Formation of Martiain Magnetic Field and Its Distribution

ELSEVIER Chinese Astronomy and Astrophysics 34(2010)406–4120275-1062/09/$-see front matter © 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.chinastron.2010.10.010Chinese Astronomy and Astrophysics 33 (2010) 406–4120275-1062/10/$-see front matter © 2010 Elsevier B.V Chinese Astronomy and Astrophysics 34 (2010) 406–412TONG Dong-sheng et al. /Chinese Astronomy and Astrophysics 34 (2010) 406–412 407 to Mars,such as Mariner,Martian Satellite No.2and Mars Global Surveyor(MGS).The observational materials of these spacecraft have revealed that,unlike the Earth,Mars itself does not possess a rather strong magneticfield[3,4].It is now generally believed that the existence of a proper magneticfield cannot be excluded,yet it is very weak and may be effective only in low altitudes.The main part of Martian magneticfield is still the induction magneticfield produced by Mars with solar wind[3].Mars can be regarded as a non-magnetic planet.Its interaction with solar wind is similar with that of Venus[5]and differs much from the interaction between the earth and solar wind.On Mars,the intense solar ultraviolet radiation can ionize the upper atmospheric layer and form a hot neutral atmospheric layer,which extends to the realm of solar wind.The ionospheric pressure,which is composed of thermal and magnetic pressures,is in equilibrium with the pressure of solar wind stream.After entering the solar wind,the neutral atmosphere is ionized.It can supplement the solar wind stream and lessen the velocity of solar wind. Due to this kind of interaction,the magneticfield,which is perpendicular to solar wind, becomes curved.To the space plasmas,the action of Mars and its ambient atmosphere is like that of an obstacle.The magneticfield around Mars is linked with this obstacle.By the method of simulation,Kallio and Janhunen[6]obtained the distribution of magneticfield on Martian equatorial plane.Starting from the above-mentioned model,one may suggest that in the Martian iono-sphere there are uniformly distributed electric currents,that the shape of magnetopause is parabolic and that the current sheets of magnetotail lie on an infinitely long plane.Because electric currents are closed,so the system of electric currents under these three background conditions satisfies a certain ly,the total current in magnetopause is equal to the sum of the total current in ionosphere and the total current of the current sheets of mag-netotail.According to Ampere’s law and via numerical calculations,we have obtained the magneticfield distribution on Martian equatorial plane and furthermore some meaningful results.2.MODEL OF MARTIAN INDUCTION MAGNETIC FIELD2.1The Structure of Martian Magnetosphere and Related ParametersThe magnetospheric structure on Martian equatorial plane is illustrated in Fig.1.The center of the coordinate system is Mars,the x-axis points to the sun,and the direction of the y-axis is opposite to that of the velocity on Martian orbit.This system of coordinates is commonly called as the Mars-centered solar orbital coordinate system(MSO).Owing to the interaction between Mars and solar wind,the ambient atmosphere of Mars is ionized into plasma.Under the action of the induction magneticfield E sw=−v sw×B sw, the plasma in ionosphere produces induction current.From Ohm’s law,i.e.,j=σE sw,the magnitude of induction current can be obtained.Here j is the density of induction current in Martian ionosphere andσis the electrical conductivity of Martian ionosphere.For the sake of convenience,we assume that the induction current is homogeneously distributed in ionosphere and the density of induction current[6]is j=10nA/m2.From Ref.[10]it is known that the thickness of Martian ionosphere is200km and its height above the ground surface of Mars is200-400km.The radius of Mars is R m=3397km.The strength of408TONG Dong-sheng et al. /Chinese Astronomy and Astrophysics 34 (2010) 406–412Fig.1Magnetospheric structure on Martian equatorial planemagneticfield of solar wind is B sw=4nT.Its direction is the same as that of y-axis.Herein we consider only the case that the solar wind magneticfield B sw is perpendicular to the velocity of solar wind v sw.As for the shape and position of Martian magnetosphere,until present there are no definite conclusions.However,according to the observational data of the satellite MGS,the Martian magnetopause can be represented with a paraboloid.Now we like to ignore the thickness of Martian magnetopause and suppose that its shape is a paraboloid which can be expressed by x=−0.5y2+1.25.For the distribution of electric current on the magnetopause we like to probe the two cases of homogeneous and non-homogeneous distributions.When the distribution of electric current on magnetopause is homogeneous,we take the electric current to be i1=0.0007A/m.When it is non-homogeneous,it is symmetric with respect to the y-axis.Its distribution can be represented with the following expressions:⎧⎨⎩i1=−0.0025.9y+5.9y+TONG Dong-sheng et al. /Chinese Astronomy and Astrophysics 34 (2010) 406–412 409 symmetric distribution with respect to the y-axis,we havei1=−0.0024.76y+4.76y+Hereμ0=4π×10−7N·A−2is the magnetic conductivity in vacuum,i1is the electric current on Martian magnetopause and i2is that on the current sheet of Martian magnetotail.The coordinates(x1,y1)represent the position of current sheet in Martian magnetopause.x1=−0.5y21+1.25,−5.9≤y1≤5.9.x2expresses the position of current sheet in magnetotail.−13≤x2≤−3,y2=0.The unit of the above magneticfield strengths is nT.In the plane rectangular coordinate system and by using the relation between the magneticfield strength B=(B x,B y)and magneticflux functionΨ,i.e.,B=∇Φ×e z=(∂Ψ∂y,−∂x),410 TONG Dong-sheng et al. /Chinese Astronomy and Astrophysics 34 (2010) 406–412In the above expression of magnetic flux function Ψ,the 2nd and 3rd terms are integrals with respect to electric current.Due to the difficulty of getting analytical solutions,we can only make numerical solutions.In the process of seeking numerical solutions,the trapezoidal method is ly,we haveb a f (x )d x =lim N →∞N 1f (x n ) x n .In this expression the width of every small trapezoid is Δx =b −a N .TFig.2Distribution of magnetic field on Martian equa-torial plane under the condition of a uniform currentdistribution in magnetopause with the configurationgiven by x =−0.5y 2+1.25Fig.3Distribution of magnetic field on Mar-tian equatorial plane under the condition of a non-uniform current distribution in magne-topause with the configuration given by x =−0.5y 2+1.25TONG Dong-sheng et al. /Chinese Astronomy and Astrophysics 34 (2010) 406–412411Fig.4Distribution of magneticfield on Martian equatorial plane under the condition of a uni-form current distribution in magnetopause with the configuration given by x=−0.8y2+1.25Fig.5Distribution of magneticfield on Martian equatorial plane under the condition of a non-uniform current distribution in magnetopause with the configuration given by x=−0.8y2+1.253.SUMMARY AND DISCUSSIONVia the physical model constructed by us,the distribution of magneticfield on Martian equatorial plane can be obtained.Because the fashion of action of Mars with solar wind differs from that of the earth,solar wind leads to the ionization of the atmosphere around Mars.Under the action of induction electricfield,there occurs an induction current in Martian ionosphere.The related characteristics of electric current and the observational data of satellites are utilized.It is assumed that there are currents in the current sheets of both Martian magnetopause and magnetotail,and that these currents satisfy certain relations.The distributions of electric currents under the background conditions described above and Ampere’s law are used to get the magneticfield strength B on Martian equatorial plane.Furthermore,from the relation between the magneticfield strength B and magnetic flux functionΨin plane rectangular coordinate system,Ψcan be solved out.For the magneticfield distribution on Martian equatorial plane,the results given by our physical model agree rather well with those of satellite observations and other methods. Nevertheless,the description of the structure of Martian magneticfield with our model is still rather coarse,and there still exist the following defects and deficiencies.(1)In our model,only the homogeneous distribution of electric currents in Martian ionosphere is taken into consideration.It is needed to probe the details of electric current distribution in Martian ionosphere.(2)The question of the shape of magnetopause–If the simulation is not made with a paraboloid,it is possible that the results which agree better with reality can be acquired.(3)Our model ignores the thickness of the current sheets in magnetopause and mag-netotail.(4)It is supposed that the direction of magneticfield in solar wind is perpendicular to that of the velocity of solar wind.In our future research we shall further refine the various parameters in our model and solve the above questions,so that the model may be successively improved.412TONG Dong-sheng et al. /Chinese Astronomy and Astrophysics 34 (2010) 406–412References1Shi Jiankui,Liu Zhenxing,Prog.Geophys.,1996,11,772Slavin J.A.,Schwingenschuh K.,Reidler W.,et al.,J.Geophys.Res.,1996,A17,112353Acuna M.H.,et al.,Science,1998,279,16764Reidler W.,Schwingenschuh K.,Lichtenegger H.I.M.,et al.,Planet.Space Sci.,1991,39,75 5Luhmann J.G.,Space Sci.Rev.,1986,44,2416Kallio E.,Janhunen P.,J.Geophys.Res.,2002,107(A3)13057Kallio E.,Hannu Koskinen,Adv.Space Res.,1998,21(4),5738Brain D.A.,Bagenal F.,et al.,J.Geophys.Res.,2003,108(A12),14249Harnett E.M.,Winglee R.M.,J.Geophys.Res.,2005,110,A0722610Whitten R.C.,Colin L.,Rev.Geophys.,1974,12,15511Halekas J.S.,Brain D.A.,Lillis R.J.,et al.,Geophys.Res.Lett,2006,33L13101。