Submit manuscript...
eISSN: 2576-4543

Physics & Astronomy International Journal

Mini Review Volume 2 Issue 4

Gravitional potential generated by an arc bent into an elliptical shape

Nour Eddine Najid

Faculty of Sciences, University Hassan II Casablanca, Morocco

Correspondence: Nour Eddine Najid, Faculty of Sciences, University Hassan II Casablanca, BP 5366, Ma, Tel 0021?2522?2306?80, Fax 0021?2522?2306?74

Received: April 25, 2018 | Published: July 12, 2018

Citation: Najid NE. Gravitional potential generated by an arc bent into an elliptical shape. Phys Astron Int J. 2018;2(4):297-298. DOI: 10.15406/paij.2018.02.00102

Download PDF

Abstract

In this Study we plane to calculate the gravitational potential generated by a ring bent into an elliptical mass distribution. In a first part we suppose that the mass distribution is homogeneous, after we take the case in which the mass distribution is inhomogeneous or anisotropic. The aim of this study is to generalise that of circular case and work out the behaviour of a test particle in the vicinity of that distribution. The rings around asteroids are recently discovered. This study will have an impact concerning the behaviour of a test particle moving around these distributions.

Keywords: potential–elliptic functions–gravitation, static case, dynamical case, anisotropic distribution, mass

Introduction

In an earlier work Najid et al.,1 we studied the gravitational potential generated by an inhomogeneous asteroid with a parabolic mass distribution. We established the close form of this potential. It allows us to study both, the static and dynamical case in which a test particle is near the distribution.

In another work Najid et al.,2 we studied the case of a circular ring with an anisotropic distribution of mass. The mass distribution depends on the direction of observation. We manage to work out the problem in terms of elliptic functions.

In this work, we plane to generalise the case of ring by scrutinizing a more realistic case. In general, around all structures the rings are elliptical, and even with inhomogeneous or anisotropic mass distribution. In the literature we find studies like ellipsoids, material points and a segment of double material were used.3,4 Fred et al.,5 1982established the expressions for both the potential and the field of a disk. They suggested a formula of a ring. Anisotropic and inhomgeneous distributions are our new ideas.

Gravitational potential generated by elliptical distribution in its plane

We will express the potential dV in a point P(x,y) generated by the infinitesimal point M of mass dm in the plane of the ellipse (Figure 1)

Figure 1 Gravitational potential generated by elliptical distribution at plane.

dV is given by:

dV=G dm MP MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadAfacqGH9aqpcqGHsislcaWG hbqcfa4aaSaaaOWdaeaajugib8qacaWGKbGaamyBaaGcpaqaaKqbao aafmaakeaajuaGdaWhcaGcbaqcLbsapeGaamytaiaadcfaaOWdaiaa wEniaaGaayzcSlaawQa7aaaaaaa@49A6@

with

FM=r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOraiaad2eacqGH9aqpcaWGYbaaaa@3D09@ and FP=ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOraiaadcfacqGH9aqpcqaHbpGCaaa@3DD4@ .
( r,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajuaGqaaaaaaaaaWdbmaabmaak8aabaqcLbsapeGaamOCaiaacYca cqaH4oqCaOGaayjkaiaawMcaaaaa@3F16@ are the polar coordinates of M.
( ρ,φ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajuaGqaaaaaaaaaWdbmaabmaak8aabaqcLbsapeGaeqyWdiNaaiil aiabeA8aQbGccaGLOaGaayzkaaaaaa@3FE6@ are the polar coordinates of P.
MP = FP FM MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajuaGdaWhcaGcbaqcLbsaqaaaaaaaaaWdbiaad2eacaWGqbaak8aa caGLxdcajugib8qacqGH9aqpjuaGpaWaa8HaaOqaaKqzGeWdbiaadA eacaWGqbaak8aacaGLxdcajugib8qacqGHsisljuaGpaWaa8HaaOqa aKqzGeWdbiaadAeacaWGnbaak8aacaGLxdcaaaa@4A0F@ .
dm=λrdθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaad2gacqGH9aqpcqaH7oaBcaWG YbGaamizaiabeI7aXbaa@419A@ , λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaeq4UdWgaaa@3B23@ is the linear mass of the ellipse.

We deduce

dV=G λrdθ r 2 + ρ 2 2ρrcos( θφ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadAfacqGH9aqpcqGHsislcaWG hbqcfa4aaSaaaOWdaeaajugib8qacqaH7oaBcaWGYbGaamizaiabeI 7aXbGcpaqaaKqba+qadaGcaaGcpaqaaKqzGeWdbiaadkhal8aadaah aaqcbasabeaajugWa8qacaaIYaaaaKqzGeGaey4kaSIaeqyWdi3cpa WaaWbaaKqaGeqabaqcLbmapeGaaGOmaaaajugibiabgkHiTiaaikda cqaHbpGCcaWGYbGaam4yaiaad+gacaWGZbqcfa4aaeWaaOWdaeaaju gib8qacqaH4oqCcqGHsislcqaHgpGAaOGaayjkaiaawMcaaaWcbeaa aaaaaa@5E00@

The origin of the coordinates is at the focus

If we take the origin at the focus F, we are getting :

r= a( 1 e 2 ) 1+ecosθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOCaiabg2da9Kqbaoaalaaak8aabaqc LbsapeGaamyyaKqbaoaabmaak8aabaqcLbsapeGaaGymaiabgkHiTi aadwgal8aadaahaaqcbasabeaajugWa8qacaaIYaaaaaGccaGLOaGa ayzkaaaapaqaaKqzGeWdbiaaigdacqGHRaWkcaWGLbGaam4yaiaad+ gacaWGZbGaeqiUdehaaaaa@4D32@

Where a is the semi axis of the ellipse and e is the eccentricity

V=Ga( 1 e 2 ) θ=0 θ=2π λ( θ )dθ a 2 ( 1 e 2 ) 2 + ρ 2 ( 1+ecosθ ) 2 2ρa( 1 e 2 )( 1+ecosθ )cos( θφ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOvaiabg2da9iabgkHiTiaadEeacaWG Hbqcfa4aaeWaaOWdaeaajugib8qacaaIXaGaeyOeI0IaamyzaSWdam aaCaaajeaibeqaaKqzadWdbiaaikdaaaaakiaawIcacaGLPaaajuaG daWdXbqaamaalaaapaqaaKqzGeWdbiabeU7aSLqbaoaabmaapaqaaK qzGeWdbiabeI7aXbqcfaOaayjkaiaawMcaaKqzGeGaamizaiabeI7a Xbqcfa4daeaapeWaaOaaa8aabaqcLbsapeGaamyyaSWdamaaCaaaju aibeqaaKqzadWdbiaaikdaaaqcfa4aaeWaa8aabaqcLbsapeGaaGym aiabgkHiTiaadwgal8aadaahaaqcfasabeaajugWa8qacaaIYaaaaa qcfaOaayjkaiaawMcaaSWdamaaCaaajuaibeqaaKqzadWdbiaaikda aaqcLbsacqGHRaWkcqaHbpGCl8aadaahaaqcfasabeaajugWa8qaca aIYaaaaKqbaoaabmaapaqaaKqzGeWdbiaaigdacqGHRaWkcaWGLbGa am4yaiaad+gacaWGZbGaeqiUdehajuaGcaGLOaGaayzkaaWcpaWaaW baaKqbGeqabaqcLbmapeGaaGOmaaaajugibiabgkHiTiaaikdacqaH bpGCcaWGHbqcfa4aaeWaa8aabaqcLbsapeGaaGymaiabgkHiTiaadw gajuaGpaWaaWbaaeqajuaibaqcLbmapeGaaGOmaaaaaKqbakaawIca caGLPaaadaqadaWdaeaajugib8qacaaIXaGaey4kaSIaamyzaiaado gacaWGVbGaam4CaiabeI7aXbqcfaOaayjkaiaawMcaaKqzGeGaae4y aiaab+gacaqGZbqcfa4aaeWaa8aabaqcLbsapeGaaeiUdiabgkHiTi aabA8aaKqbakaawIcacaGLPaaaaeqaaaaaaKqbGeaajugWaiabeI7a Xjabg2da9iaaicdaaKqbGeaajugWaiabeI7aXjabg2da9iaaikdacq aHapaCaKqzGeGaey4kIipaaaa@A299@

The origin of the coordinates is at the center

In this case r is given by (Figure 2)

r= b 1 e 2 co s 2 θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOCaiabg2da9Kqbaoaalaaak8aabaqc LbsapeGaamOyaaGcpaqaaKqba+qadaGcaaGcpaqaaKqzGeWdbiaaig dacqGHsislcaWGLbWcpaWaaWbaaKqaGeqabaqcLbmapeGaaGOmaaaa jugibiaadogacaWGVbGaam4CaSWdamaaCaaajeaibeqaaKqzadWdbi aaikdaaaqcLbsacqaH4oqCaSqabaaaaaaa@4C2D@

Figure 2 Gravitational potential generated by elliptical distribution at origin.

b is the semi–minor axis of the ellipse.
In this case
OM=r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaam4taiaad2eacqGH9aqpcaWGYbaaaa@3D12@ and OP=ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaam4taiaadcfacqGH9aqpcqaHbpGCaaa@3DDE@
( r,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajuaGqaaaaaaaaaWdbmaabmaak8aabaqcLbsapeGaamOCaiaacYca cqaH4oqCaOGaayjkaiaawMcaaaaa@3F16@ are the polar coordinates of M.
( ρ,φ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajuaGqaaaaaaaaaWdbmaabmaak8aabaqcLbsapeGaeqyWdiNaaiil aiabeA8aQbGccaGLOaGaayzkaaaaaa@3FE6@ are the polar coordinates of P
MP = OP OM MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajuaGdaWhcaGcbaqcLbsaqaaaaaaaaaWdbiaad2eacaWGqbaak8aa caGLxdcajugib8qacqGH9aqpjuaGpaWaa8HaaOqaaKqzGeWdbiaad+ eacaWGqbaak8aacaGLxdcajugib8qacqGHsisljuaGpaWaa8HaaOqa aKqzGeWdbiaad+eacaWGnbaak8aacaGLxdcaaaa@4A21@

In this case we get :

dV=Gb λ( θ )dθ b 2 + ρ 2 ( 1 e 2 co s 2 θ )2bρcos( θφ ) 1 e 2 co s 2 θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadAfacqGH9aqpcqGHsislcaWG hbGaamOyaKqbaoaalaaak8aabaqcLbsapeGaeq4UdWwcfa4aaeWaaO Wdaeaajugib8qacqaH4oqCaOGaayjkaiaawMcaaKqzGeGaamizaiab eI7aXbGcpaqaaKqba+qadaGcaaGcpaqaaKqzGeWdbiaadkgal8aada ahaaqcbasabeaajugWa8qacaaIYaaaaKqzGeGaey4kaSIaeqyWdi3c paWaaWbaaKqaGeqabaqcLbmapeGaaGOmaaaajuaGdaqadaGcpaqaaK qzGeWdbiaaigdacqGHsislcaWGLbqcfa4damaaCaaaleqajeaibaqc LbmapeGaaGOmaaaajugibiaadogacaWGVbGaam4CaSWdamaaCaaaje aibeqaaKqzadWdbiaaikdaaaqcLbsacqaH4oqCaOGaayjkaiaawMca aKqzGeGaeyOeI0IaaGOmaiaadkgacqaHbpGCcaqGJbGaae4Baiaabo hajuaGdaqadaGcpaqaaKqzGeWdbiabeI7aXjabgkHiTiabeA8aQbGc caGLOaGaayzkaaqcfa4aaOaaaOWdaeaajugib8qacaaIXaGaeyOeI0 IaamyzaSWdamaaCaaajeaibeqaaKqzadWdbiaaikdaaaqcLbsacaWG JbGaam4Baiaadohal8aadaahaaqcbasabeaajugWa8qacaaIYaaaaK qzGeGaeqiUdehaleqaaaqabaaaaaaa@8199@

Simplified solution–Particular case.2 In the case of a circular ring of radius a and λ( θ )= λ 0 ( 1+dco s 2 θ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaeq4UdWwcfa4aaeWaaOWdaeaajugib8qa cqaH4oqCaOGaayjkaiaawMcaaKqzGeGaeyypa0Jaeq4UdW2cpaWaaS baaKqaGeaajugWa8qacaaIWaaajeaipaqabaqcfa4dbmaabmaak8aa baqcLbsapeGaaGymaiabgUcaRiaadsgacaWGJbGaam4Baiaadohal8 aadaahaaqcbasabeaajugWa8qacaaIYaaaaKqbaoaalaaak8aabaqc LbsapeGaeqiUdehak8aabaqcLbsapeGaaGOmaaaaaOGaayjkaiaawM caaaaa@54B4@ . Where λ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaeq4UdW2cpaWaaSbaaKqaGeaajugWa8qa caaIWaaajeaipaqabaaaaa@3DB9@ and d are constants, in the plane xy the potential is given by:

V= 4G λ 0 a p k 2 [ ( k 2 +d )K( k )dE( k ) ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOvaiabg2da9iabgkHiTKqbaoaalaaa k8aabaqcLbsapeGaaGinaiaadEeacqaH7oaBl8aadaWgaaqcbasaaK qzadWdbiaaicdaaKqaG8aabeaajugib8qacaWGHbaak8aabaqcLbsa peGaamiCaiaadUgajuaGpaWaaWbaaSqabKqaGeaajugWa8qacaaIYa aaaaaajuaGdaWadaGcpaqaaKqba+qadaqadaGcpaqaaKqzGeWdbiaa dUgal8aadaahaaqcbasabeaajugWa8qacaaIYaaaaKqzGeGaey4kaS IaamizaaGccaGLOaGaayzkaaqcLbsacaWGlbqcfa4aaeWaaOWdaeaa jugib8qacaWGRbaakiaawIcacaGLPaaajugibiabgkHiTiaadsgaca WGfbqcfa4aaeWaaOWdaeaajugib8qacaWGRbaakiaawIcacaGLPaaa aiaawUfacaGLDbaaaaa@61C3@

p:The largest distance between P and the ring.
K(k):The complete elliptic integral of first kind
E(k ): The complete elliuptic integral of second kind (Handbook of Mathematics)

k: k 2 =1 q 2 p 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaam4AaiaacQdacaWGRbWcpaWaaWbaaKqa GeqabaqcLbmapeGaaGOmaaaajugibiabg2da9iaaigdacqGHsislju aGdaWcaaGcpaqaaKqzGeWdbiaadghajuaGpaWaaWbaaSqabKqaGeaa jugWa8qacaaIYaaaaaGcpaqaaKqzGeWdbiaadchajuaGpaWaaWbaaS qabKqaGeaajugWa8qacaaIYaaaaaaaaaa@4B7E@ .

q:The smallest distance between P and the ring.

Conclusion

The calculation of V depends on the expression of λ( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaeq4UdWwcfa4aaeWaaOWdaeaajugib8qa cqaH4oqCaOGaayjkaiaawMcaaaaa@3FB2@ . According to this we expect to have an elliptical integral, in a close form. With V we study the behaviour of a test particle in the vicinity of the distribution.6

Acknowledgements

None.

Conflict of interest

Author declares there is no conflict of interest.

References

Creative Commons Attribution License

©2018 Najid. This is an open access article distributed under the terms of the, which permits unrestricted use, distribution, and build upon your work non-commercially.