtungen der Sterngeschwindigkeit mit der Dunkle -Materie-Theorie in Übereinstimmung zu bringen.Um das Geheimnis dieser rätselhaften Dunklen Materie endgültig zu klären, wurde mit einem Aufwand von einigen Milliarden Euro an der Forschungseinrichtung CERN in Genf eine Beschleunigungsanlage gebaut. Ein internationales Team von Wissenschaftlern will versuchen, mit diesem LHC, einem der größten Experimente aller Zeiten, „Dunkle Materie“ selbst herzustellen. Dabei wird schon vorausgesetzt, dass es die Dunkle Materie tatsächlich geben muss.Kaum jemand, der sich für die Astronomie interessiert, hat wirklich daran geglaubt, dass mit den vorgesehenen Experimenten „eines der letzten Rätsel der Menschheit gelöst“ werden könnte, denn es gibt noch genügend andere ungelöste Fragen. Möglicherweise wird man gar keine Dunkle Materie herstellen können. Sind dann die Milliarden umsonst ausgegeben worden? Die Erforschung der Dunklen Materie war das wichtigste, aber nicht das einzige Anliegen der CERN-Forscher. Auf jeden Fall wird die Wissenschaft durch die Experimente bereichert werden, und wir können uns schon auf viele dadurch entstehende neue Rätsel freuen.Gleichgültig, was gefunden wird: Um die Bahngeschwindigkeit der Sterne in einer Spiralgalaxie zu erklären, brauchen wir keine Dunkle Materie. In der vorliegenden Abhandlung wird gezeigt, dass die vorhandene Materie (Sterne, Gas- und Staubwolken) und ihre gegenseitige Anziehung eine vollständige Erklärung für die beobachtete Geschwindigkeit ergeben. Dies geschieht auf der Grundlage von bekannten physikalischen Gesetzen, geometrischem Grundwissen und astronomischen Beobachtungsdaten.Die Kosmologen sagen, dass es die Dunkle Materie schon deshalb geben müsse, weil man sonst nicht plausibel erklären könne, wie sich ohne Dunkle Materie nach dem Urknall die Galaxien entwickelt hätten.Wenn jetzt der eine oder andere Leser das Gefühl hat, er wisse immer noch nicht so ganz genau,was die Dunkle Materie eigentlich ist, hier die Antwort darauf:1. So genau weiß das niemand.2. Es gibt sie nicht.

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Vorwort an aufgeschlossene Studenten und Einsteiger der Astrophysik

Die Tatsache, dass die Bahngeschwindigkeit der Sterne in einer galaktischen Scheibe nicht kontinuierlich zum Rand hin abfälltunddie scheinbar zu hohe Geschwindigkeit von Galaxien in Galaxienhaufenwerden üblicherweise als überzeugende Beweise für die Existenz der nicht-baryonischen Dunklen Materie angesehen.Die folgende Arbeit zeigt in Teil I die eigentliche Ursache für die beobachtete Geschwindigkeit der Sterne. Auf der Grundlage bekannter Tatsachen wird eine neue Methode zur Berechnung der lokalen Gravitation an jeder beliebigen Position der galaktischen Scheibe eingeführt. In Teil II werden neue Gesichtspunkte zur Beurteilung der Geschwindigkeit von Galaxien vorgestellt. Die oben genannten „Beweise“ für die Dunkle Materie halten einer genauen Betrachtung nicht stand.Die bisherige Standard-Methode rechnet bei der Interpretation der Rotationskurven mit einer Unbekannten, der Dunklen Materie. Im Gegensatz zu dieser eher spekulativen Methode handelt es sich bei der hier vorgestellten Berechnungsart um eine rechnerische Methode, ohne dass unbekannte Größen verwendet werden müssen. Auch muss die Newton'sche Gravitation nicht modifiziert werden. Mit diesem neuen Zugang zu einem alten Problem erhalten wir durch Berücksichtigung der Gravitationsvektoren einzelner Gravitationsbereiche der bekannten Materie eine Gravitationskurve (lokale Gravitation gegen Entfernung vom Zentrum), die der bekannten Rotationskurve (Bahngeschwindigkeit gegen Entfernung vom Zentrum) entspricht. Damit haben wir zum ersten Mal eine klare, kausale und vollständige Erklärung für die Bahngeschwindigkeit der Sterne.Es ergibt sich kein Hinweis auf eine „fehlende Masse“ oder die Notwendigkeit einer Dunkle-Materie-Hypothese. Ein Hauptargument für die Existenz der nicht-baryonischen kalten Dunklen Materie löst sich auf. Warum wird dies in der Astrophysik bisher nicht so gesehen?Kritische Bemerkungen, die ein gewisses Unbehagen an der bisherigen Theorie erkennen lassen, gab es schon vereinzelt, ohne dass daraus Konsequenzen gezogen wurden. So formuliert H. Lesch aus München: „Die Dunkle Materie ist eine Katastrophe für die Wissenschaft.“Und im Jahresbericht 2007 des Max-Planck-Instituts für Astronomie in Heidelberg heißt es:„Alle theoretischen Kurven fallen zu größeren Radien hin ab, während die gemessenen Kurven dies nicht tun. Außerdem geben alle Modelle für die Zentralbereiche stets zu hohe Geschwindigkeiten an. Theorie und Beobachtungen passen also nicht zusammen. Somit besteht nach wie vor eine erhebliche Diskrepanz zwischen den kosmologischen Simulationen von Halos Dunkler Materie und der Wirklichkeit.“ (Walter, Bigiel & Leroy 2008)Diese Formulierung aus dem Jahr 2007 beschreibt den derzeitigen Stand der Wissenschaft.Die astronomischen Beobachtungen passen nicht mit den Erwartungen, die sich aus den Gravitationsgesetzen ergeben, zusammen. Aber nicht die von Kepler und Newton entdeckten Planeten- und Gravitationsgesetze sind falsch, sondern ihre Anwendung ist nicht immer korrekt.Das Problem der Standard-Wissenschaftler ist, dass sie das mehr als 300 Jahre alte Schalen-Theorem Newtons, das die Gravitationswirkung in kugelförmig verteilter Materie beschreiben sollte, auf die Materieverteilung in einer Galaxie anwenden. Eine Galaxie ist jedoch komplizierter aufgebaut als eine Kugel. Aus diesem geometrisch-dynamischen Missverständnis ergibt sich ein Berechnungsfehler, der genau der Größenordnung der postulierten Dunklen Materie entspricht.

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Eine weitere Fehlermöglichkeit ist in der Masse-Licht-Relation enthalten. Weil die meiste Helligkeit von der Mitte einer Galaxie ausgeht, wird argumentiert, dass in der Scheibe zuwenig Materie sei, um die Rotationskurve zu erklären. Aber : Die Masse-Licht-Relation wird teilweise überbewertet und sollte relativiert werden. Wenn man zu einer leuchtenden Masse eine zusätzliche Masse von beispielsweise 30 % hinzufügt, wird die Gesamtmasse auf 130 % erhöht; wenn diese zusätzliche Masse aber aus lichtschluckender Materie besteht (man denke an die Staub- und Gaswolken der galaktischen Scheibe), wird die resultierende Gesamtleuchtkraft vermindert. Deshalb kann man keine große Präzision der Wo-viel-Licht-da-viel-Masse-Regel erwarten.Zweifel an der Dunkle-Materie-Hypothese sind in der wissenschaftlichen Gemeinde unerwünscht, vor allem, wenn sie von unabhängigen Wissenschaftlern geäußert werden. Und die Redakteure der großen Fachzeitschriften haben mehr Angst vor dem Kopfschütteln der Gutachter als davor, eine möglicherweise wichtige Arbeit zurückzuweisen.Auf diese Weise wird die wissenschaftliche Diskussion stark auf die offizielle Standardmeinung eingeengt. Ein Paradigmenwechsel, der von Zeit zu Zeit notwendig wird, um Fehlentwicklungen zu korrigieren, kann so nur mit unnötiger Verspätung durch eine neue Generation von Wissenschaftlern durchgesetzt werden. Effektiver für das Vorankommen der Wissenschaft wäre es, wenn einerseits die standardmäßig vorgegebenen „Beweise“ für eine Theorie (z.B. „fehlende Masse“ für die Dunkle Materie) kritisch hinterfragt würden: Stimmt das überhaupt? Oder ist es nur eine Verlegenheitserklärung, weil niemand eine bessere Idee hatte? Andererseits sollten alternative Vorschläge nicht schon deshalb abgelehnt werden, weil es Zeit kosten würde oder unbequem wäre, sich damit zu beschäftigen. Widerlegbare Konzepte sollten widerlegt werden, nicht widerlegbare auf den Grad ihrer Plausibilität geprüft und beurteilt werden.In diesem Sinne werden alle kreativen Köpfe hiermit aufgerufen, die zentrale These der vorliegenden Arbeit plausibel zu widerlegen, dass keine nicht-baryonische Dunkle Materie benötigt wird, um die Bahngeschwindigkeit der Sterne zu erklären; wem dies gelingt, der darf sich mit dem Titel „Retter der Dunklen Kosmologie“ schmücken.

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Preface to open-minded students and young professionals of astrophysics

The fact that the velocity of stars within the disc of a galaxy does not decrease continuously towards the edgeand the velocity of galaxies in clusters seeming too highare usually taken as convincing proofs for the existance of non-baryonic dark matter.The enclosed paper shows in part I the real reasons for the observed velocity of stars.Based on well-known facts a new method for calculating the local gravitation of any position at any distance from the centre of the galaxy is introduced, called the method of gravity areas.In part II new points on behalf of the velocity of galaxies are introduced.Those “proofs“ for dark matter mentioned above don't stand up to close examination.Up to now the standard method deals with an unknown, the Dark Matter. In contrary to this rather speculative method the new method of gravity areas is calculative, without any unknown.Newtonian gravitation has not to be modified.By this new approach to an old issue considering gravitational vectors of gravity areas we get a gravitation curve (local gravitation versus distance from centre) corresponding to the well-known rotation curve (velocity versus distance from centre). Here we have for the first time a clear, causal and complete explanation for the velocity of stars.There is no evidence for any “missing mass“. Neither any need for dark matter theory. A main argument for non-baryonic cold dark matter is vanishing.Why astrophysicians do not see it this way up to now?Some uneasy feeling on behalf of the standard theory might be supposed because of occasional critical remarks without taking the appropriate step. For example H. Lesch (Munich) said: “Dark Matter is a desaster for astrophysics“.And in the Annual Report 2007 of the Max-Planck-Institute for Astronomy Heidelberg you can read: “All theoretical curves are decreasing towards greater radii, whereas the measured curves are not. In addition, all theories result in too high velocity in the central areas. The theory doesn't fit the observations. There still exists a considerable discrepancy between the cosmological simulations of dark matter haloes and reality.“ (Walter, Bigiel & Leroy 2008)This statement from the year 2007 describes the cutting-edge of this scientific issue.The astronomical observations are not corresponding to the amounts expected from gravitational laws. But the laws of gravitation and planetary motion of Newton and Kepler are not wrong, they should be used correctly.Standard cosmologists use Newton's Shell Theorem, originally describing gravitation within a globe, with regard to spiral galaxies. But spiral galaxies are built somewhat more complicated than globes. From this geometric dynamic misconception derives a fault of exactly the amount of the supposed dark matter.Another obstacle for successful calculations may be the inaccuracy of the mass-to-light ratio. Inferring the number of solar units from the observed brightness comprises some relevant inaccuracy. (If you took for example a quantity of 100 % of shining matter and you added another quantity of 30 % of interstellar matter extinguishing light, the total mass will result in 130 %; but the total brightness will decrease.)Any doubts about Dark Matter Theory are not welcome in scientific community, especially when expressed by independent scientists. And the editors of big specialist journals do first fear the referees shaking their heads and only second they are afraid of refusing an article possibly important. Thus the scientific discussion is reduced on official standard opinion. Therefore a change of paradigm, sometimes necessary to correct things going off course, may only be carried through by a new generation of scientists, with some delay. More effective on behalf of advance in science would be to examine critically any standard “proof“ of standard theories. For example on behalf of the “missing mass“ one could ask: Is this correct at all? Or is it a stopgap because no one has got any better idea?

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On the other hand alternative proposals should not be refused for the reason it would be difficult or time-consuming to have a critical look at them. Disprovable proposals should be disproved. Not disprovable proposals should be judged on behalf of their degree of plausibility.Thus all capable brains are called upon to disprove the enclosed paper in a plausible way, especially the main thesis: There is no need for any dark matter to explain the velocity of stars. Who will succeed in this, may be called “The Hero of Dark Cosmology“.

Preface to professionals and standard cosmologists

Based on well-known facts a new method for calculating the local gravitation of any position at any distance from the centre of the galaxy is introduced considering the effects of several gravity areas on any object dependent on their mass (stars and interstellar matter) and dependent on their distance (more exactly: on the square of the distance of their centres of gravity).Newton's law of universal gravitation serves as a basis for this paper.Another basis are astronomical observations of the number of solar units in the surroundings of the sun and the distribution of matter in the whole Galaxy.The result are accurate amounts of local gravitation.The resulting gravitation curve (gravitation versus distance from centre) is corresponding to the well-known rotation curve (velocity versus distance from centre).There is no evidence for any missing mass. The basis of dark matter theory calls for careful consideration.With regard to the far-reaching significance it is to emphasize that Newtonian gravitation is not modified. Neither any additional baryonic matter is required nor any non-baryonic dark matter.The formula v² = GM/r is not called into question; hitherto M is enlarged to explain the observed velocity, but now we adjust r to the real distance.All over the world enormous computer capacities are brought into action to solve the problem of the missing mass by improved distribution of dark matter to get a correct correlation between gravitation (caused by normal baryonic matter and non-baryonic Cold Dark Matter) and the velocity of stars. Now you may use it to perfect the method of gravity areas. For example you will find the clear and causal explanation for the fluctuations of the rotation curve of the Galaxy by considering the mass distribution of the spiral arms.You shouldn't find too hard to accept the non-existance of the non-baryonic missing mass.

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Part I

Calculations of local gravitation within spiral galaxies

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ABSTRACT

The observed velocity of stars orbiting the galactical centre was up to now not completely corresponding to the assumed gravitational force in spiral galaxies. Applying the new method of gravity areas by using Newton's law of universal gravitation we get the following results:1) In bright galaxies like the Milky Way the local gravitation in the flat disc is nearly constant

irrespective of the distance from the centre of the galaxy.2) There is a high gradient of local gravitation and a relative maximum in the internal section of the

bulge.3) In less bright galaxies the local gravitation is increasing towards the edge without a flat phase.These results are fitting the observations. This method of gravity areas is based on geometric facts, astronomical observations and on well-known physical laws without speculating on unknown matters or unknown forces.

KEYWORDScosmology:dark matter; galaxies:spiral; gravitation

1 INTRODUCTION

The observation that the velocity of stars orbiting the centre of a galaxy is not decreasing towards the edge of the disc corresponding to Kepler's second law, is usually considered as a convincing argument or proof for the existance of the non-baryonic dark matter, (e.g. Einasto 2009, Weijmans et al. 2008).In the year 1609 Johannes Kepler published the second law of planetary motion. He showed that a planet orbiting the sun at a short radius must have higher velocity than a planet at a greater radius.The reason is that the velocity of an orbiting object causes the counterpoise to gravity by fitting the tangential impulse of the orbiting object to the local effective gravity. If the velocity was too low the gravitational force would pull the object towards the gravitational centre, if it was too high the object would drift off. There is only one possible velocity to stay in a stable orbit. Velocity is balanced with gravity.Gravity force decreases in proportion to the square of the distance, that's the square rule of gravity. For calculating the gravity of a global object like the sun one may use Newton's rule considering the total mass in the very centre of gravity. Using Newton's rule to a galaxy – all the mass into the centre – one should expect decreasing velocity of stars at greater distance from the centre according to Kepler's law. In the 1970s Vera Rubin found that in Andromeda Galaxy the stars at great distance had nearly the same velocity. Up to now the observation of not decreasing velocity was confirmed again and again in other galaxies including our Milky Way. Within less bright galaxies velocity paradoxically even increases towards the edge of the galaxy.Why?The usual explanation goes like that: There is some “missing mass“ (strictly speaking “missing gravity“) to explain the velocity; therefore some kind of invisible mass must exist – the dark matter.But the results of this assumption are disappointing, because all theoretical computer simulations don't fit the observations, (Walter, Bigiel & Leroy 2008).This raises the question if there may be another approach to this issue. First of all let us consider if it is correct to use Newton's rule with regard to a galaxy instead of a globe. In actual fact, the idea of replacing a spiral galaxy by a globe is somewhat misleading because spiral galaxies are built somewhat more complicated than globes.In the following sections we will persue the question of whether you can get better results by calculating the effects of gravity vectors to get the local gravity at any single position in the galactical disc.

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2 GRAVITATION AND DISTANCE

2.1 The aspect of the near mass.We know for sure that the earth is orbiting the sun. There is an obvious correlation with the gravity force of the sun.In 1694 the theologian Richard Bentley embarrassed Isaac Newton by the question: If there is an infinite number of stars in any direction of the universe, how can it be possible that the earth is orbiting the sun and is not bound by gravity on its very place?Newton's answer: Because of a permanent miracle of God.The correct physical answer would have been: Because the near mass of the sun has a stronger force of gravity than the far away masses of all the other stars.Let's make now an imaginary simulation. We assume we could take the tenth part of the sun's mass and place it upon the connection line between the sun and the earth, and regard the gravitational effect. The gravity of 1/10 mass within the sun is, of course, 1/10 of the sun's total gravity. 1/10 mass at half the distance causes 4/10 of total full distance gravity – because of the square rule of gravity. And this 1/10 mass at a quarter of the distance causes gravity of 16/10 of the total sun mass in full distance; together with the remaining 9/10 mass in full distance, causing 9/10 of gravity force, we get a total gravity force of 25/10 effecting on the earth. You would get the same result by adding one and a half sun mass (as “missing mass“) into the very sun.That is: to get a higher gravitational force on a certain orbit you may add some more mass to the centre, or you may split up a certain mass into portions and distribute it as described. The additional near mass would pull the earth towards the sun. Unless the earth would get a higher velocity. Because velocity is balanced with gravity.In other words: If there is an additional near mass towards the centre, a certain orbit can only be stable in case the velocity of an object in this orbit is higher than without the additional mass. One may call this the rule of velocity of the internal near mass.You may suggest now and you will see that the effect of the near mass is the crucial step to explain the velocity of stars in the galactical disc.

2.2 Principles of the following reflections.We know that not the total matter of a spiral galaxy is concentrated in the very centre of the galaxy. A part of the matter is forming the galactical disc. We want to know if a certain star in the disc is considerably influenced by the gravity force of the stars in direction to the centre. We want to calculate the amount of gravity effected by the stars between sun and galactical centre in relation to the total effective gravity on the sun. If we can find a considerable influence, the missing gravity at least partially will be explained. Or perhaps the total enigma of “missing mass“ will vanish?For the following calculations, in contrary to the imaginary simulation above, we are not using arbitrarily positioned matter but the results of astronomical observations. We are starting out from the mass distribution of the galactical surroundings of the sun because here counting galactical objects such as stars and interstellar clouds of gas and dust is most exact. To simplify the calculations we suppose equal distribution of matter within the galactical disc from the transition to the bulge until the edge of observable shining matter. We accept that mass distribution in reality is diminishing towards the edge. The fault deriving therefore is not as big as the fault deriving from concluding that mass would be proportional to observable light. Because a part of the matter, the interstellar matter, is rather absorbing light and therefore the brightness of shining matter is reduced in a variable way.

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2.3 Introduction of the astronomical gravitation unit (agu).Based on astronomical observations we assume within a distance of 20 light years (ly) from the sun towards the galactical centre 20 solar units ( sun masses, consisting of stars and interstellar matter). We suppose this as the average density near the plane of symmetry.To get a measuring unit for gravitational effects we define an astronomical gravitation unit (agu) in the following way:1 solar unit ( the mass of our sun, that's 1.989 x 1030 kg )at the distance of 1 light year ( the distance the light covers within 1 year, that's 9.461 x 1015 m ) causes a local gravity effect of 1 agu .The gravitational effect is increasing in proportion to the mass and decreasing in proportion to the square of the distance, this means: mass is divided by the square of the radius, or m/r² . Therefore one solar unit at 100 ly causes 1/ 10,000 agu. 1 sun mass at the distance 8 light-minutes (the distance sun to earth) causes more than 4.4 billion agu, whereas the central force, 10 billion solar units (1010 sun masses, the content of the galactical bulge) at 26,000 ly (the distance centre to sun)cause only 14.79 agu. (That's why the earth is bound to the sun.)Now we ask for the gravitational force of the near stars on the sun.All matter which pulls the sun primarily towards the centre is included within a spherical angle of 90° from the sun to the centre. We call it the effective angle enclosing a gravity area.According to Newton's rule for simplifying the calculation we define the effective point of a gravitational area ( strictly speaking of the conic spherical sector ) at half of the radius of the area in the connecting line to the galactical centre. Here the total mass of the gravity area is assumed to be concentrated. Therefore we calculate 20 solar units at 10 ly ( the half of the radius 20 ly mentioned above ) and get

20 solar units / ( 10 ly )² = 0.20 agu .Comparing the central force ( 14.79 agu ) with the gravitational force of the internal near mass within 20 ly ( 0.20 agu ) we state that this part of the near mass has an effect of more than 1 per cent of the central force. This may be regarded as a noteworthy intermediate result.

3 AREAS OF GRAVITY

The Milky Way (and any typical spiral galaxy) can be roughly divided into the zone of the flat disc and the central vaulted zone, the bulge. The total visible disc is 100,000 light years in diameter. It is supposed that the total mass of the Galaxy is at 100 to 300 billions of solar units.In the central vaulted zone there are supposed roughly 10 billions of solar units.The altitude of the bulge is supposed at 13,000 to 15,000 ly. In the disc the altitude is about 3,000 to 5,000 ly ; the matter is more concentrated in the fifth near the plane of symmetry.The transition from bulge to disc is supposed at a radius of 15,000 to 16,000 ly .The sun is orbiting the centre at a distance of 26,000 ly, that means approximately at half the radius of the total galactical disc, and near the plane of symmetry. At a velocity of 220 km s-1 a total orbiting will take 240 million years.

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Now we divide several areas of different gravity effects on the sun ( figure 1 ) :

Fig. 1: Simplified illustration of gravity areas.A distant mass, B central mass, C internal near mass, D external near mass

(Point of intersection = position of a regarded star)

3.1 Distant massThe distant mass containing primarily the stars of the opposite part of the disc is situated within the effective angle of 90° beyond the area of the bulge and beyond that of the internal near mass. The effective point of the distant mass is in the opposite half of the disc. When moving the point of intersection and the effective angle towards the edge of the disc, the area of the distant mass will be enlarged on both sides; the gravity force of the initial distant mass will decrease by the square of the distance of its effective point; this effect is compensated by increasing quantities of solar units which fill on both sides the removed effective angle, also in proportion to the square of the distance. So we get because of geometric reasons a (roughly) constant amount for any distance of a regarded star. This amount is calculated at 55.56 agu.

3.2 Central massThe central mass of the total bulge, consisting of 10 billion solar units, causes the central force, decreasing by square of distance.

3.3 Internal near massThe internal near mass fills the sector between the sun (or any regarded star of the disc) and the transition to the bulge. In section 2.3 we calculated the gravity force for a distance of 20 ly from the sun and got 0.20 agu. When doubling the radius we get, within the spherical sector, increasing mass by cube and additional gravity force of 0.15 agu. At the radius of 1,280 ly the summed up gravity forces come to 10.00 agu .Reaching the top and the bottom of the galactical disc we continue to calculate further effects of additional solar units only by the square of the distance. Each further doubling adds now 4.27 agu, and at the distance of 10,240 ly from the sun, that's approximately the transition to the bulge, we get 22.80 agu for the summed up gravitational force of the internal near mass.We get as intermediate result: There is an additional effect of gravity areas in the same direction.

3.4 External near massNow we regard the external near mass in the conic spherical sector of 90°, starting from the sun, in the opposite direction of the centre.Regarding the gravity effects of two masses on a third mass upon the connecting line it is to

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emphasize that the opposite gravitational vectors are diminishing the effect of each other. So for any point of the connecting line we have to subtract the local gravitational vectors. For example in the middle of the connecting line between two objects of the same mass the local effective gravity is zero, because the gravity forces of both objects are balanced out by subtraction of the vectors in opposite direction. (On the flight to the moon the zero gravitation point is near to the moon because of the predominant gravity of the earth. At the surface of the earth the gravity force of the moon diminishes the gravity force of the earth, so we get the tides.)Assuming the gravitational effect of a central/internal mass is diminished by the gravitational effect of an external mass we conclude that the velocity of an orbiting object within a stable orbit is corresponding to the diminished gravitational effect resulting from subtraction of opposite vectors. (It is self-evident, that gravitational force in rectangular direction is not the immediate reason for higher or lower tangential velocity of an orbiting object, but it is the counterpoise of the balanced velocity of an object in a certain orbit.)We now can formulate the rule of velocity of the external near mass:If there is an additional near mass opposite to a gravity centre, a certain orbit will only be stable in case the velocity of an object in this orbit is lower than without the additional mass.A critical reader might say now:“I don't understand all the excitement : If the internal mass works accelerating, and the external mass in the same way slowing down, there remains an effect of zero and Kepler's law should be in force. What's the whole point?“Kepler's law will stay in force. The point is the additional effect mentioned above and the composite effect of the different areas of gravity you will see in the next section.

4 ADDITION OF GRAVITATIONAL VECTORS

4.1 Composite effect of the gravitational vectors in the discNow we calculate the composed effects of several gravity areas with regard to the sun and other stars in the galactical disc. The effects of the gravity areas may be regarded as physical vectors directed towards the centre or opposite. All gravitational forces effecting towards the centre are summed up and the negative gravitational effect of the external near mass is subtracted. We sum up the effects of gravity with regard to the sun towards the centre consisting of the distant mass (55.56 agu), the central force (14.79 agu) and the internal near mass (23.43 agu, distance to the bulge assumed at 11,000 ly) and get an amount of 93.78 agu. The external near mass between the sun and the edge of the disc (distance 24,000 ly) effects 28.42 agu.All in all we get on behalf of the sun (distance from the centre 26,000 ly) a total effect of gravitational vectors of 93.78 – 28.42 = 65.63 agu. See table 1 .The calculation of a star at 21,000 ly comes to 68.16 agu. At double distance of 42,000 ly we get 69.01 agu, that's roughly the same amount. Therefore the velocity of a star in double distance must also be roughly the same. This fits the observations.(In contrast, the method of putting all mass into the centre would result in only 70 % of velocity at double distance, according to Kepler's law. Evidently this method is not accurate.)

Table 1 : Local gravity by composite effects of gravity areasDistance from centre 20 25 30 35 40 45 (x 1,000 ly)

distant mass 55.56 55.56 55.56 55.56 55.56 55.56 (agu)central mass + 25.00 16.00 11.11 8.16 6.25 4.94 (agu)internal near mass + 18.38 22.82 25.41 27.25 28.68 29.85 (agu)sum 98.94 94.38 92.08 90.97 90.49 90.35 (agu)external near mass - 29.85 28.68 27.25 25.41 22.82 18.38 (agu)local gravity 69.09 65.70 64.83 65.56 67.67 71.97 (agu)

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Result N° 1 : When applying the method of gravity areas to the galactical disc you see that the local effective gravity is nearly constant. Because gravity is balanced with velocity, stars in stable orbits also must have nearly constant amounts of velocity. This is corresponding to the observed rotation curves.The causal explanation derives from the geometrical distribution of gravitational forces:By increasing radius the centre force will diminish in proportion to the square of the distance;at the same time the effect of the enlarged internal near mass will increaseand the antagonistic effect of the external near mass will decrease. See figure 2.These effects are roughly compensating. (The weakly increasing amounts towards the edge are due to the diminishing influence of the external near mass.)At figure 2 you see the curve of local gravity versus distance from centre of galaxy as result of several gravitational vectors (gravity area forces). The resulting curve is corresponding to the well-known rotation curves (velocity versus distance).Figure 2 : Interactions of the gravitational vectors.

A distant mass, B central mass, C internal near mass, D external near mass,E composite effect (that's A plus B plus C minus D)

For calculating the local gravitational force at a certain position (radius r1 from the centre), you first take the amount of the central force for this position from table 5 . Then you add the constant amount of the distant mass of 55.56 agu for any position. And you add the force of the internal near mass for a radius r2 (distance from the centre r1 minus 15,000 ly of the bulge) from table 4 . Finally you subtract the force of the external near mass for a radius r3 (that's 50,000 ly radius of the disc minus r1) also from table 4 . Gravitational objects of identical mass and distance have identical gravitational effects on a certain position. Two identical objects extinguish the gravitational force of each other in the very middle of the connecting line. And the same quantity of objects at the same distance in opposite areas also extinguish the gravity of each other. Of course this effect works also with regard to the internal and external near masses. But the summed up partial areas of the internal and the external near masses show a surplus to one or the other side, except one single distance. This surplus is interacting with the central force and the constant distant mass.

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4.2 A walk through the bulgeFor calculating local gravitation within the flat disc we supposed equal density allover. But within the bulge we haven't got exact amounts of density and the mass distribution is more complicated. Therefore exact calculations of local gravitation are not possible. Using available figures we are able to make qualitative and semi-quantitative reflections.Now we imagine a walk through the bulge using an instrument that showed us the local gravity effect resulting from the gravity force towards the centre diminished by the gravity force opposite to the centre.

Result N° 2: Using the method of gravity areas a high gradient of gravity and velocity and a relative maximum within the interior region of the bulge is expected. Also this result is corresponding to the measured rotation curves.

4.3 Low brightness galaxiesObservations of galaxies of low brightness show an increasing rotation curve without a flat phase. Low brightness galaxies will contain less solar units in the central region.We imagine a low brightness galaxy by taking away 90 % of the central mass of the galaxy. Now we calculate with only 1 billion solar units of central force. See table 2 .

Table 2 : Local gravity within low surface brightness galaxies distance from centre 20 25 30 35 40 45 (x 1,000 ly)effective gravity 46.59 51.30 54.83 58.22 62.05 67.52 (agu)

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Result N° 3: When applying the method of gravity areas to galaxies with low surface brightness we get increasing gravity force (and therefore increasing velocity) towards the edge corresponding to the effects of the near masses. This result also fits the observations.

These three results we have got without using any dark matter.One or other of the readers, getting gradually suspicious, may ask now: “Could it be possible that there is something rotten about the Theory of The Dark Matter?“The answer: “You don't need even a microgramm of dark matter to explain the velocity of stars.“

5 METHODS

5.1 Newton's Shell Theorem and its restrictionTo calculate the gravity effect of all stars within the bulge with regard to the sun, we use Newton's rule: You may consider the total mass of a globe in its very centre of gravity. This way one gets a certain distance to calculate with. We accept that the bulge is not a globe of even distribution of matter, but an ellipsoid figure of increasing density towards the centre. The inaccuracy in using Newton's rule is neglectable because of the great distance of the gravity effecting objects from the sun.There would be a danger to be misled by using Newton's rule uncritically with regard to the stars of the disc between the bulge and the sun. Because of the square rule of gravity and Newton's gravitation law we know for sure that the gravity effect of an object is inversely proportional to the square of the distance.For simplifying calculation one could superficially consider the masses of two exactly opposite objects of identical mass and distance being combined in the very centre of the connecting line. But it depends on the specific issue if one gets a considerable factor of inaccuracy or not.We take for example a star of 1 solar unit at the distance of 2,000 ly from the sun and 24,000 ly from the centre, and a corresponding star of 1 solar unit exactly in the opposite direction at a distance of also 24,000 ly from the centre. Because of the square rule of gravity the gravitational force of the star at the distance 2,000 ly (square 4 millions) from the sun is 1 divided by 4 x 106 agu.The opposite star at the distance of 50,000 ly from the sun causes gravitational force of 1 divided by 2,500 x 106 agu. 1 solar unit in the very centre of the Galaxy at 26,000 ly causes 1 divided by 676 x 106 agu; therefore 2 solar units cause gravity of 1 divided by 338 x 106 agu. Considering 2 distant, corresponding stars in the very centre, we get in this case the following result :The gravity force of 2 solar units at the distance of 26,000 ly is smaller by the factor 84.5 than the gravity force of 1 solar unit at 2,000 ly. This inaccuracy factor of 84.5 is not acceptable. (The gravitational effect of the opposite star at 50,000 ly from the sun causes less gravity by the factor 625 compared with the near star and therefore may be neglected.) In contrary we regard two opposite stars in the bulge at a distance of each 4,000 ly from the centre, one of them at the distance of 22,000 ly from the sun, the other at 30,000 ly. The first star causes a gravitational force of 1 divided by 484 x 106 agu on behalf of the sun, and the second 1 divided by 900 x 106 agu. We compare again with 2 stars in the very centre ( 1 divided by 338 x 106 agu) and get an inaccuracy factor of much less than 1.5 . This means a permissible tolerance on behalf of the calculation of the bulge mentioned above. (An observer in Andromeda Galaxy who wants to calculate the gravitational force of the total Milky Way sums up all stars in the middle; but if he wants to know the amount of gravitational forces which are effecting upon a star at the distance of 26,000 ly from the centre, he will try to take into account the real distribution of matter in the disc, otherwise he would be surprised by the “too high“ velocity of the sun.)These simple and definite calculations are to some extent contrary to Newton's Shell Theorem: All objects within a sphere of a radius which touches a regarded object will cause the same force at this object as if all of the mass enclosed by this radius was

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For example the distances from the sun (millions of km) and velocities (km s-1) of three planets:Mercury 57.91 / 47.87Earth 149.60 / 29.79Jupiter 778.40 / 13.07Doubling the distance of an orbiting object of known velocity, you get the velocity of the distant object by the factor 1/ root of 2, so you have to divide by 1.414, you get the result of 0.7 . Therefore the velocity of an object in double distance is at 70 per cent of the velocity of the near object.

6 DISCUSSION AND FUTURE POSSIBLE APPLICATIONS

Hitherto it is supposed that you will get the correct amount of gravity force by considering the whole baryonic mass within the internal shell of a regarded point of the disc being concentrated in the very centre of the galaxy, according to uncritically misapplication of Newton's Shell Theorem; the velocity of the stars should be explained by non-baryonic dark matter within a halo surrounding the galaxy.But the results of this assumption are disappointing.Therefore we try another approach to this issue, based on well-known facts and physical laws and on pure, unmodified Newtonian gravitation.By the method of gravity areas and interaction of gravitational vectors we get a causal explanation on behalf of the velocity of stars. The results are fitting the observations. With regard to cosmology there may be a far-reaching significance, because there is no evidence for any missing mass or need for any dark matter theory.To get clear calculations, the density of matter in the flat disc was assumed to be constant. In more elaborated tables it would be possible to include the tendency of increasing density of the galactical disc towards the centre and the decreasing height towards the edge of the disc; and to consider more precisely the decreasing density of matter dependent on the distance from the plane of symmetry; as well as the gravitational effect of gas clouds and dust clouds beyond the edge, enlarging the external near mass. In this paper the transition of increasing mass from cube to square is supposed as one step to simplify the calculations accepting that it is in reality rather continuously. The effective angle may be splitted up into smaller sectors to get more precision.If the border of the bulge is supposed to be nearer to the centre, the table of the central force has not to be changed; the radius and the effect of the internal near mass would be increased. Nevertheless the conclusions of the method would not be disproved.The average density of matter in the disc might be higher than in the surroundings of the sun; there may be areas of higher density, for example in the regions of the spiral arms. Also at this point the final results of the method will not be challenged.In addition, the method may be used to conclude from the measured velocity of stars on the nearby solar units, for example at the edge of the disc or within the bulge.

Acknowledgement:For mathematical advice and support I express my thanks to Antje Westenberger,University of Heidelberg, Department of Mathematics.

References :Einasto J., 2009 arXiv:0901.0632v1Weijmans A.-M. et al. 2008 MNRAS p.1343Walter F. , Bigiel F. , Leroy A., 2008 in Jahresbericht 2007, Max-Planck-Institute for Astronomy Heidelberg, p. 57