Research Article A Note on the Observational Evidence for ...downloads.hindawi.com/journals/tswj/2013/204315.pdf · A Note on the Observational Evidence for the Existence of Event

Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 204315 4 pageshttpdxdoiorg1011552013204315

Research ArticleA Note on the Observational Evidence for the Existenceof Event Horizons in Astrophysical Black Hole Candidates

Cosimo Bambi12

1 Arnold Sommerfeld Center for Theoretical Physics Ludwig Maximilians University Munich 80333 Munich Germany2Department of Physics Fudan University Shanghai 200433 China

Correspondence should be addressed to Cosimo Bambi bambifudaneducn

Received 18 April 2013 Accepted 30 May 2013

Academic Editors V Beckmann S Covino and V Van Elewyck

Copyright copy 2013 Cosimo Bambi This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Black holes have the peculiar and intriguing property of having an event horizon a one-way membrane causally separating theirinternal region from the rest of the Universe Today astrophysical observations provide some evidence for the existence of eventhorizons in astrophysical black hole candidates In this short paper I compare the constraint we can infer from the nonobservationof electromagnetic radiation from the putative surface of these objects with the bound coming from the ergoregion instabilitypointing out the respective assumptions and limitations

1 Introduction

A black hole (BH) can be defined as the regionB of the totalspacetime M which does not overlap with the causal past offuture null infinity 119869minus(I+) [1]

B =M minus 119869minus

(I+

) (1)

The event horizon of a BH is the boundary delimiting theBH Everything falling onto the BH and crossing the eventhorizon is lost forever and it cannot affect events happeningoutside the BH any more However it may be possible thatevent horizons never form in nature but that only apparenthorizons can be created [2ndash4] An apparent horizon is a closedsurface of zero expansion for a congruence of outgoing nullgeodesics orthogonal to the surface [1] Outward-pointinglight rays behind an apparent horizon actually move inwardsand therefore they cannot cross the apparent horizon Inthe special case of a stationary spacetime an event horizonis also an apparent horizon but the reverse is not true ingeneral In particular the event horizon is determined by theglobal properties of the spacetimewhile the apparent horizondepends on the observer

Astronomers have discovered at least two classes ofastrophysical BH candidates [5] stellar-mass objects in X-ray binary systems and supermassive objects at the center

of every normal galaxy These objects are thought to beBHs because they cannot be explained otherwise withoutintroducing new physics the stellar-mass BH candidates aretoo heavy to be neutron stars for any reasonable matterequation of state [6 7] while at least someof the supermassiveobjects in galactic nuclei are too heavy compact and old tobe clusters of nonluminous bodies [8] There is also a set ofobservations suggesting that BH candidates have really anevent horizon [9ndash13] Basically these objects seem to be ableto swallow all the accreting gas without emitting any kindof electromagnetic radiation from their putative surface Inthe case of low-mass X-ray binaries we can compare systemsin which the primary is thought to be a BH and the onesin which the primary is thought to be a neutron star In thequiescent state we can observe thermal radiation from thesurface of neutron stars while no such a radiation is observedfromBH candidates [9 10] Neutron star systems show type-IX-ray bursts (as outcome of compression and heating of thegas accumulated on their surface) while the phenomenonhas never been observed in binaries with BH candidates [11]There are also strong constraints on the radiation emitted bythe possible surface of the supermassive BH candidate at thecenter of our Galaxy [12 13]

This body of observations can be easily explainedwith thefact that BHs have no surface and that the gas crossing the

2 The Scientific World Journal

event horizon cannot be seen by distant observers any more(see however ref [14]) Strictly speaking the confirmationfor the existence of an event horizon would require theknowledge of the future null infinity of the Universe whichis clearly impossible for us On the contrary the nonobser-vation of electromagnetic radiation emitted by the gas afterfalling into the compact object nicely meets the definition ofapparent horizon However the geometry of the spacetimearound astrophysical BH candidates is practically stationaryfor the timescale of our observations and that may make itimpossible to discriminate an event horizon froman apparenthorizon

2 Electromagnetic Constraint

Let us image a BHas a gas of particles packed in a small regionby the gravitational force (The model of BH I will considermay remind of the one discussed in [15] The radius of thecompact object 119877 is larger than the one corresponding tothe event horizon of a (classical) BH with the same mass andspin) As this gas has a finite temperature it must radiateHowever if the object is very compact the emitted radiationis strongly redshifted when it reaches a distant observer andthe object can appear very faint Here I relax the quitecommon assumption of steady state 119871 = 119888

2 [9 10 12 13]where 119871 is the surface luminosity and is themass accretionrate That would require that the accreting gas hits the ldquosolidsurfacerdquo of the object and then radiates to infinity all itskinetic energy If this were the case a very compact objectwould not be able to increase its mass or at least the processwould be very inefficient likely in contradiction with theobservations of the supermassive objects in galactic nucleiMoreover there are no reasons to assume that BH candidateshave a solid surface In the picture in which we have a gas ofparticles packed in a small region by the gravitational forcethe accreting gas enters into the compact object and bothits rest-mass and kinetic energy contribute to increasing themass of the BH candidate

Let us now see the constraint we can obtain in this picturefrom the nonobservation of thermal spectrum from BHcandidates The specific energy flux density of the compactobject (often measured in erg cmminus2 sminus1Hzminus1) as detected by adistant observer is as follows

119865 = int 119868119900

119889Ω (2)

where 119868119900

is the specific intensity of the radiation as measuredby the distant observer and 119889Ω is the element of the solidangle subtended by the image of the object on the observerrsquossky 119868119909

]3119909

= const (Liouvillersquos Theorem) where ]119909

is thephoton frequency measured by any local observer on thephoton path and

119889Ω =119889119909119889119910

1198632 (3)

where 119909 and 119910 are the Cartesian coordinates on the observerrsquossky and 119863 is the distance of the compact object from

the observer The equivalent isotropic luminosity of the BHcandidate is thus

119871 = 4120587int1198923

119868119890

119889119909 119889119910119889] (4)

Here 119892 = ]119900

]119890

is the redshift factor ]119900

is the photonfrequencymeasured by the distant observer and ]

119890

and 119868119890

arerespectively the photon frequency and the specific intensityof the radiation measured by an observer located at the pointof emission of the photon on the surface of the compactobject and corotating with the surface of the compact objectThe emission should be like the one of a blackbody that is

119868119890

=2ℎ]3119890

1198882

1

exp (ℎ]119890

119896119861

119879119890

) minus 1 (5)

where119879119890

is the temperature of the surface of the BH candidatemeasured by a locally corotating observer

For the sake of simplicity we now consider a spherically-symmetric nonrotating object The geometry of the space-time around the BH candidate will be described by theSchwarzschild solution which is valid till the radius of thecompact object 119877 The luminosity becomes as follows

119871 = 41205901198924

1198794

119890

int119889119909119889119910 (6)

where 120590 is the Stefan-Boltzmann constant and

119892 = (1 minus2119872

119877)

12

(7)

Here 119892 is a constant but it would be a function of 119909 and 119910in a more general backgroundThe integrand in (6) is simplythe area of the apparent image of the BH candidate on theobserverrsquos sky

int119889119909119889119910 = 1205871198772

app =

271205871198722

119877 lt 3119872

1205871198772

1198922119877 gt 3119872

(8)

The radius 119903 = 3119872 is the capture photon radius of theSchwarzschild spacetime Inside such a radius the gravita-tional force is so strong that any light ray coming from infinityis captured by the compact object

A distant observer sees therefore an object with anapparent temperature

119879app = 119892119879119890 asymp 119879119890(120575

2119872)

12

(9)

where I wrote 119877 = 2119872 + 120575 and assumed 120575 ≪ 1 and positiveThe most stringent constraint on 120575 can be inferred from theobservations of the supermassive BH candidate at the centerof our Galaxy Infrared and near-infrared data require 119879app lt001 eV [12 13] If we assume a local temperature as high as119896119861

119879119890

sim 119898119901

1198882

sim 1GeV (roughly the gravitational bindingenergy of a proton) we find the following

120575 lt 10minus10 cm (10)

The Scientific World Journal 3

as 119872 asymp 6 sdot 1011 cm With a lower temperature 119879

119890

theconstraint would be weaker while a higher temperatureseems to be unlikely as the object is old and the accretinggas would have already cooled it down The proper distanceof the boundary of the BH candidate from the event horizonof a Schwarzschild BH with the same mass is as follows

Δ asymp120575

radic1 minus (2119872119877)asymp radic2119872120575 lt 10 cm (11)

Such a result should not change significantly if we consider arotating object

3 Stability Constraint

The existence of event or apparent horizons in astrophysicalBH candidates is also suggested by considerations concerningthe stability of these objects It is well known that rapidlyrotating very compact objects may be affected by the ergore-gion instability [16ndash19] In the ergoregion 119892

119905119905

gt 0 (if themetric has signature minus + ++) and the frame dragging is sostrong that stationary orbits are not allowedThat implies thatin the ergoregion there are excitations with negative energywith respect to a stationary observer at infinity These excita-tions can be seen as quasi-bound states they are trapped bythe gravitational potential on the one side and by the surfaceof the object (or by the center of the object if the latter ismade of matter noninteracting with the excitations) on theother side As some modes can escape to infinity carryingpositive energy negative energy modes in the ergoregion cangrow indefinitely thus generating an instability Objects witha horizon may instead be stable because there may not bequasi-bound states in the ergoregion any excitation in theergoregion is swallowed by the BH Let us notice howeverthat the existence of a horizon is not sufficient in general toprevent the ergoregion instability [20]

Roughly speaking the instability timescale 120591 decreasesas the angular velocity and the compactness of the compactobject increase For rotating very compact objects onetypically finds that the instability is strong and occurs on adynamical timescale 120591 sim 119872 [21] that is sim1 s for objectswith a mass 119872 sim 10119872

⊙

and sim107 s if 119872 sim 108

119872⊙

While there are counterexamples in which rotating compactobjects can be stable or very long living [22] it seems difficultthat the latter can still meet observations requiring thatastrophysical BH candidates can rotate very rapidly [23ndash27]and have a high radiative efficiency [28ndash31] Let us noticehowever that the issue of the ergoregion instability canbe discussed only within a well-defined theoretical model(gravity theory internal structure and composition of thecompact object etc) and that it has been studied only fora very limited number of specific cases Considerations onthe nonobservations of electromagnetic radiation from thesurface of BH candidates are much more model independentand rely on a set of assumptions that can be violated onlyinvoking very exotic new physics

Here I will discuss the ergoregion instability within thefollowing picture I assume that the geometry around anastrophysical BH candidate is exactly described by the Kerr

solution up the radius of the compact object 119877 Considera-tions on the ergoregion instability indeed require a specificbackground and we may think that possible deviations fromthe Kerr metric can be tested with other approaches [32]In the case of a reflecting surface the timescale for scalarinstabilities can be estimated as follows [33]

120591 sim 119860 (119872 119886lowast

)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ln( 119877 minus 119877119867

2119872radic1 minus 1198862lowast

)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(12)

where119877119867

= 119872(1+radic1 minus 1198862lowast

) is the radius of the event horizonof a Kerr BH with mass119872 and spin parameter 119886

lowast

119860(119872 119886lowast

)

is a function of 119872 and 119886lowast

For moderate values of the spinparameter 119886

lowast

119860 sim 119872 that is the instability occurs on adynamical timescale For high values of 119886

lowast

119860 decreases veryquickly In the case of a Kerr BH 119877 = 119877

119867

and the object isstable On the other hand if 119877 = 119877

119867

+ 120575 the fact that weobserve long-living rapidly rotating BH candidates demands

120575 Δ ≪ 119871Pl asymp 10minus33 cm (13)

where Δ is the physical distance encountered in the previoussection Equation (13) essentially rules out the possibility thatcurrent BH candidates have no horizon or at least somethingthat behaves verymuch like a horizon for the unstablemodesThe possibility of an exact Kerr background with 120575 so largethat there is no ergoregion seems to be unlikely as we knowobjects that when the spacetime around them is describedby the Kerr solution would have an accretion disk with inneredge inside the ergosphere [23ndash25]

4 Conclusions

In conclusion we have observations suggesting that BHcandidates have a horizon or at least putting constraints onthe possible distance between the boundary of these compactobjects and the event horizon of a BHwith the samemass andspin Such a distance can be seen as a measurement of howmuch close the formation of the horizon is From the nonob-servation of thermal radiation from the putative surface ofastrophysical BH candidates one can infer the constraint in(10) and (11) actually such a bound is not so stringent asone may argue that new physics can show up at much shorterscales However the result seems to be quite robustmdashit is justsupposed that the compact object must emit electromagneticradiation due to its finite temperaturemdashand very exotic newphysics is necessary to change these conclusions or to get adifferent bound Considerations on the ergoregion instabilityare instead to be taken with cautionThe timescale instabilitystrongly depends on the exact model that is gravity theoryinternal structure and composition of the object and soon which we do not know However we can optimisticallyarrive at the following conclusion If the geometry aroundastrophysical BH candidates is very close to the Kerr solutionthe existence of stable or long-living objects likely requiressome kind of horizon Otherwise we can probably hopeto discover deviations from the Kerr background with testsalready proposed in the literature and possible in a near futurewith new observational facilities


Acknowledgment

This work was supported by the Humboldt Foundation

References

[1] E Poisson A Relativistrsquos Toolkit The Mathematics of Black-HoleMechanics Cambridge University Press Cambridge UK 2004

[2] A Ashtekar and M Bojowald ldquoBlack hole evaporation aparadigmrdquo Classical and Quantum Gravity vol 22 no 16article 3349 2005

[3] A Ashtekar and M Bojowald ldquoQuantum geometry and theSchwarzschild singularityrdquo Classical and Quantum Gravity vol23 no 2 article 391 2006

[4] S W Hawking ldquoInformation loss in black holesrdquo PhysicalReview D vol 72 no 8 Article ID 084013 4 pages 2005

[5] R Narayan ldquoBlack holes in astrophysicsrdquo New Journal ofPhysics vol 7 no 1 article 199 2005

[6] C E Rhoades and R Ruffini ldquoMaximum mass of a neutronstarrdquo Physical Review Letters vol 32 no 6 pp 324ndash327 1974

[7] V Kalogera and G Baym ldquoThe maximum mass of a neutronstarrdquoTheAstrophysical Journal Letters vol 470 no 1 article L611996

[8] E Maoz ldquoDynamical constraints on alternatives to supermas-sive black holes in galactic nucleirdquo The Astrophysical JournalLetters vol 494 no 2 article L181 1998

[9] J E McClintock R Narayan and G B Rybicki ldquoOn the lack ofthermal emission from the quiescent black hole XTE J1118+480evidence for the event horizonrdquo The Astrophysical Journal vol615 no 1 article 402 2004

[10] R Narayan and J E McClintock ldquoAdvection-dominated accre-tion and the black hole event horizonrdquoNewAstronomy Reviewsvol 51 no 10ndash12 pp 733ndash751 2008

[11] R Narayan and J S Heyl ldquoOn the lack of type I X-ray bursts inblack hole X-ray binaries evidence for the event horizonrdquoTheAstrophysical Journal Letters vol 574 no 2 article L139 2002

[12] A E Broderick and R Narayan ldquoWhere are all the gravastarsLimits upon the gravastar model from accreting black holesrdquoClassical and Quantum Gravity vol 24 no 3 article 659 2007

[13] A E Broderick A Loeb and R Narayan ldquoThe event horizon ofsagittarius AlowastrdquoThe Astrophysical Journal vol 701 no 2 article1357 2009

[14] M A Abramowicz W Kluzniak and J P Lasota ldquoNo obser-vational proof of the black-hole event-horizonrdquo Astronomy andAstrophysics vol 396 no 3 pp L31ndashL34 2002

[15] G Dvali and C Gomez ldquoBlack holersquos quantum N-portraitrdquohttparxivorgabs11123359

[16] J L Friedman ldquoErgosphere instabilityrdquo Communications inMathematical Physics vol 63 no 3 pp 243ndash255 1978

[17] H Sato and K I Maeda ldquoInstability of a quantum field inthe curved space-time of a rotating starrdquo Progress of TheoreticalPhysics vol 59 no 4 pp 1173ndash1187 1978

[18] N Comins and B F Schutz ldquoOn the ergoregion instabilityrdquoProceedings of the Royal Society A vol 364 no 1717 pp 211ndash2261978

[19] S Yoshida and Y Eriguchi ldquoErgoregion instability revisitedmdasha new and general method for numerical analysis of stabilityrdquoMonthly Notices of the Royal Astronomical Society vol 282 no2 pp 580ndash586 1996

[20] P Pani E Barausse E Berti and V Cardoso ldquoGravitationalinstabilities of superspinarsrdquo Physical Review D vol 82 no 4Article ID 044009 15 pages 2010

[21] V Cardoso P Pani M Cadoni and M Cavaglia ldquoErgore-gion instability of ultracompact astrophysical objectsrdquo PhysicalReview D vol 77 no 12 Article ID 124044 15 pages 2008

[22] C B M H Chirenti and L Rezzolla ldquoErgoregion instability inrotating gravastarsrdquo Physical Review D vol 78 no 8 Article ID084011 11 pages 2008

[23] J E McClintock R Shafee R Narayan R A Remillard S WDavis and L X Li ldquoThe spin of the near-extreme Kerr blackhole GRS 1915+105rdquo The Astrophysical Journal vol 652 no 1article 518 2006

[24] L W Brenneman and C S Reynolds ldquoConstraining black holespin via X-ray spectroscopyrdquoTheAstrophysical Journal vol 652no 2 article 1028 2006

[25] L W Brenneman C S Reynolds M A Nowak et al ldquoThe spinof the supermassive black hole in NGC 3783rdquoThe AstrophysicalJournal vol 736 no 2 article 103 2011

[26] R Narayan and J E McClintock ldquoObservational evidence fora correlation between jet power and black hole spinrdquo MonthlyNotices of the Royal Astronomical Society vol 419 no 1 pp L69ndashL73 2012

[27] C Bambi ldquoTesting the Kerr nature of stellar-mass black holecandidates by combining the continuum-fittingmethod and thepower estimate of transient ballistic jetsrdquo Physical ReviewD vol85 no 4 Article ID 043002 8 pages 2012

[28] J M Wang Y M Chen L C Ho and R J McLure ldquoEvidencefor rapidly spinning black holes in quasarsrdquo The AstrophysicalJournal Letters vol 642 no 2 article L111 2006

[29] S W Davis and A Laor ldquoThe radiative efficiency of accretionflows in individual active galactic nucleirdquo The AstrophysicalJournal vol 728 no 2 article 98 2011

[30] C Bambi ldquoConstraint on the quadrupole moment of super-massive black hole candidates from the estimate of the meanradiative efficiency of AGNrdquo Physical Review D vol 83 no 10Article ID 103003 4 pages 2011

[31] C Bambi ldquoTowards the use of the most massive black holecandidates in active galactic nuclei to test the Kerr paradigmrdquoPhysical Review D vol 85 no 4 Article ID 043001 11 pages2012

[32] C Bambi ldquoTesting the Kerr black hole hypothesisrdquo ModernPhysics Letters A vol 26 no 33 p 2453 2011

[33] S Mukohyama ldquoIs the brick-wall model unstable for a rotatingbackgroundrdquo Physical Review D vol 61 no 12 Article ID124021 9 pages 2000

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


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Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

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Biophysics


ThermodynamicsJournal of


event horizon cannot be seen by distant observers any more(see however ref [14]) Strictly speaking the confirmationfor the existence of an event horizon would require theknowledge of the future null infinity of the Universe whichis clearly impossible for us On the contrary the nonobser-vation of electromagnetic radiation emitted by the gas afterfalling into the compact object nicely meets the definition ofapparent horizon However the geometry of the spacetimearound astrophysical BH candidates is practically stationaryfor the timescale of our observations and that may make itimpossible to discriminate an event horizon froman apparenthorizon

2 Electromagnetic Constraint

Let us image a BHas a gas of particles packed in a small regionby the gravitational force (The model of BH I will considermay remind of the one discussed in [15] The radius of thecompact object 119877 is larger than the one corresponding tothe event horizon of a (classical) BH with the same mass andspin) As this gas has a finite temperature it must radiateHowever if the object is very compact the emitted radiationis strongly redshifted when it reaches a distant observer andthe object can appear very faint Here I relax the quitecommon assumption of steady state 119871 = 119888

2 [9 10 12 13]where 119871 is the surface luminosity and is themass accretionrate That would require that the accreting gas hits the ldquosolidsurfacerdquo of the object and then radiates to infinity all itskinetic energy If this were the case a very compact objectwould not be able to increase its mass or at least the processwould be very inefficient likely in contradiction with theobservations of the supermassive objects in galactic nucleiMoreover there are no reasons to assume that BH candidateshave a solid surface In the picture in which we have a gas ofparticles packed in a small region by the gravitational forcethe accreting gas enters into the compact object and bothits rest-mass and kinetic energy contribute to increasing themass of the BH candidate

Let us now see the constraint we can obtain in this picturefrom the nonobservation of thermal spectrum from BHcandidates The specific energy flux density of the compactobject (often measured in erg cmminus2 sminus1Hzminus1) as detected by adistant observer is as follows

119865 = int 119868119900

119889Ω (2)

where 119868119900

is the specific intensity of the radiation as measuredby the distant observer and 119889Ω is the element of the solidangle subtended by the image of the object on the observerrsquossky 119868119909

]3119909

= const (Liouvillersquos Theorem) where ]119909

is thephoton frequency measured by any local observer on thephoton path and

119889Ω =119889119909119889119910

1198632 (3)

where 119909 and 119910 are the Cartesian coordinates on the observerrsquossky and 119863 is the distance of the compact object from

the observer The equivalent isotropic luminosity of the BHcandidate is thus

119871 = 4120587int1198923

119868119890

119889119909 119889119910119889] (4)

Here 119892 = ]119900

]119890

is the redshift factor ]119900

is the photonfrequencymeasured by the distant observer and ]

119890

and 119868119890

arerespectively the photon frequency and the specific intensityof the radiation measured by an observer located at the pointof emission of the photon on the surface of the compactobject and corotating with the surface of the compact objectThe emission should be like the one of a blackbody that is

119868119890

=2ℎ]3119890

1198882

1

exp (ℎ]119890

119896119861

119879119890

) minus 1 (5)

where119879119890

is the temperature of the surface of the BH candidatemeasured by a locally corotating observer

For the sake of simplicity we now consider a spherically-symmetric nonrotating object The geometry of the space-time around the BH candidate will be described by theSchwarzschild solution which is valid till the radius of thecompact object 119877 The luminosity becomes as follows

119871 = 41205901198924

1198794

119890

int119889119909119889119910 (6)

where 120590 is the Stefan-Boltzmann constant and

119892 = (1 minus2119872

119877)

12

(7)

Here 119892 is a constant but it would be a function of 119909 and 119910in a more general backgroundThe integrand in (6) is simplythe area of the apparent image of the BH candidate on theobserverrsquos sky

int119889119909119889119910 = 1205871198772

app =

271205871198722

119877 lt 3119872

1205871198772

1198922119877 gt 3119872

(8)

The radius 119903 = 3119872 is the capture photon radius of theSchwarzschild spacetime Inside such a radius the gravita-tional force is so strong that any light ray coming from infinityis captured by the compact object

A distant observer sees therefore an object with anapparent temperature

119879app = 119892119879119890 asymp 119879119890(120575

2119872)

12

(9)

where I wrote 119877 = 2119872 + 120575 and assumed 120575 ≪ 1 and positiveThe most stringent constraint on 120575 can be inferred from theobservations of the supermassive BH candidate at the centerof our Galaxy Infrared and near-infrared data require 119879app lt001 eV [12 13] If we assume a local temperature as high as119896119861

119879119890

sim 119898119901

1198882

sim 1GeV (roughly the gravitational bindingenergy of a proton) we find the following

120575 lt 10minus10 cm (10)



119890


Δ asymp120575





119905119905



⊙


119872⊙




120591 sim 119860 (119872 119886lowast

)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ln( 119877 minus 119877119867


)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(12)

where119877119867



lowast

119860(119872 119886lowast

)



lowast


lowast


119867


119867




4 Conclusions



Acknowledgment


References







































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119890


Δ asymp120575





119905119905



⊙


119872⊙




120591 sim 119860 (119872 119886lowast

)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ln( 119877 minus 119877119867


)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(12)

where119877119867



lowast

119860(119872 119886lowast

)



lowast


lowast


119867


119867




4 Conclusions



Acknowledgment


References







































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Acknowledgment


References







































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Documents

Research Article A Note on the Observational Evidence for ...downloads.hindawi.com/journals/tswj/2013/204315.pdf · A Note on the Observational Evidence for the Existence of Event