Comets' Astrophysical & Chemical Analysis

7/28/2019 Comets' Astrophysical & Chemical Analysis

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Copyright 2005-2013 - Ibrahim Ziazadeh - http://www.linkedin.com/pub/ibrahim-ziazadeh/66/569/465

i

Burgrdens Lrcentrum

Ibrahim ZiazadehProject work course n1201:

A scientific review project to be handed in to Ulf Jonasson

Gothenburg

Done: week 5 - week 24Handed in: June 15, 2005

Dedicated to all those who adorethe astonishing world of the

outer space.


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Table of Contents

Page

1. Summary........................................................................................... 1

2. Introduction ................................................................................... 1-5

2.1. Background & Objective ................................................... 1-22.2. Question State ................................................................... 2-32.3. Method .................................................................................. 3

2.4. Source Review ................................................................... 3-42.5. Typographical Conventions ................................................ 42.6. Structure ............................................................................ 4-5

3. Comets in General....................................................................... 5-29

3.1. Defining a comet ................................................................ 5-6

3.2. The Morphology of Comets:3.2.1. The Visible Parts of a Comet ................................ 6-7

3.2.2. The Invisible Parts of a Comet .............................. 7-83.3 The Chemical Abundance of Comets ............................... 8-9

3.4. The Mathematical & Physical Aspect of Comets:3.4.1. The Physics of the Ion Tail .................................. 9-103.4.2. The Physics of the Dust Tail ............................. 10-123.4.3. The Physics of the Nucleus .............................. 12-133.4.4. The Conicals ...................................................... 13-173.4.5. The Orbits of Comets ........................................ 17-24

3.5. The Non-gravitational Forces ....................................... 24-25

3.6. The Origin & Formation of Comets:3.6.1. Preface .................................................................... 263.6.2. The Formation Mechanism ............................... 26-27

3.7. The Luminosity of Comets ............................................ 28-29

3.8. Nomenclature of Comets ................................................... 29


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4. The Comet Hale-Bopp............................................................... 30-45

4.1. The Discovery ..................................................................... 30

4.2. The Magnitude ............................................................... 30-31

4.3. The Molecular Composition .......................................... 31-374.4. The Nucleus & the Coma:

4.4.1. Diameter of the Nucleus .................................... 37-384.4.2. Gas Emission from the Nucleus ....................... 38-394.4.3. Rotation of the Nucleus .................................... 39-40

4.5. The Dust Tail .................................................................. 40-41

4.6. The Ion Tail ......................................................................... 41

4.7. The Sodium Tail ............................................................. 41-43

4.8. The Orbit & Orbital Elements ........................................ 44-45

5. Conclusion................................................................................. 46-47

6. Glossary..................................................................................... 48-50

7. Bibliography .............................................................................. 51-57


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1. Summary

Most of the people, classified as dummies of comets, may well have a hard timefinding and gathering the information they need on comets in general. They might

find the information they need, however they will still not be able to select themost important facts concerning a comets common components for instance.Others not belonging to the dummies category are more interested in the visualapplications of a comet. Therefore, this astronomical report will inform readers onthe amazing physical and the chemical aspects of comets. The physical andchemical knowledge that you already have from the Physics A and Chemistry A-courses will, for the first time, lead to the main idea of why one used these twofields in order to analyze comets; without taking into account the fact that youwill acquire even more knowledge on these scientific applications during yourreading. Have you ever heard about a comet having an average diameter nucleusof 60 km separate its nucleus into different parts? What about a comet so bright

that no telescope or any other device is needed to observe it during more than aperiod of a year? These are one of the few reasons why the astonishing cometHale-Bopp will be covered. The remarkable and unusual facts regarding thisexceptional comet, discovered in the mid 90s, has resulted into the intensive andcompetitive research lead by professional scientists around the world and even byany other amateurish observer. Therefore, think of the shocking effect that thesecoming facts on an object such as a comet will have on you as a reader!

2. Introduction

2.1. Background & ObjectiveThe word comet used in most European languages was first named by the ancientGreeks. This word is derived from the Greek word, kometes, which means the

hairy one. However, the earliest existent annals of observations of comet goback to around 1,000 BC in China.These comets were regarded by most people as holy omens - for good or bad. Forexample, the Aztec ruler of Mexico believed that the coming of the Spaniards wasan achievement of prophecies brought by comets. Whereas, in 1066, Harold theruler of the Saxon England found out that his kingdom got invaded by William ofNormandy. People then thought that the observed comet was a bad sign.However, after the assassination of Harold in the Battle of Hastings, Williamclaimed that the comet had foreseen his success; which was the opposite of what acomets significance had earlier for mankind.It wasnt until 1577, when the Danish astronomer Tycho Brahe showed that

comets were instead celestial corpses. In the 17 th century, Isaac Newton provedthat the movements of the comets were referred to the same laws that control theplanets in their orbitthe law of gravitational forces.


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The science of comets revealed by the modern sophisticated devices used inthe 19th and 20th century has made it possible for us to understand the purpose ofstudying comets. People might still wonder: Why do astronomers andastrophysicians invest so much money for researching on an insignificant celestial

body such as a comet?Well, there are very good reasons to this. Many scientists have reasons to believethat the study of comets can lead to the understanding of the formation of ourSolar System, since comets were initially formed at the same time as the Sun forabout 4.5 billions years ago. A clue to the success of this research is for instance,the study of the chemical species ejected by a mobile comet. This research canalso in the future lead to confirm the theory that states that earths water isoriginated by comets that bombarded the Earth as falling rain, which mostscientists agree on.These are two of many other reasons why I have decided to inform readers aboutcomets characteristics including the different parts of a comet, the chemical

abundance, the origin & formation, the luminosity and how comets are named;where most of these aspects include subsections for aim to analyze each partdeeper. Moreover, the general section of comets will also comprise themathematical & physical aspect of comets, and the non-gravitational force aspect.Eventually, the amazing chemical composition aspect of the remarkable cometHale-Bopp will be examined, including the most fundamental facts concerningHale-Bopps discovery, the magnitude, the nucleus & coma, the orbits & orbitalelements and its dust, ion and above all the unusual sodium tail.

2.2. Question State

As mentioned earlier, one of the most crucial sections was the mathematical andphysical aspect of comets, which is therefore one of the most significant sectionsto understand how our knowledge of mathematics and physics can be applied to acelestial body, instead of lab applications that we are used to apply in school forinstance. Hence, the fundamental question to be answer will be:

What kinds of mathematical and physical applications can be done oncomets?

The section of Hale-Bopp is extremely essential when bearing in mind that thecomet Hale-Bopp was considered as one of the most astonishing comets everobserved by mankind. The comet Halley was considered as the comet of the

century before the observation of the comet Hale-Bopp. However thanks to thehigh technology that scientists have had since the early 90s, it has been possiblefor astronomers and astrophysicians to analyze the comet Hale-Bopp to a fullextent, thus more than any other observed comet. As a result, one can take thefollowing question into account:


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What is so special with the comet Hale-Bopp, bearing in mind that it gotthe title of the comet of the century?

2.3. Method

As a method, the information used in this report has mainly been taken from theinternet. The reason is quite obvious. It is more preferable to use websites forastronomical information, especially for specific celestial bodies such as thecomet Hale-Bopp that is constantly being analyzed than using books that cannotbe updated as often in comparison.Besides the use of websites a PowerPoint presentation has also been madecontaining images and diagrams that cannot be well seen on printed paper becauseof their color or large size that is compulsory to maintain in order to not lose anycolor and symbol details.The information found in the first chapter - comets in general - of the report has

been primarily used from astronomical books as well as internet, since the generalinformation on comets cannot, and will not be modified in the future, due to thefact that the information mentioned in these books are primarily related onscientific research that involves any observed comet.

2.4. Source Review

The use of websites has been selected with awareness taking for example, in mindthat the writers name must occur on the website. Besides that, the website must

belong to a well-known organization rather than an individual. Hence, thisinformation found in this report is primarily taken from NASAs website and then

the European Southern Observatory (ESO) home page, which have been the mostuseful source of information:

http://www2.jpl.nasa.gov/comet/

and

http://www.eso.org/outreach/info-events/hale-bopp/

In addition, other websites belonging to the scientific observatory, such as theEuropean Space Agency's Infrared Space Observatory (ISO), many privatewebsites (with .edu in the URL address, meaning education) have been used.Here are their home pages:

http://www.iso.vilspa.esa.es/outreach/esa_pr/in9708.htm

respectively

http://www.richland.edu/james/lecture/m116/conics/translate.html

http://burro.astr.cwru.edu/Academics/Astr221/SolarSys/Flotsam/comettail.html
http://www2.jpl.nasa.gov/comet/http://www2.jpl.nasa.gov/comet/http://www.eso.org/outreach/info-events/hale-bopp/http://www.eso.org/outreach/info-events/hale-bopp/http://www.iso.vilspa.esa.es/outreach/esa_pr/in9708.htmhttp://www.iso.vilspa.esa.es/outreach/esa_pr/in9708.htmhttp://www.richland.edu/james/lecture/m116/conics/translate.htmlhttp://www.richland.edu/james/lecture/m116/conics/translate.htmlhttp://burro.astr.cwru.edu/Academics/Astr221/SolarSys/Flotsam/comettail.htmlhttp://burro.astr.cwru.edu/Academics/Astr221/SolarSys/Flotsam/comettail.htmlhttp://burro.astr.cwru.edu/Academics/Astr221/SolarSys/Flotsam/comettail.htmlhttp://www.richland.edu/james/lecture/m116/conics/translate.htmlhttp://www.iso.vilspa.esa.es/outreach/esa_pr/in9708.htmhttp://www.eso.org/outreach/info-events/hale-bopp/http://www2.jpl.nasa.gov/comet/


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http://ase.tufts.edu/astroweb/print_images.asp?id=12http://abob.libs.uga.edu/bobk/ccc/cc112399.html

The reason why Ive chosen these websites is to get trustworthy information from

very famous observatories and companies. As a result of this manner of selectingworthy information I have been able to be 100% certain that all the facts used inthis report are absolutely truthful.

In order to get information on comets mathematical and physical field I haveconsulted both internet websites but mostly astronomical books, such as bySerway, Louis and James; due to the fact that there was a lack of information onthis aspect on the internet.

2.5. Typographical Conventions

This report has a quite many topographical conventions, used in order to make theunderstanding easier for reader.The fact that most readers might not be familiar with astronomical terms, whichalso includes chemical and physical terms, has been taken into consideration.Therefore any sophisticated term which is mentioned in italic will be founddefined in the glossary.Additionally, there will also be typographical convention applied on mathematicalformulas found in both chapters. The equations will have Courier New as font,a centered and bold with a font size of 14 and having a black cornet line (exceptfor those being less important)surrounding them. The aim with this format is tocatch the readers attention on the formulas, which is very essential for those whoare interested in the physical and mathematical applications regarding comets.Besides, all the units and symbols of the given equations will be mentioned in

Arial. In addition, all the key words related to their appropriate aspect will have abold font face with the same font size.

Finally, the reader might encounter on some words being underlined that helpingthe reader to pay for attention on this simple word that has a major role in a givensentence.

2.6. Structure

As it already has been noticed by viewing the table of contents page, I havedivided my informative report in two chapters, mainly comets in general and thecomet Hale-Bopp with their corresponding pages. The purpose of this is simplybecause of the fact that the information found in these two chapters has absolutelynothing in common.Within each chapter an example of the following subdivision has been used:
http://ase.tufts.edu/astroweb/print_images.asp?id=12http://ase.tufts.edu/astroweb/print_images.asp?id=12http://abob.libs.uga.edu/bobk/ccc/cc112399.htmlhttp://abob.libs.uga.edu/bobk/ccc/cc112399.htmlhttp://abob.libs.uga.edu/bobk/ccc/cc112399.htmlhttp://ase.tufts.edu/astroweb/print_images.asp?id=12


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3.3.1.3.2.

3.2.1

and so forth.This very same subdivision has been used for the chapter 4.This subdivision has been arranged given that the information used in some casesbelonged to more or less the same item. In other words, they were sort of relatedto each other. As a result, this arrangement has made it possible for readers toread this report in one shot, without the need to read back and forth in order tonote and unite information needed. Thus, the use of this report is an excellentchoice for those who want to write a term paper on comets or comet Hale-Bopp oreven to be tutored.

3. Comets in General

3.1. Defining a comet

Before starting with this topic, it is compulsory to know the difference betweenthe following five celestial corpses: an asteroid, a comet, a meteor, a meteoroidand a meteorite; which many people have the tendency to confound.

First of all, asteroids and comets are both classified as nearEarth objects.Asteroids are composed of rock or metal and are believed to have been created inthe warmer inner solar system, the asteroid belta zone between the orbits ofMars and Jupiter. An asteroid can be regarded as a small planet with a size

below 1,000 km. However, most asteroids have a diameter that is comprisedbetween one and several hundreds of kilometersComets, though, are made of gigantic snowballs or ice balls, dust, rockyfragments and organic materials reaching a diameter of ten or dozen kilometers.On the other hand, some comets can have a much greater diameter, such as thecomet Hale-Bopp with a diameter range of 30-40 km. But they are invisibleduring ten or a dozen, hundreds or even thousands of years on account of theirsmall diameter relative to their massive surrounding the outer space. Suchcelestial corpses are thought to have been formed in the cold outer solar system.However these two objects do have something in common. Scientists havereasons to believe that they both are the earliest remains of the opening period of

the formation of our solar system for more than 4 billion years ago.You might already be familiar with the term shooting star. Well, a shooting staris the same as a meteor. A meteor is the streak of light seen during the nightcaused by debris (from a comet or asteroid) from space through the Earths

atmosphere.


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However the debris itself is called a meteoroid. They are formed when a cometpasses near the sun where the solar heat releases dust particles from the cometsicy tail. If a meteoroid entering through the Earths atmosphere doesnt vaporize

completely before reaching the ground, its name gets changed to a meteorite.

Read on further in order to find more information on the amazing world ofcomets.

3.2. Morphology of Comets

3.2.1. The Visible Parts of a Comet

As a drawing is always better than a long speech, we indicate on this image thedifferent parts of a comet.

The Head (or Coma):

The coma is a nebulosity having a sphericalform and centered on the nucleus. The comareaches a diameter of hundreds of thousandsof kilometers and sometimes up to 1 AU. It isdependent on the comets distance from thesun and the size of the nucleus. But how doesthe coma form?When the comet approaches the sun it getsgradually warm; where the ice sublimatesand the frozen gas evaporates by sweepingalong the rocky fragments and dusts. At thevery same time a diffuse nebulositylike a

cloud - is formed: the coma (orhead). This coma becomes luminous by the solarlight and diffuses by the dust and itsfluorescence in contact with gases, and getssurrounded by an enormous hydrogen envelope detectable in ultraviolet.

The Ion (or Plasma) Tail:

The ion tail, consisting mostly of ions such as CO+, N2+ and CO2

+, is straight and

several hundred million kilometers long. This type of tails is sometimes calledplasma tail, due to the fact that is composed ofplasma. But how does this type oftail form?As mentioned earlier, the coma surrounded by an enormous hydrogen envelope isdetectable in ultraviolet. Pushed back by the solar wind, the ions formed in thecoma engender - with a velocity up to 100 km/s - always an elongated bluish

Ion tail

Dust tail

Coma

Fig.3.2.1.1. Different visible parts of acomet.


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rectilinear tail positioned in the opposite direction of the sun. This tail, colored inblue, is due to the presence of carbon monoxide ions (CO+) from the sublimationof gases that occur in the nucleus. However, there is a more accurate reason tothis. The CO+ absorbs the UV radiation and fluorescence by emitting energy at a

wavelength of 0.42m (or 420 nm).** For information on the physical applications of the ion tail consult thesection

3.4.1.**

The Dust Tail:

This type of tail, also pointing away from the sun, is the easiest part of the cometto see and isnt as long as the ion tail. The dust particles ejected from the nucleus

that is already pushed back due to the pressure of the solar wind, forms ayellowish dust tail, which is larger, more diffuse and curved than the ion tail.This dust tail, containing silicates, carbon and carbon compounds, is usually white

but gets yellowish while it reflects the sunlight to us.

Furthermore, the dust tail normally has an orientation that differs from the iontails one. In fact, once they have been expulsed from the coma, the dust particles

in the tail will be governed by the motion of the comet along its orbit around theSun, causing the dust tail to be curved as the comet follows its trajectory. Besidesthe dependence of the motion of the comet, the dust tails orientation in space is

also depended of the size of the dust particles and the speed of the ejection fromthe coma.

** For information on the physical applications of the dust tail consult thesection

3.4.2.**

3.2.2. The Invisible Parts of a Comet

The Hydrogen Cloud:

The neutral hydrogen cloud is very large atmillions of kilometers in diameter.This type of cloud wasnt discovered until 1970in comets Tago-Sato-Kosako and Bennett fromspectroscopy by satellites. The reason why thehydrogen cloud was not discovered earlier is

because it is not visible from Earth.

Fig.3.2.2.1. Picture on the neutral hydrogencloud surrounding the coma of acomet


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Moreover, the main reason is that the atomic hydrogen is emitted in the ultraviolet(UV), however the ozone layer in the ionosphere stops these waves frompenetrating and reaching land. Therefore, the hydrogen cloud can only beobserved with satellites.

The Nucleus:

** For information on the physical applications of the nucleus consult thesection3.4.3.There is also thesection 3.5.,on the nucleus movement in space. **

The Sodium Tail:

This tremendous sodium tail was observed for the first time in the comet Hale-

Bopp and will therefore be covered uniquely in the section 4.7. of the Hale-Bopp-chapter.

3.3. The Chemical Abundance of Comets

The comet becomes active when the acquired solar energy is superior to thesublimation energy of the water - since a comet is composed of 75-80 % water.This position in space corresponds to a distance of 2.5 AUfrom the Sun with atemperature of -70C. Yet one has also observed active comets found in greaterdistances, which had lead to expect the presence of more volatile corpses thanwater in the nucleus; such as carbon monoxide (CO), carbon dioxide (CO2),

hydrocyanide (HCN) and methanol (CH3OH).

The nucleus, a compact corpse of an irregular formhaving a diameter varying between 1 and 100 km, isessentially composed of ice and dust. The ice sublimatesand the gas released engendering the dust. Much of thecometary materials ejected are very volatiles since thenucleus is already active from a distance of five, eight oreven of 15 AU from the sun. It follows that even at farheliocentric distance, the sunlight reflected from the

comet does not come from the solid nucleus, instead fromthe halo of materials surrounding it; thus making itimpossible to measure the diameter of the nucleus.Nevertheless, if the comet approaches sufficiently theEarth one can then observe its nucleus.

Fig.3.2.2.2. The nucleusof the cometHalley, takenin 1986.


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Subjected to solar radiation, the corpsesof the sublimated molecules (parentmolecules = molcules mres)undergo chemical transformations where

the produced products are free radicals,atoms or ions (simple molecules =molcules filles).

One can easily identify the simplemolecules by their emission spectrum,which is found in the visible area. Theidentification of the parent molecules ismore difficult since their spectrum isindeed found in the infrared, which isblocked by the atmosphere.Thus, it is not possible to determine the

composition of comets in a directmanner. Experts can on the other handexpress suppositions thanks to theacquaintance of the simple molecules.

3.4. The Mathematical & Physical Aspect of Comets

3.4.1. The Physics of the Ion Tail

The magnetic field lines of the occurrence of the solar wind are compressedwhen they penetrate into the comets atmosphere, which captures the ions from

Fig.3.3.1. Illustration showing the producedsimple molecules from parent

molecules, while the comet

enters in the phase of activity.

Fig.3.4.1.1. Illustration showing the interaction between a comet and the solar wind thatresult in the engendering of the ion tail, where the ions flow away from the Sunbetween oppositely directed magnetic field lines in the tail. When the cometenters in a region - at far heliocentric distance - where the magnetic field of thesolar wind changes direction, the comet loses its ion tail.


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thesolar wind. By accumulating these ions, the mass of the solar wind increasesand thus slowing down in accordance with the physical law of conservation oflinear momentum. It is this phenomenon that induces the compression of themagnetic field lines of thesolar wind.

After penetrating into the atmosphere of the comet, the quantity of ions capturedfinishes to get enough large so that the ionic pressure moving away from thenucleus of the comet compensates the ionic pressure carried by the solar wind.This equilibrium prevents the plasma and the lines of force of the solar wind tocontinue their progression in the atmosphere, which represents a magnetic fieldzone of zero in the comet.

The solar wind and its magnetic field are directed towards the comet with avelocity of around 400 km/s. However, the ions from the comet recently formedare directed towards the sun with a velocity of only about 1 km/s. These reasonsare enough to predict what will occur under the collision:When the ions try to leave the atmosphere of the comet by penetrating into the

solar wind, they are trapped by the magnetic fields of the solar wind. Thereforethese ions are forced to return towards the comet in the same direction as thesolarwind, in other words by moving away from the sun.As mentioned earlier, when the mass of the solar windincreases with the supplyof ions from the comet, the velocity of the solar wind diminishes because thequantity of the movement remains constant according to the law of conservationof linear momentum.

This physical law can also be expressed mathematically:

p = mv

where p is the quantity of movement (in kgm/s), m is the mass in movement (inkg) and v, the velocity (in m/s).

3.4.2. The Physics of the Dust Tail

The dust emitted by the nucleus is subjected to two different and opposite forces:

1) As mentioned earlier, the solar radiation pressure acting on the dustparticles - due to the solar wind - that tries to push it away from the Sun(Frad).

2) The other, force of solar gravity trying to drag it towards the Sun (Fgrav).

By taking the difference of the two forces in mind we represent the ratio of theradiation pressure to gravitational force by (1 - ) where:

(1 - ) = FradFgrav

Eq. 3.4.1.2.

Eq. 3.4.2.1.


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Assuming that the dust particles are spherical we can write the formulae for Fradand Fgrav as follows:

Frad = d2 Qpr FS

4 c 4r

2

and

Fgrav = GMSgd3

r2 6

where FS stands for the solar radiation field imposing on the grain (dust particle)of diameterd and density d and Qpr, the efficiency factor for radiation pressure.Thus Eq. 2 and 3 can be solved in Eq. 1, giving the following:

(1 - ) = Frad = 3QprFS 1

Fgrav 8cGMS dd

Where this equation can be simplified by using C, as a constant:

(1 - ) = C (dd)-1

where

C = 3QprFS = 1.210-4 Qpr

8cGMs

The value of(1-) tells us on the nature of the orbit of the particle ejected fromthe nucleus.There are two major situations to consider.First of all, it is crucial to be aware of the fact that the particles are constantlyejected from the nucleus as a function of time. The locus of these particles havingthe same value of (1-) is called the Syndyname (orSyndyne) curve; knowingthat the ejected particles possess the same size - thus the same value of d(diameter).Secondly, the case of the spreading of particles that are ejected at any given timeis to consider as well. The occurrence of an outbreak is an ideal example to this

concern. Hence, these particles will have different values of (1-)

where thelocus of the particles describing this case is called the Synchrone curve.

Figure: This figure will be drawn during the oral presentation. It will show thetrajectories of the particles ejected from the nucleus and the formation ofthe dust tail. Thus, all information concerning this figure will not be

Eq. 3.4.2.3.By applying Newtons

gravitational law

Eq. 3.4.2.4.

Eq. 3.4.2.5.

Eq. 3.4.2.6.

Eq. 3.4.2.2.


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mentioned any further in this report. It will instead be included in myspeech.

3.4.3. The Physics of the Nucleus

It has been proven by experts that it is not possible to determine the size of acomets nucleus since it cannot be done by pure calculation. The most common

method utilized is by the use of the photometry of the comet at farheliocentricdistances. Why is this method used only at far distances from the Sun? First of all,in such far distances the surrounding of the comet is located at very lowtemperatures. Thus, this makes it impossible forsublimation of gases from thenucleus to occur, which needs enough kinetic energy from the solar radiation.Therefore, such farheliocentric distances have been chosen in order to minimizethe uncertainty of the determined dimensions of the nucleus.It is thanks to the incident solar radiation reflected from the nucleus that helps

scientists to determine the size of a nucleus. This observation is done by takingthe farheliocentric distances into account. It is important to notice that these fardistances is referred to as when the comet is before and after the stage of activity -thus before and after theperihelion.

Before moving forward to the equation of the magnitude of the nucleus, wepresent the following formula of the flux, Fc, of reflected solar light from thenucleus:

Fc = ()pvSFSr22

where the flux, Fc, measured on Earth at a distance in AU from the comet isproportional to the cross section of the nucleus, S, and the geometric albedo, pv.In addition, () is the phase function, which is stabilized to the phase angle = 0 and FS is the solar flux at r= 1 AU. Note that physically the phase angle() is the Earth-comet-Sun angle in space.

The preceding equation can be written in a more appropriate manner in order tobring forward the magnitude of the nucleus V(1,1,0), meaning at r= = 1 AUand = 0 and determining the radius of the nucleus, RNalso called the nuclearradius, as follows:

log(RN) = 2.14 0.2 V(1,1,0) 0.5 log(pv)

It is crucial to notice that the albedo, pv, and the phase function, () are difficultto determine. Scientists state that this is due to the presence of very dark dustcovering large parts of the nucleus, which do not allow the solar incident radiation

Eq. 3.4.3.1.

Eq. 3.4.3.2.


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to be reflected as normal. As a result, only 10 % or even less of the incident lightwill be reflected to the Earth.The method of photometry, named earlier, has unfortunately disadvantages. It isthe photographic plates that arent sensible enough to the enormously low

surface-brightness, which the coma possesses at farheliocentric distances.This method was then hopefully replaced by the modern CCD (Charge-CoupledDevice) detector. It became the most effective device determining the nuclearradius, RN-value. This measurement could be done from the phase of inactivenucleus at far heliocentric distances from the sun by seeing 9 to 10 times fartherout into space than the previous photographic plates. This lead to the registrationof a very high effectiveness of about 90 to 95 %, compared to the insignificant1 % by the earlier photometry technique. 1

3.4.4. The Conicals

Before starting with this section, it is good to remind readers that this unit willhelp you to understand the next section: the orbits of comets, by analyzing thedifferent trajectories celestial corpses can take. In fact, these ones are theparticular objects of obeying the mathematical laws: the conicals.

The orbits are characterized by their eccentricity. In a mathematical point ofview, one can say that the eccentricity is a parameter associated to every conicsection. In other words, it can be seen as the measure of how much the conicsection deviates from being a circular.The figure below shows conic sections, which result from the intersecting of acone with a plane:

1http://mcdonaldobservatory.org/research/glossary/index.php?sort=a&f=c&l=c

Fig. 3.4.4.1. This cone showsthe differentexisting orbitsin space.

Circle

Ellipse

Parabola

Hyperbola
http://mcdonaldobservatory.org/research/glossary/index.php?sort=a&f=c&l=chttp://mcdonaldobservatory.org/research/glossary/index.php?sort=a&f=c&l=chttp://mcdonaldobservatory.org/research/glossary/index.php?sort=a&f=c&l=chttp://mcdonaldobservatory.org/research/glossary/index.php?sort=a&f=c&l=c


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Now we can express mathematically the orbits characterized by their eccentricity,as follows:

e = c/a

where e is the eccentricity, c, the half distance between the focus (singular offoci) and a is the semi (half)-major axis. These variables are of course notobvious to understand, although they will be illustrated shortly in this unit.

By the eccentricity one can classify the trajectories in five different groups:

Circles, having an eccentricity value: e = 0 Ellipses, having an eccentricity value: 0 < e < 1 Parabolas, having an eccentricity value: e = 1 Hyperbolas, having an eccentricity value: e > 1 Straight line, having an eccentricity value: e (infinity)

However, not all of these groups are referred to cometary trajectories. Comet canonly have parabolic, elliptical orhyperbolic orbits. Therefore, the mathematicalexpressions that these conics will be explained:

1) Parabolas:

A parabola is the set of points in a plane where the distance from a fixed point F(called the focus) and a fixed line (called the directrix) are equal. This definitionis illustrated by the next figure, where the point halfway between the focus andthe directrix lies on the parabola is called the vertex. The line through the focusperpendicular to the directrix is called the axis (oraxis of symmetry).

If we place the vertex of this parabolic graphat the origin O and its directrix parallel to thex-axis we can obtain a particular simpleequation.Thus, if the focus of this graph is the point (0,p), then the directrix will get the equationy = -p, so that the parabola will have thefollowing equation:

x2

= 4pyFig.3.4.4.3. Figure of any givenparabola illustratingwhere focus, thedirectrix, vertex andthe axis of symmetryare.

Eq. 3.4.4.2.

Eq. 3.4.4.4.


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This equation can also be arranged to a simpler one, where if we puta = 1/(4p) this formula will become:

y = ax2

A fact to this equation is if a>0 the parabola will open upward (as the figureabove shows) or else, ifa0 and to the left ifa


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x2 + y2 = 1a2 b2

Again, we also have to consider ellipses in a more general form.

So if the center of an ellipse is at (h, k) and the distance between the verticalmajor axis is 2a, then the formula is modifies to:

(x-h)2 + (y-k)2 = 1a2 b2

however if the major axis is horizontal, then the ellipse gets the following form:

(y-k)2 + (x-h)2 = 1a2 b2

where the a-value is always larger than b-value.

As you might have noticed again, we basically altered the variables x to y (andinversely) of the before last equation in order obtain the last given formula.

Please note that while solving mathematical problems involving analyzingellipses, the question might not provide you the equation as shown below. Instead,the equation can be modified where the (x - h)2 and (y - k)2 have been expandedand then simplified, ending up in the following form:

Ax

2

+ Cy

2

+ Dx + Ey + F = 0whereA and C have the same sign.

F2

Fig.3.4.4.7. Figure of an ellipse having ahorizontal major axis of a fixeddistance 2a, illustrating the foci,the vertex and the point P and

obeying the rule d1 + d2 = 2a.

F1

Eq. 3.4.4.8.

Eq. 3.4.4.9.

Eq. 3.4.4.10.

Eq. 3.4.4.11.


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3) Hyperbolas:

A hyperbola is the geometrical set of pointswhere the distance between two fixed points,foci, has a constant difference in absolute

value.As with the ellipse, the line going through thefoci is called the principal axis, the pointswhere the hyperbola intersects the principalaxis are called the vertices (orvertex) and thepoint halfway between the vertices is calledthe center. Note that the center here is not atthe origin.A hyperbolic graph has an auxiliaryrectangle formed by two diagonalasymptotes intersecting the center (h, k).These two asymptotes have the followinglinear equation:

y = (b/a) x and y = - (b/a) x

The standard form of the equation of a hyperbola is the exactly the same as theone for the ellipse but with a minus sign instead of a plus.

So if the center of a hyperbola is at (h, k) and the distance between the vertixes is2a, then the equation of the hyperbola is of the following form:

(x-h)

2

(y-k)2

= 1a2 b2

if the principal axis is horizontal, and

(y-k)2 - (x-h)2 = 1a2 b2

As one can see, the graph of a hyperbola consists of two curves. But where doesthe second curve come from?

The view of the two coming figures will explain this.

giving the following hyperbolic graph

Fig.3.4.4.12. This illustrates ahyperbola that has ahorizontal transverseaxis where two foci,the conjugate axis,vertices andauxiliary rectanglewith its two diagonalasymptotes occur.

Fig.3.3.4.17. Two figures illustrating the origin of the second symmetrical curve in a hyperbola.

Eq. 3.4.4.13-14.

Eq. 3.4.4.15.

Eq. 3.4.4.16.


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3.4.5. The Orbits of Comets

Preface:

What is the purpose of measuring the orbit of comets?First of all, the analytic geometry of the orbits of comets has been analyzed. Inother words, the mathematical properties of orbit of comets have been carried outby intend to make the physical calculations of the orbits of comets much easier tounderstand and to apply. Thus, this section will comprise astrophysics applied tothe orbits of comets in movement.

The Fundamental Relations of the Orbital Movements:

In the absence of the non-gravitational effects and of planetary disturbances, thecomet, of negligible mass in comparison with the Sun, traverses a conical sectionwhere the Sun occupies one of the foci.Before going any further, notice that this fact was first discovered by the German

astronomer Johannes Kepler. Keplers first law states the following:All planets move in elliptical orbits with the Sun at one of the focal points.

His statement isnot only appliedto planets orbitingin ellipticallyaround the Sun,but even comet orany other celestialcorps in space.

Please note that the following facts are applied on comet, whereas they were firstapplied on orbiting (closed system) planets by Newton and Kepler. In case thesystem is not closed, such as parabolic of hyperbolic orbits, then one must specifywhich celestial corps is being analyzing.

The total energy E - i.e. the sum of both the kinetic and potential energy(E=K+U) - of a corps, such as a comet, determines the type of orbit. This can bedescribed by the following equation:

E = mv

2

GMm2 r

where r represents the heliocentric distance, v is the velocity and G, thegravitational constant. M and m, are the masses of the Sun and of a moving bodywith a velocity v in the vicinity of the massive mass M respectively, where M ismuch smaller than m (M >> m). Note that when the E-value is less than 0 (E


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then we have an elliptic orbit and when the E-value is above 0 (E>0) we get ahyperbolic orbit.

Now lets assume that a planet, such as the Earth, moves along its orbit form point

P to Q. We can prove that the moving mass m from point P has exactly the sametotal energy as later on point Q, in three-dimensional space. Mathematically, it isshown as follows:

E = mvi2 GMm = mvf

2 GMm2 ri 2 rf

This equation shows that E may be positive, negative, or zero, depending on thevalue of the velocity v. For the case of the Earth-Sun, the E-value is less thanzero. Newtons second law can be applied to m in a system where E> m, givingthe following:

GMm = mv2r2 r

Multiplying both sides by r and dividing by 2 gives:

mv2 = GMm2 2r

Substituting this into equation 2.3.1., we obtain:

E = GMm GMm2r r

simplified to

E = - GMm2r

One can clearly see now that the total energy is always negative in the case ofcircular orbits. Note that the total energy is also negative in the case of ellipticalorbits, such as for many planets and comets. Hence, the expression for E forelliptical orbits is the same as equation 7 with rreplaced by the semi-major axislength, a, as follows:

E = - GMm2a

The semi-latus rectump of the ellipse is expressed by the following equation:

p = a(1 e2)

Eq. 3.4.5.7.

Eq. 3.4.5.8.

Eq. 3.4.5.6.

Eq.3.4.5.3.

Eq. 3.4.5.4.

Eq. 3.4.5.5.

Eq. 3.4.5.9.


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Similarly, for a parabola the semi-latus rectum p is given by:

p = 2q

where q is theperihelion point, coming up soon.

Heres the polar equation for the radius position r(distance sun-comet) along itselliptic or parabolic orbit making the angle - the true anomaly - with the semi-major axis:

r = p/(1 + e cos)

or by using the angle E - the eccentric anomaly - which is the angle between thesemi-major axis and the imaginary circle in the same vertical level as some pointon the elliptic orbit:

r = a(1 e cosE)

Before moving forward to other equations describing orbits, we can give therelation between the two angles E and as follows:

tan(/2)= [(1 + e)/(1 - e)]tan(E/2)

In elliptical orbit we have two foci along the major axis, where the Sun is locatedat one of the foci. Thus there exist two very important distances along the orbit tothe Sun: theperihelion and aphelionpoint.

The distance q from the Sun to the closest point along the elliptic orbit

(perihelion point) we have the following expression:q = a(1 - e)

and the furthest point Q along the orbit to the Sun that is situated being at one ofthe foci (aphelion point) is given by:

Q = a(1 + e)

A parabola having also aperihelion point q has the following equation:

q = p/2

taken from the rearranged equation 10.

The aphelion point Q for a parabolic orbit is at infinity:

Q =

Eq. 3.4.5.14.

Eq. 3.4.5.15.

Eq. 3.4.5.10.

Eq. 3.4.5.11.

Eq. 3.4.5.12.

Eq. 3.4.5.16.

Eq. 3.4.5.17.

Eq. 3.4.5.13.


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From the equations 2 and 8, the expression for the orbital speed of the particlealong an ellipse can be obtained and it is given by:

v2= (2/r - 1/a)

where = GM and 1/a, is positive, zero or negative if the orbit is ellipse, parabolaor hyperbola respectively.

The corresponding equation for a parabola, which has only one focus, is thefollowing:

v2 = 2r

However, the velocity at the perihelion point for an ellipse is given as:

vp2 = [(1 + e)/(1 - e)]

ain the same way at the aphelion point, the equation for the velocity is given by:

va2 = [(1 - e)/(1 + e)]

a

By the expression in the brackets of both equations 20 and 21 it is noticeable thatthe velocity of the particle is maximum at perihelion and minimum at aphelionand varies along its orbit.

Now it is possible for us to state Keplers secondlaw of equal areas:The violet-shaded area are all equal to eachother if and only if the planet moves from forinstance B to C in the same time that it movesfrom L to A.

In other words, this law supports also the fact that the planet moves fastest when itis closest to the Sun (ex. at perihelion point) and slowest when it is furthest fromthe Sun (ex. at aphelion point). Please note that even this law is applied to comets.

Johannes Kepler has even a third law, called Keplers third law) that can be

applied to the elliptic orbit of a comet. Assuming that we have a circular orbitwhere the Earth is revolving around the Sun. Since it is assumed that readersalready have the basis for the laws of gravity from Physics Course B, this 3rd lawwill be derived.

Since the gravitational force exerted on Earth by the Sun is equal to the centralforce needed to keep the planet moving along its orbit:

Eq. 3.4.5.18.

Eq. 3.4.5.20.

Eq. 3.4.5.21.

Eq. 3.4.5.19.

Fig. 3.4.5.22. Keplers 2nd law of the areas for an elliptic orbit.


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GMsMp = Mpv2

r2 r

where Mp stands for the mass of a planet.

Using the physical equation: v = rone can expand it to: v = 2r/T where = 2/T

Taking T to be the period of the circular orbit the equation 21 above becomes:

GMs = (2r/T)2

r2 r

and reduces to:

T2 = r3[(42)/(GMs)] = Ksr3

where Ks is a constant given by: Ks= (42) / (GMs) = 2.97 10

-19s

2/m

3

The equation 25 is Keplers third law.

Note that this law is also valid for elliptical orbits if we replace rby the length ofthe semi-major axis, a.

It has surely been noticed that all the equations of this section are related toelliptic and parabolic orbit. This does not imply that comets traveling alonghyperbolic orbits do not exist; instead it simply means they are very rare. This isthe reason why formulas for such an orbit were not pointed out. However, youmay consult Krishna Swamys book for more information on the equations

involving hyperbolic orbits.

The Orbital Elements:

The orbit is computed from the brightest point of the cometary coma, whichoverlaps with the nucleus being some thousands of kilometers apart. The orbit ofa comet can be described with the help of seven parameters. The use of theseparameters can be compared to the mathematical application of the x, y, z planes,of a graph, in order to position the coordinates of a certain point in three-dimensional space. As a result, there are many more specifications (parameters)involved in determining the location of a comet in space, due to its hugesurrounding, where the presence of the Sun playing the major focus role, forinstance.

Here are the seven parameters with their clarification and a descriptive figure:

1) e = eccentricity:

An orbital element represents the eccentricity of the elliptical orbit, in otherwords, the extent to which an elliptical orbit departs from a circular one. It isusually expressed as a decimal fraction.

Eq. 3.4.5.23.

Eq. 3.4.5.24.

Eq. 3.4.5.25.


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An eccentricity of 0 corresponds to a perfect circular orbit and an eccentricitycomprised between 0 and 1 has an elliptic orbit, where just less than 1 theeccentricity is an extremely elliptical orbit. A parabolic orbit owns an eccentricityof 1. An eccentricity superior to 1 corresponds to a hyperbolic orbit. Note that the

parabolic and hyperbolic orbits are no longer periodic orbits. In other words,when a coming comet reaching its perihelion point - by following any of thesetwo preceding orbits - is observed from the Earth, it will then no longer be seen,since it will continue away along its non-periodical orbit and never ever comeback.Mathematically one can even describe the eccentricity as the amount by which theelliptical orbit deviates from circularity:e =c/a, where c, is the distance from the center to a focus and a, is the semi-major axis.

2) a = semi-major axis:

The semi-major axis, a, is an orbital element representing half the length of themajor axis of an ellipse.

3) i = inclination:

The inclination is an orbital element representing the inclination of the celestialcorpses (ex. a comets) orbital plane to the ecliptic plane (or a referenceplane).

4) = longitude (or argument) of the perihelion:

The longitude of the perihelion is an orbital element representing the angle

measured along the orbital plane from the ascending node to the perihelion point.

5) = longitude of the ascending node:

The longitude of the ascending node is the angle measured from the vernalequinox along the ecliptic plane to the ascending node - the point of intersectionof orbital plane with the ecliptic plane.

6) T = epoch:

The epoch (or time) is an orbital element representing the time of perihelionpassage.

7) P = period:

The period is an orbital element representing the time required to complete anorbit.


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** The records of some of the parameters of the comet Hale-Bopp will be giveninchapter 4.**

3.5. The Non-Gravitational Forces

By the use of familiar methods that are based on Newtons gravitational law theorbit of a comet can be calculated. The more observation done for a comet, themore precise the determined orbit will be. Thus it is much easier to obtain a moreaccurately determined orbit for short-period comets than long-period ones. Once acomet enters inside the solar system its orbit gets usually disturbed by the planetsrevolving along their orbit. Thanks to the availability of high speed computersscientist can include such perturbations, or other types of complications, in theirorbit calculations. However experts have noticed a peculiar thing: that the position

of the comet varies after each revolution along its orbit. In other words, thepredicted calculations of the future comet position along its orbit are not the sameas the observed one. For instance, the short-period comet Encke having a periodof 3.3 years continues to come 2.5 hours too soon after each revolution.

Besides the comet Encke, it has been observed that nearly half of all the cometsstudied came earlier than the calculated time while the other half came later than

Fig.3.4.5.26. This three-dimensional space figure shows some of the seven orbital elements,such as the argument of perihelion, longitude of the ascending node andinclination


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the predicted. In other words, this means that half of the observed cometsaccelerate and the other half decelerate respectively. This lead to the fact thatthere must be an additional non-gravitational force acting on the comet, whichwas not included in scientists orbit computation. After observing a comets

movement along its orbit it was found that the non-gravitational force diminishedsignificantly with an increase in the distance between the comet and the Sun.

The non-gravitational force acting on the comet has an influence on its orbit-period. However, this fact may still not seem understandable, yet.

By physical means, what makes the comet accelerate or decelerate?In order to clarify the influence of the non-gravitational force one has to take thematter ejected from a rotating nucleus.

When the solar radiation - from the solar wind - imposes on the nucleus, thematerial evaporates from the surface and moves out in the direction of theincident radiation. This ends up in a jet event (orjet action), where the comet ispushed away from the Sun by a force. Whenever the nucleus rotates a time delaybetween the heating and the ejection of the gas from any point on the surface willtake place. Now, by taking the orbit comet into consideration the comet can occursooner or later than the calculated time due to the direction of rotation of thenucleus:

Case #1:The orbit-period of the comet will increase whenever the rotation is in the

same direction as its motion - i.e. along its orbit - around the Sun; where this isdue to the belated jet action which has an advance factor. Therefore the orbitcomet will become superior leading to the fact that the comer will appear later

than the predicted time, thus decelerating.Case #2:

In contrast, if the rotation of the nucleus is in the opposite direction to themotion around the Sun the jet action will have a backward factor and shrink theorbit-period so that the comet occurs earlier than the predicted time, thusaccelerating.

As a result, nearly half of the existing comets are accelerated and other halfdecelerated when it is presumed that the axes of rotation of the nucleus arearbitrarily distributed.


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3.6. The Origin & Formation of Comets

3.6.1. Preface

As we have seen earlier, comets are corpses having a very small dimension to its

surrounding essentially composed of ice where a part of it sublimates when thecomet passes by the perihelion point, by forming the coma and the dust and iontails. At every passage the nucleus loses its dimension by some meter of itsexterior layer. Therefore, the periodic comets size shrinks more the more itrevolves several hundreds or thousands of times along its orbit near the Sun. Thisprocess continues until the comet dies and losses all its matter. This event is

called the ablation process.But which comets can hold on longer?First of all, it is compulsory to know that the periodic comets are usually dividedinto short-period comets (those with periods of less than 200 years) and long-period comets (those with periods of more than 200 years). Technically, since the

diameter of a nucleus is in general measured between one and ten kilometers itcannot survive longer than hundred to three hundred passages, i.e. at most ten

thousand years for short-period comets. However it takes one to about ten millionyears for long-period comets.

This process introduces the question of where the comets come from. Thus,comets must have originated from some permanent source that compensates theirdisappearance and maintains a certain number of activities; otherwise, it would bea long time since there would no longer be any comets left.

3.5.2. The Formation Mechanism

In 1950 the Dutch astronomer Jan Hendrick Oortdiscovered that comets originate from very distantregions, located between 50,000 and 100,000 AUfrom the Sun, thus outside our solar system. Henamed this zone the Oort cloud.This spherical cloud contains around 1062 comets,mainly long-period comets. Most of these long-period comets have a parabolic orbit, thusappearing once.

There are also long-period comets having elliptic orbits such as the comet Hale-Bopp, which will be coming back.Oort showed that the movement of the comets in the Oort cloud was controlled bythe gravitational disruptions and induced by the neighboring stars. Some of thesedisturbances expel comets outside the solar system and others, on the contrary, are

Fig.3.5.2.1. Scaled figure showing the Oort cloud enveloping

our solar system.


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ejected toward the inside of the solar system so that they can be observed from theearth when they approach the Sun.

The Oort cloud is said to be a remainder of the original nebula that failed to formthe Sun and the planets for five billion years ago. There are theories stating that

the Oort cloud was first formed closer to the Sun and that is got ejected toextremely long heliocentric distances away due to the effect gravitationalinteractive forces of the giant planets such as Jupiter. This theory implies thespherical shape of the cloud along with the gravitational interactions of the nearbystars.

From 1943 to 1949, the Americanastronomer Gerard Peter Kuiper stated thatthe comets, owning an analogous chemicalcomposition as the one of the giant planets,could have been formed at some tens of

astronomical unities from the Sun, at thelevel of the orbits of Neptune and Pluto.This zone became called the Kuiper belt.Formed in this belt, comets might have thenbeen ejected by the disruptions of the giantgaseous planets to much more stable orbitsat 50,000 AU, thus constituting the Oortcloud. However, comets in this region areshort-period ones.

In order to confirm this theory scientists are doing research on fossil comets

that have not yet been ejected into the Oort cloud. So far only some thirty cometshave been identified but this research is still taking place.

This formation mechanism of comets explains why the comets give rise to suchan interest for professional astronomers, interest that isnt uniquely visual. The

comets have passed the biggest part of their life in the Oort cloud, far from thesun with a temperature a few degrees above absolute zero (-273C or 0 K), due tothe absence of the solar radiation at such far heliocentric distances. Very little

modified during their journey, comets could be true fossils of the solar system,rich with information on the physical conditions that could have been needed forour solar system to form. The analysis on fossil comets can be done thanks to

the fact that they havent undergone any chemical modifications at all.

Fig.3.5.2.2. Scaled figure showing the

Kuiper belt both fromabove and sideways.


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3.6. The Luminosity of Comets

The classical relation giving the visual (apparent) magnitude of a cometaccording to the distances to the Sun and the Earth, expressed in AU is:

m = m0 + 5 log() + 2.5n log(r)where m represents the apparent magnitude and m0 the absolute magnitudeproperly corresponding to r== 1 AU and a phase angle of 0. Thus, is thegeocentric distance Earth-comet, ris the heliocentric distance and n is the cometactivity factor.

The 5 log() part of the equation takes the comets variation of brightness withchanging Earth-comet distance () into account. This part of the equation holdsan inverse-squared relationship, where if the geocentric distance increases by afactor of 3 the brightness will decrease by a factor of 3 and squared, thus 9.

The last tem 2.5n log(r), represents the change in brightness due to the variationof the heliocentric distance of the comet. The reflected light from the produceddust and gas (from the coma or the dust tail) and the solar radiation causing thegas to shine, produce a speedily change in brightness with changing heliocentricdistance. In other words, the n value defines how rapid this change is. Therefore,it is not possible to find a single value of n that covers the entire range of theheliocentric distance. The value of n varies from before and after the perihelionpassage. Scientists have so far concluded that the n-value for new comets isabout 2, since their brilliance is quite low when they approach the Sun comparedto when they are at farther heliocentric distances by being relatively brighter.Conversely, the n-value of the old comets is comprised between 4 and 6.

The term apparent magnitude is employed a lot in astronomy; signifying a scaleused by an astronomer to determine the brilliance of a star or any celestial bodyfrom Earth. By the term apparent one means the brightness measured in thevisible part of thespectrum, in other words, the part one can see with naked eye.The scale of the magnitude is so that each magnitude is about 2.5 times brighterthan the newt brighter magnitude. This implies that a variation of 5 magnitudeunits (ex. from 1 to 6) corresponds to a change in brilliance of 100 times.However astronomers do not round off the determined apparent magnitudevalues, instead they use the precise ones with decimal values.A bright star like the star Vega has an apparent magnitude of 0. Visual

magnitudes higher than the one of the star Vega will turn out to go in negativevalues. Thus the more negative value the apparent magnitude is the more it isbrilliant.

Eq. 3.6.1.


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Thus this leaves us to know that the two major terms are the absolute magnitudeand the n-value of the comet, where both determine the intrinsic variation of acomets brilliance.

3.7. Nomenclature of Comets

By tradition a comet is normally named for its (or their) discover(s). This methodof designation is used by the International Astronomical Union on its IAUCirculars and in astronomical literatures written by professional scientists.Initially, the prefix C/ is used as a scientific abbreviation for long-period comets,i.e. comets having an orbit-period superior to 200 years. The second possibleprefix, P/, is utilized for short-period comets. The two last ones, X/ and D/, areused for comets having an unknown orbit and comets that have disappeared,respectively.

Lets take the great comet Hale-Bopp, which will be covered in the next chapter,

as an example. The scientific name of this comet is C/1995 O1. As mentionedearlier the prefix C/ designs the fact that this comet is a long-period one. 1 By1995 it is meant that is was discovered in year 1995. The letter0 shows that thiscomet was discovered in the second half-month of July; where A stands forJanuary 1st to 15th, B for January 16th to 31st, C for February 1st to 15th etc...It is important to note that the I is omitted and the Z isnt needed.At last, the number 1 after the letterO implies that this was the first comet tobe discovered in the second-half of July.

On the roadto chapter 4

As a reader, you have learned all the basics needed and even more in order to beable to both understand and analyze any informative article, magazine, book onany given comet. In other words, all the well-known characteristic components ofa comet were mentioned. However, as you might already know, a comets activitymight not always follow mankinds knowledge, whereas the physical laws arealways obeyed by the laws of physics. Thus, some observations can be quitepeculiar to scientists although knowing a lot on comets already. Therefore, I haveput the amazing facts of the comet Hale-Bopp after this studied chapter.


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4. The Comet Hale-Bopp

4.1. The Discovery

During the night of July 23

rd

and 24

th

1995, two amateurish Americanastronomers, Alan Hale, in New-Mexico and Thomson Bopp, in Arizona, decidedto observe the constellation of Sagittarius. Although they did not know eachother, they perceived about at the same time a luminous dot that did notcorrespond to any known object. At that moment, they transmitted the perceivedinformation to the Central Bureau for Astronomical Telegrams. In less than 12hours, after the objects first observation, the discovery of Hale-Bopp comet wasannounced to the scientific community, where it acquired later C/1995 01 as ascientific name.

During that night, having a magnitude of

around 10.5, the comet Hale-Bopp wasalready 250 times more brilliant than thefamous comet Halley when at anequivalent distance to earth. Moreover, thecomet Hale-Bopp was the most distantcomet yet discovered - at 7 AU from theSun. This amazing comet had one of thestrongest luminosity of all comets of the19th century. In addition, it was possible toobserve this comet without any difficultyand it did, in deed, astonish astronomers,astrophysiciansand many more by its

amazing characteristics.

4.2. The Magnitude

The first time that one could observe the comet Hale-Bopp with naked eye wasMay 20 1996, with a magnitude of 6.7. The comet H-B was visible for a period ofmore than 17 and a half month, till October-December 1997, which beat therecord of all the other comets by their visibility. On April 1997, at the perihelion

date, Hale-Bopp had -1.5 as magnitude, which indicates that the comet was asclear as the planet Jupiter!The Hubble Space Telescope determined that in the year 2020, the comet H-Bwill have a magnitude of 30, i.e. it will be 1.5 billion times weaker than what thenaked eye can see. However in average, the comet Hale-Bopp had a maximalbrilliance of -0.4 to -1.2 in clear sky and -0.8 in dark sky.

Fig.4.1.1. Showing the comet Hale-Bopp with the two lines,taken on July 24, 1995.


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Heres a list of the top ten brightest comets recorded of the past three centuries in

comparison to the comet Hale-Bopp:

1729 Comet Sarabat -3.01577 Comet Tycho -1.8

1997 Comet Hale-Bopp -1.31747 Comet De Cheseaux -0.51811 Comet Flaugergues 0.01744 Comet De Cheseaux +0.51882 Comet Cruls +0.81914 Comet Delavan +1.11433 Great Comet +1.21962 Comet Humason +1.35

As a result, one can notice that the Comet Hale-Bopp is the brightest comet everrecorded of the 19th century and the third brightest one of the last three centuries.

This graph, called aLight Curve, pointsup the magnitudeobserved fromSeptember 1995 toSeptember 1997.Observe that thevertical violet lineshows the perihelionpassage on April 1,1997, where the

magnitude was -1.3.

Fig.4.2.2.: Consult the figure 4.4.3.2. of the section 4.4.3. in order to see anexample of how bright the magnitude of the comet Hale-Bopp canreach.

4.3. The Molecular Composition

It is very important to know the composition of the comets and especially thecomet Hale-Bopps, since comets come from much removed regions (about50,000 AU) where a temperature close to absolute zero prevails. Comets, which

Fig.4.2.1. A Light Curve illustrating the magnitude of the CometHale-Bopp, where the black curve indicates the averagemagnitude obtained from September 1995 to September1997.


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have not undergone any chemical transformation and the chemical analysis oftheirnucleus, inform us on the composition of the primitive interstellarcloud.

The activity of a comet increases when it approaches the sun and depends on the

nature of the nucleus. Far from the sun, one expects that the types that are mostvolatile (those of which the transition temperature of the solid state to the gaseousis the lowest) aresublimatedfirst, and being therefore more abundant in the coma.This was actually what was observed in the comet Hale-Bopp by a team of Frenchand English astronomer, to the assistance of the Nanay Decimetric RadioTelescope (NRT) and the James Clerk Maxwell Telescope (JCMT), wherebeyond 4 AU, the coma was dominated by more volatile species than water(presence of mostly carbon monoxide (CO) that sublimated at 24 K (-249 C) andwater that sublimated in vacuum at 200 K). The maximal water production ratewas astonishingly 250,000 kg/s, taking in mind that it was about 10 times higherthan the famous comet Halley, being at its most active state. 2

Carbon monoxide (CO) and hydrogen sulfide (H2S) are two very volatilemolecules, which are effectively overabundant at 4 AU, just as the toxichydrocyanide (HCN).But two other molecules, such as methylcyanide (CH3CN) and methanol(CH3OH), which have nevertheless the same sublimation point as the HCN, arepresent in a normal amount.Thus, these observations give evidence of the complexity of the chemicalprocesses that are taking place in the heart of the nucleus.

Closer to the sun (at 1AU), the group of twenty molecules, already recognized inearlier comets spectrum have all been identified in the spectrum of the comet

Hale-Bopp. Seven new molecules were for the first time made obvious: sulfurmonoxide (SO), sulfur dioxide (SO2), formic acid (HCOOH), acetic acid -methyl formate - (CH3COOH), nitrogen sulfide (NS), formamide NH2CHOand cyanoacetylene (HC3N).

Other astromolecules, such as OCS, H2CS and isocyanic acid, HNCO havingdifferent rotational transitions occurred for the second time, i.e. they were firstidentified, a year earlier, in comet Hyakutake.Furthermore, the Infrared Space Observatory (ISO) detected the presence ofcrystalline olivine (Mg2SiO4), rich with magnesium, in the spectrum of thecomet Hale-Bopps dust tail.

2http://www.eso.org/outreach/info-events/hale-bopp/comet-hale-bopp-summary-mar25-97-rw.html
http://www.eso.org/outreach/info-events/hale-bopp/comet-hale-bopp-summary-mar25-97-rw.htmlhttp://www.eso.org/outreach/info-events/hale-bopp/comet-hale-bopp-summary-mar25-97-rw.htmlhttp://www.eso.org/outreach/info-events/hale-bopp/comet-hale-bopp-summary-mar25-97-rw.htmlhttp://www.eso.org/outreach/info-events/hale-bopp/comet-hale-bopp-summary-mar25-97-rw.html


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The picture to the left shows theejection of the parent molecule,

hydrogen cyanide (HCN) from theinner coma of the comet Hale-Bopp

taken by the James Clerk MaxwellTelescope (JCMT) on April 1996. Theidea behind the use of the x and y axis,which arent well seen here, is to showthe extraordinary 100,000 km wideregion of HCN gas streams that havinga speed of at least 1km/s.

When the comet Hale-Bopp approaches the sun the solar radiation heats up theparent molecules that absorb this solar energy (photons) and gets transformed todaughter molecules. The hydrogen cyanide (HCN) is a perfect example whe rethis parent molecule gets dissociated to the two following ionic daughtermolecules: cyanogen (CN) and the element hydrogen (H).

Thespectrum of the comaof the comet Hale-Boppidentifies the strongestmolecular emissioncharacteristics. From thewavelengths below 4000to over 5000 angstroms

the species cyanogen(CN) and molecularcarbon (C2) arenoticeable in the coma ofthe comet, thus in thevisible wavelength regionof the spectrum

The sodium (NaI) emission line around the wavelength 5800 angstroms is linkedto the sodium tail that G. Cremonese along with the European Hale-Bopp teamreported on April 18, 1997. More on the sodium tail will be covered in section4.7.

Fig.4.3.1. Image showing the distribution ofthe HCN in the coma, taken bythe JCM telescope on April 1996.

Fig.4.3.2. Figure showing the spectrum of the coma of thecomet Hale-Bopp within 20 arcsec of the nucleustaken on April 13, 1997 with the Steward

Observatory 2.3-m telescope and CCD cassegrainspectrograph.


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For more information on the spectrum detection of other molecules in the cometHale-Bopp you may consult the following website:http://quasar.la.asu.edu/comets/hale-bopp.html

These photos taken by the POLAR satellite lets different molecules appear.Clockwise from top to bottom: the image to the left shows the atomic carbon, ata wavelength of 165.7 nm on its spectrum scale; where the coma is the dark-bluecolor around the yellow zone in the middle.The next photo to the right shows a weaker concentration of carbon monoxidewith a wavelength of 156 nm. The four dots on the photo around the cometillustrate four different ultraviolet stars. As a result, one can notice that the cometis amazingly brighter than the four stars surrounding it!

The filtered photo below shows a concentration ofhydroxide ion (orradical) at awavelength range of 308.5 nm. Thanks to the scale used, one can measure theextent of the coma, the yellow zone in the middle, to about 3 millions kilometers;which is enormous compared to the 40 km dimension of the nucleus. Without theenergy from the solar wind hydroxide radicals would not exist, since they arederived from water that absorbs photons and dissociates to both hydroxide andhydrogen ion.

Fig.4.3.3-4-5. The two photos above areobtained by ultraviolet-technique imaging, allowingdifferent molecules appearby using the POLARsatellite imaging system,while the comet was near itsperihelion point. The lastphoto to the left revealsanother molecule by filter-

technique.
http://quasar.la.asu.edu/comets/hale-bopp.htmlhttp://quasar.la.asu.edu/comets/hale-bopp.htmlhttp://quasar.la.asu.edu/comets/hale-bopp.html


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The diagram to the leftdemonstrates the amount of gasreleased by the nucleus of thecomet Hale-Bopp as a function of

the heliocentric distance fromradio observations at the IRAM,JCMT, CSO, SEST and Nanaytelescopes. Note that the left sideof the diagram represents the pre-perihelion facts and the right sidethe post-perihelion facts, wherethe decreasing heliocentricdistance makes the gas productionincrease.

By performing these observationsscientists can extract valuableinformation in the physical stateof the icy nucleus containingthese chemical compositionsalong with their interior model.

The list below gives most of the molecules in the comet Hale-Bopp detected inradio signal (R), infrared (IR), visible (V), and ultraviolet (UV) technique:

Molecules already detectedMolecular Detection Nomenclature Molecular Detection Nomenclature

Formula method Formula method

H2O IR Water CH3OH R, IR Methanol

OH-R, IR,UV

Hydroxide H2CO R Formaldehyde

H2O+ V Ionized water HCOOH R Formic acid

HDO R Deuterated water CH3OCHO R Acetic acid

COR, IR,UV

Carbon monoxide HCN R, IRHydrogencyanide

CO2 IR Carbon dioxide CH3CN R Methyl cyanide

Fig.4.3.6. A graph illustrating the gas productioncurve of the main molecules found in thecomet Hale-Bopp as a function ofheliocentric distance, by the use of radioobservations.


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CO+ V, RCarbon monoxide

cationHC3N R Cyanoacetylene

HCO+ R Formyl cation HNCO R Isocyanic acid

H2S R Hydrogen sulfide CN V, RCyanogen

radicalSO R sulfur monoxide NH3 R AmmoniaSO2 R sulfur dioxide NH2

- V Amide ionOCS R Carbonyl sulfide NH- V ?

CS R, UVCarbon

monosulfideCH4 IR Methane

H2CS R Thioformaldehyde C2H2 IR AcetyleneC2H6 IR Ethane K V Potassium

C3 V Carbon trimer O+ V Ionized oxygen

C2 V Carbon dimer Na V Sodium

Unusual IsotopesMolecular Formula Detection method Nomenclature

H CN R Hydrogen cyanide acidH15CN R Hydrogen cyanide acid

34CS R Carbon sulfide

In addition to the interesting facts of the numerous detected species, did you knowthat the comet Hale-Bopp releases a mass equivalent to more than 5.5 billion carsa day? 3 And this is only a matter of the ejection of the carbon monoxide (CO),regardless of the other species released by the comet!

As mentioned earlier, long-period comets are created in the Oort cloud. Howeveraccording to NASA, the comet Hale-Bopp may have been transformed in theregion between Jupiter and Neptune. This theory is based on the massive quantityof carbon monoxide found in the comet Hale-Bopp. Dr. Michael DiSanti ofCatholic University and NASA's Goddard Space Flight Center stated thefollowing: we would have seen even more carbon monoxide emission fromHale-Bopp. The amount of carbon monoxide ice compared to water (12 percent)indicates that these comets formed somewhere between the orbits of Jupiter andNeptune4

Even the presence of the noble gas, argon, was revealed by scientists on June 5,2000, on the fact that the comet Hale-Bopp may had been formed in our solar

3http://www.solarviews.com/eng/halebppr.htm4http://www.solarviews.com/eng/halebppr.htm

Table.4.3.7. A table showing most of the species chemical formulas, nomenclatures and

detection methods, detected in the comet Hale-Bopp.
http://www.solarviews.com/eng/halebppr.htmhttp://www.solarviews.com/eng/halebppr.htmhttp://www.solarviews.com/eng/halebppr.htmhttp://www.solarviews.com/eng/halebppr.htmhttp://www.solarviews.com/eng/halebppr.htmhttp://www.solarviews.com/eng/halebppr.htmhttp://www.solarviews.com/eng/halebppr.htmhttp://www.solarviews.com/eng/halebppr.htm


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system near Neptune. This work was lead by Alan Stern, leader of a team ofresearchers and announced the discovery of the argon at the AmericanAstronomical Societys meeting in Rochester, New York. 5

It is good to know that noble gases do not interact with other chemical species,

since they already are stable molecules having an octet (eight valence electrons).The argon, a noble gas, in ice-state found in the comet Hale-Bopp can reveal a lotof information on the life of a comet, such as the temperature variations it hadexperienced during its lifetime, thus where in the universe it might have beenlocated. Note that the reason why the argon was identified at the solid state inspace was due to the fact that its melting point is about above 35 K (or -238 C).

Other theories pointed out by other scientists suggested that the comet Hale-Boppmay have been transformed near Jupiter. However this is not possible since thetemperature in the Jupiter region is above 35K, thus forcing the argon of thecomet Hale-Bopp to melt.

4.4. The Nucleus & the Coma

4.4.1. Diameter of the Nucleus

When the comet Hale-Bopp used to get too far from earth it did not becomepossible to examine its nucleus by radar. The luminosity of the comet is notalways linked to the diameter of the nucleus since its only a part of the nucleus

that is active.For example, it is possible that the comet has a small nucleus with a large part ofits area releasing gas and dust. It is also probable that the comet with a large

nucleus has only a small segment that is active. Consequently, there is no way forscientists to determine the size of the nucleus by this manner.The latest estimations of the nucleus were carried out on the basis of theobservation of the amount of material ejected by the nucleus. Hence, theAmerican astronomer Harold Weaver and the French astronomer Philippe Lamydetermined the diameter of the nucleus of Hale-Bopp in the range of40 to 80 kmby using seven different method measurements. They concluded that the diametercould still lie above 80 kilometers due to the possibility that the nucleus might beelongated, since a nucleus is in theory as seen an irregular corps. 6 Besides thesize the nucleus has an amazing density of only 0.5 g/cm3, thus half of ices.

5http://www.space.com/scienceastronomy/comet_hale_000605.html6http://www.eso.org/outreach/info-events/hale-bopp/report-rw-hbitp98.html
http://www.space.com/scienceastronomy/comet_hale_000605.htmlhttp://www.space.com/scienceastronomy/comet_hale_000605.htmlhttp://www.space.com/scienceastronomy/comet_hale_000605.htmlhttp://www.eso.org/outreach/info-events/hale-bopp/report-rw-hbitp98.htmlhttp://www.eso.org/outreach/info-events/hale-bopp/report-rw-hbitp98.htmlhttp://www.eso.org/outreach/info-events/hale-bopp/report-rw-hbitp98.htmlhttp://www.eso.org/outreach/info-events/hale-bopp/report-rw-hbitp98.htmlhttp://www.space.com/scienceastronomy/comet_hale_000605.html


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4.4.2. Gas Emissions from the Nucleus

These two image sequences show the evolution of the coma in a three day period.The first sequence reassembles three images obtained by a telescope behind afilter on May 7, 8 and 9 year 1997. On the second sequence presents thecorresponding rotational gradients. On May 8 at 19h44, an enormous

The nucleus does small

Hale-Bopps:In close-up, one sees thenucleus (below to the

right) in process ofemitting a piece of thecrust that seems biggerthan the nucleus itself,where it is broken into abrilliant cloud of particles.

Quite remarkable, isnt

it!?

Fig.4.4.1.1. This picture illustrating the nucleus dividingitself is taken by the Hubble Space Telescopeon September 26th 1995.

Fig.4.4.2.1-2. Two images showing the comas evolution: the first shown by a filter-techniqueand the lower one showing the corresponding rotational gradients.


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concentration of dust is released from the nucleus. Twenty-four hours later, thedust is elongated over 50,000 kilometers behind the nucleus. A fact is that thisdust ejection does not occur at every nucleus rotation. Thats why scientists

consider this certainly as a unique phenomenon.

The next image shows the processing of arotational gradient, which brings at least 10jets to the fore.

Anim. 4.4.2.4.: View the animation that shows the movement of the jets. This canbe seen in the PowerPoint presentation.

4.4.3. Rotation of the Nucleus

When the rotational axis is parallel to the line aimed by the observer one candetect spirals around the nucleus. This occurs since the jets of dust and gas thatreturn periodically from the active side (face to the sun) of the comet formsuccessive layers of dust.Thanks to this phenomenon astrophysicians could determine the periodical

rotation of the nucleus of comet Hale-Bopp to 11.34 0.03 hours.

Treated by the technique of the fuzzy mask, thispicture shows the complex structure that thecoma represented on February 7th 1997 at 5.30h. Eight to nine envelopes surround the nucleusaccording to a parabolic form locally disruptedby the powerful jets of dust. These envelopesare separated by a distance of around 300kilometers.

Fig.4.4.2.3. This illustration of the processing of arotational gradient is taken in the beginningof the month of February 1997.

Fig.4.4.3.1. The comas composite

structure shown onFebruary 7, 1997.


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This image indicates the abnormal size of thecoma of comet Hale-Bopp. The large luminous

p

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