astro-ph/9505014 19 Mar 96 b ecause binaries either on circular or loweccentricity orbits cannot b e the source of observations MKG data of galaxy pairs in analysis have total masses and systems are generated by means of Monte CarloMOND simulations of binary orbits Mo del pairs pro duce Abstract Received accepted Email dsoaressicaufmgbr Brazil Belo Horizonte MG Departamento de FsicaICEX UFMG CP DSL Soares MannheimKazanas I Binary fraction of wide pairs determined with velocityblindbinary selection galaxy dynamics procedures is will b investigated e only p ossible when eects are at work distribution of separations is inconsistent with the observed data unless strong selection Send oprint requests to Your thesaurus co des are will b e inserted byAA hand manuscript later no General features A denite answer to the issue whether one or another mo del is suitable to explain real A binaries and they qualitative envelopes Binary MannheimKazanas require galaxies on higheccentricity of galaxies of synthetic extremely high R R mass ratios similar to pairs in the  DSL Soares  application V are V and L L mo deled samples are MKG gravity orbits L M L L alternative theories as and values space Both MOND and MannheimKazanas p ointmasses consistent derived of reasonable a large sample containing a signicant of  MOND gravity solar units Both from a comparison with observed with cataloged sample used ob eying M L the Random samples physics values observations and the nonNewtonian MOND solar but of in the units such and the ASTROPHYSICS ASTRONOMY AND

under a rigorous statistical analysis taking into account sample selection biases and

contamination by nonphysical pairs

Key words binary galaxies alternative physics Monte Carlo simulations

Introduction

The dynamics p ertaining to Newtonian gravity are unable of avoiding mass discrepancies

in astronomical systems unless some kind of yet unknown dark particles are invoked

Faber and Gallagher van Albada and Sancisi Trimble Sanders

Ashman For example the MACHO and EROS searches for dark matter in the

form of compact subluminous ob jects in our Galaxy halo Alco ck et al Bennett

et al Aub ourg et al and references therein have not detected sucient

microlensing events to explain the kinematics of the Milky Way with a spheroidal halo

p opulated by brown dwarfs Furthermore direct optical searches of the Hubble Space

Telescope have not also b een able to detect the predicted amount of matter in the form of

faint stars in the halo of the Milky Way Bahcall Another much explored approach

to the problem has b een the suggestion that Newton laws are not adequate and that the

solution might b e found in alternatives to classical gravity and dynamical laws Many of

such alternatives are empirically motivated eg Milgrom a b c Sanders

Kuhn and Kruglyak by the observations of at rotation curves but that is

not always the case Mannheim and Kazanas have derived an alternative

theory of gravity whose main motivation was the search for the correct covariant general

theory of gravity Sp ecically they found the exterior solution to conformal Weyl gravity

asso ciated with a static spherically symmetric gravitational source

The most successful of the empirical alternatives to Newtons laws has b een Milgroms

MOND Modied Newtonian Dynamics which has b een sub jected to successive exp eri

mental tests eg Gerhard but see Milgrom The theory has b een strongly

and successfully defended by Milgrom in many pap ers in recent years see also Bege

man Bro eils and Sanders Sanders Nevertheless a most serious setback is

that there is no covariant theory of gravity that in the weak eld limit reduces itself to

MOND Some attempts have b een made otherwise in this direction Bekenstein and Mil

grom Sanders but yet MOND remains as a phenomenological ad hoc theory

b eing this up to now its main source of criticism Other consequences dynamical and cosmological of MOND are discussed by Felten

Mannheim has applied the Mannheim and Kazanas theory of gravity

hereafter MKG to mo del the circular velocity proles of four spiral galaxies with fairly

reasonable ts without the requirement of dark matter He claims that EinsteinNewton

gravity proves to b e inadequate in comparison with fourth order conformal Weyl gravity

represented by the MKG solution In spite of that and of having on its foundation a

general theory of gravity with the status of a full covariant one MKG has b een ques

tioned on its ability of explaining exceptionally extended at rotation curves Sanders

and Begeman Moreover even conformal Weyl gravity has b een criticized on cos

mological grounds Elizondo and Yepes presented exact solutions to the conformal

Weyl gravity cosmological equations that failed to yield primordial nucleosynthesis abun

dances consistent with present observational constraints They conclude that conformal

cosmological mo dels are very unlikely to give a realistic description of the Universe

This is intended to b e the rst one of a series of pap ers devoted to binary galaxy

dynamics under the premises of alternative theories of gravity Here it is investigated

what binary galaxy data might suggest one ab out MOND and MKG Simple p oint

mass mo dels are used to represent b ound pairs of galaxies under MOND and MKG

prescriptions for the gravitational mutual galaxy interaction Monte Carlo simulations

of synthetic samples are p erformed by means of numerical calculations of pair orbits

They are subsequently pro jected on the sky and the relevant observed quantities derived

A qualitative comparison b etween a sample of real pairs and simulated samples is then

made The second pap er of the series will present a rigorous statistical analysis of observed

samples of binaries in the light of MOND and the third one will fo cus on MKG Biases

introduced by selection eects on sample determinations and contamination by non

physical pairs must b e considered separately for each theory of gravity It is obvious for

example that the denition of what a nonphysical pair is dep ends up on the theory that

describ es the interactions and the dynamics of galaxies in pairs

Section presents the data which come as a list of disk binary galaxies extracted

from the Catalogue of Multiple Galaxies A ducial Keplerian mo del determined in the

App endix is qualitatively tted to the data Such a mo del is meant to b e confronted with

MOND and MKG simulated samples In section the MOND gravitational p otential for

p ointmass binaries and technical details of the simulations are describ ed The same is

done in section for MKG Section is devoted to Monte Carlo simulations of MOND

and MKG synthetic samples In section results of the qualitative comparison b etween observed and synthetic binaries are discussed and the main conclusions presented

Binary galaxy data

Much eort has b een dedicated in the last years to the compilation of binary galaxy

lists Karachentsev Turner Peterson Schweizer Soares

Chengalur Salp eter Terzian Soares et al etc For the sake of unifor

mity only one list of pairs is considered namely that extracted from the Catalogue of

Multiple Galaxies CMG van Mo orsel Oosterlo o Soares

The selection of pairs was done by applying a surface density enhancement pro cedure

to the Uppsala General Catalogue of Galaxies Nilson a metho d devised by TS

van Albada A description of the metho d can b e found in van Mo orsel Soares

chapter by van Albada and Soares and Soares et al The metho d do es

not require any redshift information and this is a feature that makes it particularly

suitable for investigations concerning alternative physics The concept of a b ound pair is

dep endent on what gravity law rules the dynamics of the pair and cannot b e relied up on

simple upp er limits on lineofsight velocity dierences which is a typical characteristic

of velocitydependent selection pro cedures Still on uniformity from the whole list of

binaries in the CMG a homogeneous subsample of pairs is considered see Soares

which is formed by rotationally supp orted galaxies ranging from lenticulars to late

spirals The main reason for avoiding pairs with elliptical galaxies is that b oth MOND

and MKG have gathered their status of likely alternative theories after b eing applied on

mo deling circular velocity radial proles of spiral galaxies The whole sample is available

on machine readable form up on request dsoaresfisicaufmgbr The lineofsight

helio centric velocity dierences are normalized by the square ro ot of the total blue lu

minosities of the pairs in units of L Such a normalization allows one to scale the

envelopes given by eqs A and A Figure App endix by a prop er M L value

It has b een shown in many recent binary galaxy investigations based on conventional

physics that there might b e an extended dark halo either around the individual galaxies

or embedding the whole pair White et al Schweizer Soares Charlton

and Salp eter Bartlett and Charlton Although the presence of optical ie

nonphysical pairs in a sample could bias the analysis against p ointmass mo dels as

may have happ ened with the study by White et al as p ointed out by Pichio and

TanzellaNitti and Pacheco and Junqueira there is comp elling evidence that

Keplerian mo dels are far from b eing a go o d description of binary dynamics Nevertheless

Keplerian approximations serve as reliable upp er limits to V L for a sample of binary galaxies provided one not b e interested in the detailed distribution of pairs in that space

In the App endix upp er b ounds on prop erly normalized velocity dierences of galax

ies in pairs as a function of pro jected separation in the plane of sky are derived Such

envelopes are here confronted with real data and set denite limits up on the overall dis

tribution of the observations It must b e p ointed out that the Keplerian mo del derived

in the App endix serves only as reference

Fig Keplerian envelopes in the R V L plane curves Filled circles represent CMG

pairs There are three mo del free parameters the masstolight ratio M L the minimum

p ericentric separation r and the orbital eccentricity e The solid curve in the four panels

m

is the ducial envelope mentioned in the text with M L r kp c and e chosen

m

as a reference frame for CMG pairs L L L L

Figure shows how to ll in the binary phase space with varying the three free parameters

M L r minimum p ericentric separation and e orbital eccentricity allowed by the

m

Keplerian envelope mo dels derived in the App endix The ordinate V L is obtained

by multiplying V M by the scaling factor M L The b ottom right panel shows

the CMG pair sample

The solid curve in Fig was chosen as a ducial envelope for the subsequent analysis

Its parameters are M L r kp c and e From a Newtonian p oint of

m

view pairs ab ove it are most likely unbound pairs the socalled optical pairs b ecause

they imply to o high values of M L Incidentally this is not necessarily true for MOND

and MKG as it is shown in the Monte Carlo simulations b elow

Fig The envelope of maximum V L is shown as a histogram determined after exclusion

of outliers The number of pairs in each bin and the clipping weight are given in the right b ottom

corner of the panel The smo oth curve is the ducial Keplerian envelope

To verify whether such a Keplerian mo del is a fair reference envelope for the observations

a k clipping pro cedure see Yahil and Vidal was applied to the sample to exclude

outliers which would bias the determination of the envelope of maximum velocities

Figure shows the resulting envelope confronted with the ducial Keplerian envelope

chosen ab ove Each bin in the histogram of Fig has pairs For every bin the mean

and standard deviation of V L are calculated and pairs with V L over

are excluded from the initial bin p opulation A new mean and are calculated and

the pro cess is rep eated The iteration ends when every pair falls within Then the

highest V L is computed It is apparent from Fig that the chosen parameters for the Keplerian envelope are consistent with the numerical envelope derived from the data

MOND

The most successful phenomenological theory alternative to dark matter Newtonian ap

proach to the mass discrepancy problem has b een put forward by Milgrom originally in

three successive pap ers a b c It consists of a mo dication in Newtonian

dynamics in the regime of low accelerations In the framework of Milgroms Modied

Newtonian Dynamics MOND galaxies have no dark mass Its simplied formulation

has two basic assumptions i Newtonian dynamics break down when the accelerations

involved are small and ii the acceleration a of a test particle at a distance r from a

mass m is given by a a  Gmr in the limit Gmr  a or a  a The constant

  

a plays the role of a transition acceleration implying a transition length scale r which

 t

is a function of the mass m from the Newtonian to the MOND regime In other words

in regions where the acceleration is smaller than the limit acceleration the gravitational

eld is roughly prop ortional to the inverse of the distance from the center of the galaxy

and as a consequence spiral galaxy at rotation curves can b e obtained The value of a



was originally determined by Milgrom b to b e   cms A Hubble constant

H kms Mp c will b e assumed throughout this pap er Later a larger and b etter



sample of at rotation curves were analyzed and yielded the value of  cms

for the same H Kent Milgrom and Begeman Bro eils and Sanders



such a value is used b elow in the present MOND simulations of binary galaxies

The prop er question here is in which way MOND works with binary galaxies Milgrom

gives numerical solutions of three problems in the framework of the Mo died

Newtonian Dynamics They are the force law b etween two p oint masses rotation

curves of various mo del disk galaxies and the eld of a p oint mass in a constant

external acceleration eld Of these the rst one is used here for the account of binary

galaxy dynamics

Milgrom nds in problem that the MOND force eld is very weakly dep endent

on the ratio M M  q of the two p oint masses In the limit of large r the force F can

b e put in the form

Ga M M



F M M r Aq 

M M r

h i

Aq q q q q

This is the force eld that one exp ects from a logarithmic p otential In the range 

q  characteristic of the CMG binary sample see b elow the dimensionless function

Aq varies very little namely from A to A

The testparticle circular velocity V of an individual galaxy with mass equal to

roti

M at distances larger than r is

i t

V GM a

roti i 

and the circular velocity V r of a binary system with total mass equal to M M

circ

is given by

V r Aq GM M a

circ 

Equation is valid in the limit of low accelerations or for binary galaxy separations

much larger than the corresp onding r GM a where M is the total binary mass

t 

M M In the Monte Carlo simulations describ ed b elow the total mass of pairs will

range from  M to  M The transition radius for such systems ranges

from to ab out kp c Mo del galaxies have an assumed optical radius of ab out kp c

thus the binary p otential is always MOND p otential in the regime of r  r as long as

t

the visible galaxies do not overlap

Each synthetic binary coming out of the simulation is obtained from the asso ciated

binary orbit MOND orbits must b e calculated numerically b ecause the logarithmic p o

tential do es not admit an analytical solution The starting p oint of the orbit calculation

is the ap o centric separation Thus one needs to know the ap o centric separation r

ap o

itself and the velocity at ap o center For a given eccentricity and ap o centric separation

the velocity is univocally dened It is convenient to express V r as a fraction of the

ap o

circular velocity at ap o center ie as  V r In the simple case of a Keplerian

circ ap o

orbit with eccentricity e the parameter is given by

e e

The relationship b etween and the orbit eccentricity e for the MOND binary p otential

and galaxy separation r  r is

t

i h

e e

e ln

e

e

where e is given by its general denition e r r r r The function e

ap o p er ap o p er

valid only in the low acceleration regime is represented in Figure An articial sample

of MOND binary galaxies is generated in section by means of Monte Carlo simulations of binary orbits

Fig Ap o centric velocity in units of ap o centric circular velocity as a function of orbital

eccentricity for binary galaxy orbits Solid curve MOND p otential in the low acceleration regime

eq Dashed curve Keplerian p otential shown as a reference curve

MannheimKazanas gravity

Mannheim and Kazanas in a series of pap ers Mannheim

a b put forward the idea that gravity b e based on the fourth order conformal

Weyl theory rather than on Einsteins general relativity While Einsteins gravity has a

lot of success in a wide range of exp erimental facts it faces its unequivocal inconsistency

in the weak eld limit represented by Newtonian gravity with the mass discrepancy

problem in astronomical systems Mannheim claims that one of the advantages of MKG

conformal gravity is its ability in the weak eld limit to solve the dark matter problem

in spiral galaxies explicitly p osed by at rotation curves A conceptual p oint stressed by

MKGs authors is that while alternative physics theories such as MOND depart from the

phenomenological problem which they are meant to explain eg at rotation curves

and then go on seeking a general covariant formulation in order to get the desirable

status of an acknowledged theory of gravity Mannheim and Kazanas take the inverse

way

Instead of b eing based up on a second order action as Einsteins general relativity

MKG follows from a conformal invariant fourth order action which implies a fourth order

Poisson equation Mannheim b The latter was integrated to yield an exact exterior

solution to a spherically symmetric gravitational source Neglecting terms similar to the

standard general relativistic corrections to the Newtonian p otential and a high order

term in r see b elow the following nonrelativistic gravitational p otential is derived

valid whenever the weak eld limit is applicable

i h

Gm r c Gm

U r r

r r r

MKG

Gm

r  m M kp c

MKG

c

where c is the sp eed of light and a constant The circular velocity of a binary system

with total mass M M is then

i h

c GM M

r V r

circ

r

The linear p otential comp onent in eq is resp onsible for the go o d ts obtained for at

rotation curves eg Mannheim The actual value of for a typical galaxy is

 cm Mannheim and Kazanas which intriguingly is roughly the value

of the inverse Hubble length in the authors words In passing Milgrom has also

called the attention for one fact that may prove of prime importance ie the near

equality of a the MOND constant as determined from galaxy dynamics and that of



cH where H is the present value of Hubble constant Indeed the value of a adopted

  

in this work see section is equal to cH and the value of see b elow is



H c so one derives a c

 

The reality of such relationships b etween a and H would imply that the value of

 

H itself enters nonlinearly see section into the formalism of b oth MOND and MKG



for interpreting binary orbits

The MKG constant is not an universal one like MONDs a It must b e determined



for every single galaxy contrary to what is found with MOND Begeman Bro eils and

Sanders For the four galaxies studied by Mannheim ranges from to

 cm In the simulations of section eq was used with the average which has

the inverse value of  cm But the fact that can vary from galaxy to galaxy must

b e taken into account when a sample of binaries with a substantial fraction of wide pairs

is investigated It must b e p ointed out however that it is not clear by which mechanism a

law of gravity may have a tunable constant for every gravitational system This seems

to b e a ma jor weakness of MKG unless there is some constraint of cosmological origin

or of other kind to x as a universal constant

Orbits of p ointmasses under the p otential given by eq are numerically calculated

The auxiliary parameter describ ed in the previous section is related to the orbital

eccentricity in a more complicated way than in Keplerian and MOND cases The re

lationship includes dep endences on the total binary mass M M and on the orbital

ap o centric separation r

ap o

h ih i

xe GM M c

e r 

ap o

xe r

ap o

i h

c GM M

r 

ap o

xer

ap o

where xe e e r r Equation is depicted in Figure for M M

p er ap o

 M and r and kp c

ap o

Fig Same as Figure for the MKG p otential eq Solid curves eq for r

ap o

and kp c and M M  M Dashed curve Keplerian p otential shown as a reference

curve

In deriving eq some terms were neglected including a higher order term in r

Mannheim and Kazanas and Mannheim sp eculate that on the largest scales

the contributions to such terms and even to the linear comp onent from all galaxies

might merge and add to a preexistent general cosmological background It is dicult

to anticipate the gravitational inuence of such a cosmological background until a full

development of MKG implications is available It must b e p ointed out however that one

deals here with intermediate scales  Mp c and less which are probably only little

aected by the general cosmological version of eq

Monte Carlo simulations

The Monte Carlo simulations shown here follow the general recip e given in Soares

hereafter S where a study of binary galaxies with dark halos was made In that

study all binaries had xed r kp c Now for the sake of generality a p owerlaw

ap o

distribution of binary spatial separations is

Q

P r dr / r dr

where P r dr is equal to the number of galaxies inside the spherical shell with internal

radius r and external radius r dr The exp onent Q in the onedimensional distribu

tion P r is approximately if the twopoint correlation function which is usually

expressed as a threedimensional p owerlaw distribution with p ower eg Peebles

is considered to b e valid on scales of binary galaxies Gott and Turner van

Mo orsel White et al As in S a lower limit in r of kp c is set dictated

by the assumed size of a binary system at closest approach The mo del binaries b oth in

MOND and MKG have equal galaxies with kp c of radius An upp er limit on r of 

Mp c although rather arbitrary is also chosen It is otherwise consistent with the widest

pairs in the CMG sample and with recently determined wide pair lists eg Chengalur

et al In other words it is implicitly assumed that pairs of galaxies separated by

more than  Mp c are optical b e under the MOND or the MKG p otential The analysis

made in the next section however is by no means aected by the choice of such an upp er

limit in r

For technical reasons see b elow eq is applied to binary ap o centric separations

instead of spatial separations There is no ma jor problem with this pro cedure In case

of circular orbits the distributions of spatial and ap o centric separations are of course

coincident and in case of noncircular orbits they are not signicantly dierent due to

the fact that orbiting b o dies sp end most of their orbital time near ap o center

The simulated pairs need to have a distribution of total luminosities and luminosity

ratios similar to the observed sample The total masses of the simulated pairs are derived

assuming M L as found for MOND by Begeman Bro eils and Sanders and

characteristic of stellar p opulations Rather than adopting the distribution of observed

L L for the simulated pairs which requires computing orbits the simulated

L L are obtained from the observed ones averaged in a twodimensional grid of

q L where q L L Table shows this grid pairs of the total have q

less than Mass ratios of the order of or more are typical of satellite systems eg

Bontekoe and van Albada not representative for a study of binary dynamics so

they were avoided in the determination of the nal D distribution q  L The grid

shown in Table has cells and includes all pairs with q  with cells uno ccupied

The least massive binary system has a total mass of  M and the

most massive one has a total mass of  M Such a twodimensional

grid denes the mass parameter space of articial samples generated by Monte Carlo

simulations For the simulations of articial pairs the o ccupied cells of Table are sampled in prop ortion to the o ccupation frequency of the observed sample

Table Distribution of observed q L L L

q L L L L

Note L is primary galaxy luminosity in units of L

Three dierent orbital eccentricities are considered namely e and chosen

as representative of low medium and high eccentricity orbits For every simulation panel

shown in Figure there are articial pairs randomly distributed as describ ed

ab ove throughout the q L bins in Table each pair b eing generated through the

following steps a A value of r is obtained from the distribution given by eq b

ap o

the value e is calculated eq or eq for MOND and MKG resp ectively

c V r  V r is calculated d a test is done using the conservation

ap o circ ap o

laws of energy and angular momentum to verify if the pair will ever have a p ericentric

separation r smaller than kp c the socalled merging test adopted in S If

p er

this is true the algorithm b egins again at a Otherwise it go es to the next step e A

halforbit is calculated ie the binary path from ap o center to p ericenter The equations

of motion from MOND and MKG p otentials are numerically integrated Due to the

characteristics of the force eld this orbit segment is fully representative of the time

evolution of the system f the halforbit is rotated by a random angle in the orbital

plane This simple pro cedure mimics the evolution of the binary over a long time scale

g An orbital inclination i is obtained according to the distribution F i / sin i ie the

normal to the orbital plane is distributed at random in space h Pro jected separation

in the plane of sky R and lineofsight velocity dierence V are calculated at a

random time instant within the orbit halfp erio d as well as other relevant quantities

such as orbital eccentricity orbital p erio d etc This sequence is the same regardless of the gravitational p otential adopted

Fig Monte Carlo simulations of binary galaxies Left panels represent binaries under MOND

gravitational p otential and right panels under MKG gravitational p otential The solid curve is

the ducial envelope Keplerian orbits M L The orbital eccentricities are indicated on

the right upp er corner of each panel and M L L L L L

Discussion and conclusions

Though it is not apparent from Fig it must b e p ointed out that for a given M L the

higher is the total mass of a given pair the smaller is its V L ordinate This do es

not happ en with Keplerian mo dels see eqs A and A but b oth MOND and MKG

have approximately V L / M A similar b ehavior was also seen with the dark

halos mo dels worked out in S

Figure shows that MOND and MKG have distinct asymptotic b ehavior as R in

creases The upp er envelope of the MOND circular orbit simulation is indep endent of R

while MKGs increases as R as is exp ected from the linear p otential For noncircular

orbits MONDs upp er envelopes have an asymptotic value given by  V r L

circ ap o

and MKGs again increases with R That is to say the upp er envelope of the MOND

simulated p oints will eventually reach a plateau for large R and the MKG envelope

will increase as R since the r comp onent in the p otential vanishes in that range of

separations The asymptotic limit for the Keplerian counterpart is of course zero The

asymptotic b ehavior of the various mo dels is very promising as a way of discriminat

ing amongst them sp ecially b etween MOND and MKG The lack of a suciently large

observed sample of wide pairs is presently the main obstacle to such an achievement

From Fig in the range R kp c it is readily clear that for b oth MOND and

MKG binary galaxies in either pure circular orbits or in orbits with only intermediate

eccentricity e are not able of explaining the observations since they would require

extremely high M L  in the range R kp c for MOND and MKG standards

They also show dierent trends in the functional dep endence of V L with R which

is apparent in the range R kp c Pure high eccentricity orbits e

represent viable solutions for b oth mo dels Elongated orbits are also consistent with

the cosmological picture of binary galaxy formation in which such systems are formed

by galaxies that instead of participating individually of the expanding universe stayed

together b ecause of their high mutual gravitational attraction implying returning orbits

with relatively low angular momentum eg Schweizer A larger value of a would



improve MONDs tting since V L / a see eq for the low acceleration regime



In the case of MKG on the contrary a smaller value of would do b etter since in the

large radius range where the linear p otential comp onent b ecomes signicant V L

scales  see eq although here a smaller M L b eing still within MKG standards

would work as well on lowering the p oints on the V L R plane It is apparent that

the only way to distinguish b etween MOND and MKG lies in the investigation of the

wide pair domain A statistically representative sample of wide pairs R kp c would

certainly discriminate b etween MOND and MKG solutions

Although their envelopes are consistent with the observed ones high eccentricity orbits

MOND and MKG panels with e seem inconsistent with the observations b ecause

the corresp onding simulations predict a large concentration of pairs at intermediate and

large separations as compared to small ones and most of observed pairs have small

separations Nevertheless this may b e caused by selection eects It is obvious that a

combination of binary orbits with eccentricities ranging from small to large values is also

p ossible for MOND and MKG In this case the high V L values with R kp c

will b e accounted for by higheccentricity orbits while the low values will have contribution

for the whole range of eccentricities The way to discriminate b etween the two gravity

mo dels is an investigation of the relative frequency o ccupation of the R V L plane

The large R region wide pairs again is fundamental in this asp ect b ecause of the very

distinct predictions of MOND and MKG in that region see Fig To accomplish such

a discrimination a full account of sample selection biases and of presence of optical pairs

must b e undertaken

The ab ove study gives one a secure indication of the input parameters for a more

detailed statistical analysis justifying the initial aim of the work namely to present an

overview of sp ecic applications of alternative theories of gravity to binary galaxy dy

namics The series b eginning here will continue with a rigorous analysis of binary galaxy

samples under the assumptions of MOND mo dels A third pap er is planned on an investi

gation of binaries from the p oint of view of MKG Both works will extend and complement

the qualitative study presented here with a careful account of selection eects on the

determination of real samples of binary galaxies with a quantitative measure of sample

contamination by nonphysical pairs and with the introduction of the same eects in the

list of prescriptions of Monte Carlo simulations of synthetic samples

Acknowledgements This work has b een partially supp orted by the Brazilian Conselho Na

cional de Desenvolvimento Cientco e Tecnologico CNPq pro ject number

Drs LPR Vaz and GAP Franco are thanked for reading the manuscript for very im

p ortant suggestions and for encouragement in the nal stages of the work L Vertchenko

is also thanked for an imp ortant suggestion Dr M Milgrom is gratefully acknowledged

for useful criticisms on an early version of this pap er The author would like to thank

the pap ers referee Dr G Mamon the pap er has b een considerably improved due to his suggestions

A Keplerian mo dels

The squared relative velocity of two orbiting p ointmasses M and M is

V r GM A

r a

where G is the gravitational constant M M M r is the p olar radial co ordinate

and a is the semima jor axis of the orbit Consider now a twodimensional phase space

represented by the lineofsight pair relative orbital velocity normalized by the square

ro ot of the total pair mass and the pro jected pair separation onto the plane of sky These

two quantities are related by the projected version of eq A

V

G A R

M

where R is the pair pro jected separation and V is the lineofsight velocity dierence

is a pro jection factor that takes care of the pro jection of b oth the orbital relative velocity

and space separations of the galaxies in the pair

i h

V G

sin icos e cos A

a e

M

and

a e

sin sin i A R

e cos

The angles b etween the line of no des and the orbit ma jor axis and i the orbital plane

inclination with resp ect to the plane of sky give the orientation of the orbit in space The

true anomaly p olar angular co ordinate is the angle in the orbital plane b etween the

line joining one galaxy to the other and the ma jor axis of the orbit and e is the orbital

eccentricity According to eqs A and A the pro jection factor is given by

sin i

cos e cos 

e cos

 sin sin i A

Note that for a given eccentricity has a maximum value equal to e e

max

the conditions for a maximum pro jection are i and An immediate

conclusion from eq A is that any observed sample of pure b ound pairs would app ear

b elow the curve

V

R G G A

max

M

in a V M R diagram

One characteristic exhibited by the envelope of maximum values given by eq A is

that as R approaches zero V M b ecomes increasingly larger This would b e true if

galaxies were p ointmasses which would imply no restriction whatso ever in orbital sizes

that is to say the ma jor axes could have any value Of course this is not p ossible with real

galaxies thus one needs to put a lower limit on orbital sizes This limit can b e dened in

terms of a minimum allowed p ericentric distance r  r This implies in a minimum

p er m

semima jor axis a e r e for a Keplerian orbit with eccentricity e In fact

min m

such condition is equivalent to saying that pairs with p ericentric separations smaller

than r will merge quickly which is tantamount to introducing a merging condition on

m

Keplerian pairs

Upp er envelopes similar to eq A are now derived considering several orbital eccen

tricities and the p ericentric cuto describ ed ab ove Orbits that have a a lead to

min

maximum V M

i h

V G

sin icos e cos A

M r e

m

at

r e

m

R sin sin i A

e cos

Substituting the angular conditions for maximum pro jection i and

one gets

h i

G e V

A

r

M

max

m

that o ccurs at

i h

V

r A R

m

max

M

For a given eccentricity there is a welldened maximum V M for all eccentricities

maxima o ccur at the same R r In the case of R  r the curve that describ es the

m m

maximum values of V M is

i h

V G e

A R e

R

M

and corresp onds to orbits with xed e a a e  a r i and

min min m

In the range  R r one nds the largest p ossible values of V M by

m

xing a a i and but letting that gives the relative p osition of

min

galaxies in orbit b e a free parameter Doing that in eqs A and A one gets

i h

V G

cos e A

M r e m

and

r e cos

m

R A

e cos

Eliminating in eqs A and A yields the equation of the inner maximum envelope

i h i h

R G V

e A R e

r e r e eR

M

m m

Note that for e circular orbits eq A reduces to a line V M Gr R

m

and for R r ie the p osition of the maximum value of V M it takes the form

m

of eq A as it should Figure shows the envelopes for various eccentricities and for the

total pair mass given in units of M The most imp ortant p oint to notice is that the

merging condition introduced by only having a  a do es not allow pairs with high

min

VM at small separations see dashed curves in Fig

Fig The maximum allowed values of V M as a function of R and eccentricity e are

represented by the solid curves eq A for R  r and eq A for R r The dashed

m m

curves show the envelopes for R r if galaxies are p ointmasses without restrictions on the

m

p ericentric separation The values of the eccentricities are and The total pair

mass is given by M M M M

References

van Albada TS Sancisi R Phil Trans R So c London Ser A

Alco ck C et al Nat

Ashman KM PASP

Aub ourg E et al AA Bahcall J StScI release no PRCa

Bartlett RE Charlton JC preprint astroph

Begeman K Bro eils AH Sanders RH MNRAS

Bekenstein JD Milgrom M ApJ

Bennett DP et al preprint astroph

Bontekoe TjR van Albada TS MNRAS

Charlton JC Salp eter EE ApJ

Chengalur JN Salp eter EE Terzian Y ApJ

Chengalur JN Salp eter EE Terzian Y AJ

Elizondo D Yepes G ApJ

Faber S Gallagher JS ARAA

Felten JE ApJ

Gerhard OE in Dwarf Galaxies ESOOHP workshop G Meylan P Prugniel

eds

Gott JR Turner EL ApJ L

Karachentsev I Comm Sp ec Astrophys Obs

Karachentsev I Double Galaxies Nauka Moscow

Kent SM AJ

Kuhn JR Kruglyak L ApJ

Mannheim PD ApJ

Mannheim PD ApJ

Mannheim PD a Foundations of Physics

Mannheim PD b preprint UCONN University of Connecticut

Mannheim PD preprint astroph

Mannheim PD Kazanas D ApJ

Mannheim PD Kazanas D Phys Rev D

Mannheim PD Kazanas D General Relativity and Gravitation

Milgrom M a ApJ

Milgrom M b ApJ

Milgrom M c ApJ

Milgrom M ApJ

Milgrom M ApJ

Milgrom M Comments on Astrophysics Comments on Mo dern Physics Part C

Milgrom M ApJ Milgrom M preprint astroph

van Mo orsel GA PhD thesis University of Groningen

van Mo orsel GA AA

Nilson P Uppsala Astron Obs Ann

Oosterlo o TA PhD thesis University of Groningen

Oosterlo o TA AA

Pacheco JAF Junqueira S ApSS

Pichio G TanzellaNitti G AA

Peebles PJE The LargeScale Structure of the Universe Princeton University

Press Princeton

Peterson S ApJS

Sanders RH AA L

Sanders RH MNRAS

Sanders RH MNRAS

Sanders RH AAR

Sanders RH AA L

Sanders RH Begeman KG MNRAS

Schweizer LY ApJS

Soares DSL PhD thesis University of Groningen

Soares DSL AA S

Soares DSL de Souza RE de Carvalho RR Couto da Silva TC AAS

Trimble V ARAA

Turner E ApJ

White SDM Huchra J Latham D Davis M MNRAS

Yahil A Vidal NV ApJ

a

This article was pro cessed by the author using SpringerVerlag L T X AA style le E