ISSN 0016-7932, Geomagnetism and Aeronomy, 2020, Vol. 60, No. 2, pp. 162–170. © Pleiades Publishing, Ltd., 2020.Russian Text © The Author(s), 2020, published in Geomagnetizm i Aeronomiya, 2020, Vol. 60, No. 2, pp. 164–172.

Analytical Description of the Near Planetary Bow Shock Based on Gas-Dynamic and Magneto-Gas–Dynamic Modeling

for the Magnetic Field Parallel and Perpendicular to the Plasma FlowG. A. Kotovaa, *, M. I. Verigina, †, T. Gombosib, K. Kabinc, and V. V. Bezrukikha

aSpace Research Institute, Russian Academy of Sciences, Moscow, 117997 RussiabUniversity of Michigan, Ann Arbor, MI, 48109 United States

cRoyal Military College of Canada, Kingstown, ON, K7K 7B4 Canada*e-mail: kotova@iki.rssi.ru

Received July 10, 2019; revised September 6, 2019; accepted September 26, 2019

Abstract—An analytical semiempirical model of the bow shock based on theoretical MGD calculations,accurate analytical solutions, and experimental data continues to be developed. The model parametershave a clear physical meaning. For cases in which the magnetic field of the solar wind is directed along itsvelocity or is perpendicular to the velocity vector, analytical expressions that allow calculating the param-eters of the bow shock—the distance to the subsolar point, the radius of the curvature, and the bluntness atthe subsolar point—are obtained via renormalization of the previously developed detailed gas-dynamicmodel. For the case in which the magnetic field vector is perpendicular to the solar wind velocity vector, it isshown that it is sufficient for an analytical description of the bow shock surface to approximate its parametersin two perpendicular planes.

DOI: 10.1134/S0016793220020073

1. INTRODUCTIONThe study of physical processes in plasma near

planets often requires knowledge of the position andshape of the near planetary bow shock. Modern theo-retical magnetogasdynamic (MGD), kinetic models,and their combined use (e.g., the review by Ledvinaet al., 2008; Wang et al., 2015; Chen et al., 2017)require significant processor time. They constitute areliable base for bow shock study but cannot be used inpractice to track the movements of this boundary inreal time. Therefore, most such studies have used andcontinue to use empirical models of planetary bowshock (Fairfield, 1971, 2001; Formisano 1979; Slavinand Holzer, 1981; Nĕmeček and Šafránková, 1991;Peredo et al., 1995; Chao et al., 2002; Merka et al.,2003; Chapman et al., 2003; Jelínek et al., 2012;Meziane et al., 2014). However, such models areapplicable in the region of the solar-wind parametersused in their construction and are limited in space bythe region of the measurements. Such modeling alsodoes not take into account the influence of the poly-tropic index γ on the bow shock position.

Verigin with co-authors (Verigin et al., 1999; Veri-gin et al., 2001a, 2001b; Verigin et al., 2003a, 2003b;Verigin, 2004; Kotova et al., 2005) use a semiempirical

approach based on accurate analytical solutions thatdescribe some characteristic properties bow shock,e.g., the shape of the subsolar portion of the bow shockfront at low sonic Mach numbers Ms in the gas-dynamic (GD) approximation, the asymptotic MGDMach cone, and the properties of plasma f low imme-diately after the discontinuity.The model parametersare selected based on detailed numerical calculations.At sufficiently large Alfvén Mach numbers, the GDapproximation can be used to simulate near planetarybow shock. A GD analytical model of a bow shock forobstacles of various shapes (from a hyperbolic to a stubcylinder) is presented by Verigin et al. (2003b).

The goal of this work is to build an analytical MGDmodel of a near planetary bow shock, the parametersof which have a clear physical meaning. Since the GDanalytical model of the bow shock very accuratelydescribes its position for obstacles of various shapes(Verigin et al., 2003b), it seems appropriate to use sim-ilar analytical expressions for the parameters thatdetermine the MGD bow shock. To do this, one needsto find expressions that modify the input parametersof the GD model for its conversion to the MGDmodel. In this paper, we consider cases in which theinterplanetary magnetic field is parallel to the solar-wind flow and those in which it is perpendicular to theflow.† Deceased.

162

ANALYTICAL DESCRIPTION OF THE NEAR PLANETARY BOW SHOCK 163

Fig. 1. Definition of space variables: ro and rs are distancesto the magnetopause (obstacle) and bow shock, respec-tively; Ro and Rs are proper nose curvature radii; bo and bsare the bluntness of the magnetopause and bow shock,respectively; and Δ is the bow shock standoff distance.

Bow shockMagnetosheet

Magnetopause

VY

rs

Ro

Rs

bs

bo

Vs V

X

ro Δ

ω

Fig. 2. Example of an adaptive grid in the plane (x, y) fornumerical calculations of the bow shock position andshape near the obstacle (ro= 1, Ro = 1, bo = –1, Ms = 6,Ma = 3, γ = 2.0).

–3

–2

–1

0

1

2

3

–4 –3 –2 –1 0 1 2 3

X

Y

2. COORDINATE SYSTEM AND DESIGNATIONS

To describe the bow shock, it is convenient to usethe geocentric coordinate system associated with theinterplanetary medium GIPM (Geocentric InterPlanetary Medium). In this coordinate system, thedirection of the X axis is opposite the direction of theundisturbed solar wind. The Y axis is directed such

GEOMAGNETISM AND AERONOMY Vol. 60 No. 2

that the interplanetary magnetic field vector lies in thesecond and fourth quadrants of the plane (X, Y). TheZ axis complements the coordinate system to the right.This coordinate system was first proposed by Bieber &Stone (1979) to study the outflow of energetic elec-trons from the magnetosphere and was later used byPeredo et al. (1995) to analyze near-Earth’s bowshock.

Figure 1 shows the designations used to constructthe analytical model of the planetary bow shock. Thebluntness of an obstacle bo determines its shape nearthe nose point: bo = –1 is a sphere, bo < –1 is a “blunt”ellipsoid, –1 < bo< 0 is an “elongated” ellipsoid, bo =0 is a paraboloid, and bo > 0 is a hyperboloid. Theangle ω determines the inclination of the bow shock tothe direction of the plasma flow at large distances fromthe planet.

3. MGD CALCULATIONS OF THE POSITION AND SHAPE OF THE BOW SHOCK

To model the bow shock formed in a supersonicand super-Alfvén plasma flow near obstacles of vari-ous shapes, we used detailed MGD calculations per-formed at the University of Michigan. A grid with avariable adaptive step with ~3 × 106 cells was used.Figure 2 shows an example of a grid in the (x, y) planefor numerical calculations. The calculations werecarried out for two types of obstacles: a hemispherewith an elongated cylindrical tail (Fig. 2) and aparaboloid of revolution. All calculations were car-ried out in units of ro.

4. EQUATION DESCRIBING THE BOW SHOCK SHAPE AND POSITION

When constructing a GD analytical model of a bowshock for obstacles of various shapes, Verigin et al.(Verigin et al., 2003b) used the following expression todescribe its shape and position:

(1)

The additional parameter ds characterizes the transi-tion from the predominance of the parameters of thesubsolar region of the bow shock to the region wherethe asymptotic slope of the bow shock dominates.The slope of the bow shock to the direction of theplasma f low at large distances from the planet in GDis determined by the sonic Mach number tan2ω =1/( – 1) and does not depend on the clock angle

, where .Verigin et al.(2003a) obtained an exact analytical solution in MGD todetermine the Mach cone ωas(ϕ) for any clock angle ϕ.The asymptotic Mach cone defines the asymptotic

22

22

( 1) 11( )( ) 2 ( ) .11

s ss

s s sss

s

b Mr xy x R r x r xdM R

− −+− = − + − +−

2Ms

arccos( )yϕ = ρ 2 2y zρ = +

2020

164 KOTOVA et al.

Fig. 3. Examples of the position and shape of the bow shock formed near two different obstacles in the upcoming solar wind f lowwith a magnetic field parallel to the velocity. The points show the MGD calculation, and solid lines indicate approximation withexpression (2).

–3 –2 –1 0 1 20

1

2

3

4

5

–3 –2 –1 0 1 20

1

2

3

4

5

X X

� = 5/3Ms = 6Ma = 3

� = 5/3Ms = 6Ma = 3

� �

b0 = 0b0 = –1

Mach number Mas = 1/sin2ωas = 1/tan2ωas + 1. Todescribe the MGD bow shock replacement, formula (1)can be modified by replacing Mas with Ms:

(2)

Expression (2) can be directly used for the axisymmet-ric case in which the interplanetary magnetic field isparallel to the incident plasma flow. For all otherdirections of the magnetic field, it is necessary to takeinto account that the parameters Rs, bs, ωas, and dsdepend on the clock angle ϕ.

5. ANALYTICAL APPROXIMATION OF THE MGD BOW SHOCK CALCULATIONS FOR CASES OF MAGNETIC FIELD PARALLEL

AND PERPENDICULAR TO THE PLASMA FLOW

Figure 3 shows two examples of MGD calculationsof the position and shape of the bow shock near vari-ous obstacles and their fitting by expression (2) forcases in which the magnetic field is parallel to theplasma flow. It can be seen that the indicated analyti-cal expression makes it possible to approximate quitewell the axisymmetric bow shock surface that formswhen a magnetic field is parallel to the plasma flow.For obstacles with a distance to the subsolar point ofro= 1 and a curvature radius at this point of Ro = 1 andfor a solar wind flow with parameters Ms = 6, Ma = 3,and γ = 5/3, the following fitting parameters wereobtained: rs = 1.2, Rs = 1.96, bs = 0.93, and ds = 0.5 for a

2

22 2

( ) 2 ( )

1tantan ( ) .1

1

s s

s

asas s

ss

s

x R r xb

r xr xd

R

ρ = − + − ω+ ω − + − +

GEOMA

parabolic obstacle (left) and rs = 1.18, Rs = 1.67, bs = 0.2,and ds = 0.4 for a spherical obstacle.

When the interplanetary magnetic field is perpen-dicular to the direction of the plasma flow velocity, thenose point of the bow shock still lies on the x axis, butit is necessary to determine the dependence of themodel parameters Rs, bs, and ds on the clock angle ϕ.For this, the three-dimensional surface of the bowshock calculated in the MGD approximation wasapproximated with expression (2) in various planes ofϕ = const with a ϕ-step of 5° in the interval of 0 < ϕ < 180°.Figure 4 shows the dependencies of rs, Rs, and bs thusobtained on the angle ϕ. As expected, the distance tothe subsolar point of the bow shock does not dependon ϕ. The dependence of the curvature radius at a sub-solar point on ϕ is described well by the expression:

(3)

where the parameters Rsy and Rsz are the curvatureradii of the bow shock surface in the planes (x, y), i.e.at ϕ = 0°, and (x, z), i.e., at ϕ = 90 °, respectively. Sim-ilarly, the dependence of bluntness at a subsolar pointon ϕ is well described by the expression:

(4)with parameters bsy and bsz corresponding to the blunt-nesses of the bow shock at a subsolar point in planes(x, y) and (x, z).

Thus, formulas (3) and (4) allow the use of expres-sion (2) to approximate the three-dimensional surfaceof the bow shock. The dependence ds(ϕ) could not beidentified; therefore, in our approximations, weassume ds(ϕ) = const = ds. For cases with a magneticfield directed perpendicular to the plasma flow, Fig. 5shows examples of the approximation of the position

2 2 ,sin cos

sy szs

sy sz

R RR

R R=

ϕ + ϕ

2 2sin coss sz syb b b= ϕ + ϕ

GNETISM AND AERONOMY Vol. 60 No. 2 2020

ANALYTICAL DESCRIPTION OF THE NEAR PLANETARY BOW SHOCK 165

Fig. 4. Examples of variations in parameters rs, Rs, and bs with respect to angle ϕ when the solar wind with a magnetic field per-pendicular to the velocity f lows around obstacles. The points show the fitting parameters of MGD calculations in the planes ϕ =const with 5° intervals of ϕ, and the solid lines indicate approximation with expressions (3) and (4).

0 90 1801.40

1.45

1.50

1.55

1.60

0 90 1802.2

2.3

2.4

2.5

2.6

0 90 180

0.2

0.3

0.4

0.5

0 90 1801.40

1.45

1.50

1.55

1.60

0 90 1801.9

2.0

2.1

2.2

2.3

0 90 180

–0.2

0

0.2

0.4

� = 5/3 b0 = 0 Ms = 6 Ma = 5

� = 5/3 b0 = –1 Ms = 6 Ma = 3

rs

rs Rs

Rs bs

bs

rs = 1.45

rs = 1.52

Rsy = 2.25Rsz = 2.57

Rsy = 1.93Rsz = 2.22

bsy = 0.18bsz = 0.4

bsy = –0.16bsz = 0.33

and shape of the bow shock with expressions (2–4). Inthis case, six free parameters are determined: rs, Rsy,Rsz, bsy, bsz, and ds.

6. DETERMINATION OF THE PARAMETERS OF MGD APPROXIMATION OF THE BOW SHOCK BASED ON GD CALCULATIONS

In the case of a GD flow, the relative rate of expan-sion of the cross section S of the central current tubebehind the bow shock (for example, Biermann et al.,1967; Wallis, 1973) is

(5)

where ε is the reciprocal of the largest jump in gas den-sity at the shock front. In the case of MGD, ε = ε(γ,Ma, Ms, α

vn, ϑbn), where αvn and ϑbn are angles

between the normal to the bow shock and the direc-tion of the plasma flow or magnetic field, respectively.Petrinec and Russell (1997, Equation 12)obtained acubic equation to determine ε.

For a MGD flow, the relative expansion rate of thecross section S of the central current tube behind anaxisymmetric bow shock or having two planes of sym-metry (the magnetic field is parallel or perpendicular

1 2 1 ,s

dSS dx R

− ε= −ε

GEOMAGNETISM AND AERONOMY Vol. 60 No. 2

to the incident plasma flow, αvn = 0) is determined by

the expression

(6)

where are the radii of curvature in directionsparallel and perpendicular to the magnetic field,respectively.

When the interplanetary magnetic field is parallelto the solar-wind velocity, :

(7)

where ε* = ε/(1 – ε), and the jump in density on thebow shock ε coincides with its GD value

but, in comparison with expres-

sion (5), a new factor appears: Г = Гpar = 1 – When the interplanetary magnetic field is perpen-

dicular to the solar-wind velocity,

(8)

2

2 21 1 ,

cossy sz a

sy sz a bn

R R MdSS dx R R M

+ ε− ε= − ε ε − ϑ

sy szR R<

sy sz sR R R= =

1 2 ,*s

dSS dx R

= −ε Γ

21 2 ,1 ( 1)sM

γ −ε = +γ + γ +

εΜ21 .a

1 1 2 1 .sy sz

sy sz s

R RdSS dx R R R

+ − ε − ε= − =ε ε

2020

166 KOTOVA et al.

Fig. 5. Examples of the position and shape of bow shocks in the plane (x, y) (lower curves (a)) and the plane (x, z) (uppercurves (b)) that formed around two different obstacles in an oncoming solar wind f low with magnetic field perpendicularto the velocity. The points show the MGD calculations, and the solid lines indicate approximation with expression (2) withallowance for (3) and (4).

–3 –2 –1 0 1 20

1

2

3

4

5

–3 –2 –1 0 1 20

1

2

3

4

5

X

Y/Z Y/Z

X

b

ba

aγ = 2Ms = 6Ma = 5

γ = 5/3Ms = 6Ma = 5

b0 = 0

b0 = –1

The density jump at the bow shock in this case can also

be calculated analytically as

Factor = Гper here

implicitly enters into the quantities Rsy(Гper) and Rsz(Гper).The MGD factor Г and Masvalues can be used to

renormalize GD parameters rs, Rs, bs, and ds and todetermine the MGD parameters (Verigin et al.,2001a). The GD parameters rs = ro + Δ(ε*, γ, Ro, bo),Rs(ε*, γ, Ro, bo), bs(γ, bo, Ms), and ds(bo) are calculatedwith expressions (34)–(38) and (39)–(43) from Veri-gin et al. (2003b) (expression (36) contains a typo thatwas corrected in Verigin’s review (2004); a “+” sign,not a “–” should be before the second term).

To account for the interplanetary magnetic field inthe solar-wind plasma flow when it is directed parallelto the plasma velocity, the MGD parameters rs, Rs,and bsin expressions (2) can be obtained from the sameformulas (34)–(43) from Verigin et al. (2003b), butwith the compression parameter ε calculated in theMGD approximation and with the substitution ofε*Гpar for ε* and the substitution of Mas for Ms, as wellas the introduction of an additional factor thatdepends on Гpar:

(9)

In Fig. 6, the renormalized parameters (9) arecompared with the parameters obtained via approxi-mation of the MGD calculations of the bow shockposition and shape. It is seen that the agreement is very

γ − γε = + +γ + γ +2

12( 1) 2M ( 1)a

γ − − γ +γ +

2 2 22

1 (( 1)M ) 8M .2M ( 1)

a aa

2 3norm _ norm par par

2 3_ norm par par

_ norm

( ),

( ),

– , * , , ,

* , , ,, ,( ).

s o o o

s s o o

s s o as

r r R b

R R R bb b b M

−

−

Δ = = Γ Δ ε Γ γ

= Γ ε Γ γ= γ

GEOMA

good between rs_norm and rs, and between Rs_norm and Rs(panels a, b) and that is slightly worse for bs_norm and bs(panel c).To approximate the bluntness of the bowshock at the nose point, we used the value bs_norm +βpar, where βpar = 0.2. The GD parameter ds dependsonly on the bluntness (shape) of the obstacle bo.Therefore, we will not change it for now, although theapproximated MGD parameter and the parametercalculated with the GD formulas differ significantly.

Figure 7 shows two examples of the bow shockposition and shape in front of various obstacles asdetermined with renormalized GD formulas.

Next, we consider the formation of a bow shock ina solar wind flow with a magnetic field directed perpen-dicular to the direction of the plasma flow velocity. Asshown above, in order to calculate the position andshape of the planetary bow shock, it is sufficient in thiscase to determine the parameters of expression (2) inplanes (x, y) and (x, z).To determine the parameters rs,Rsу, and bsz, we can use expressions similar to formulas

(9) but with the parameter Г = Гper= 1/(1 – ) Inaddition, the asymptotic slope of the bow shock atinfinity depends on the clock angle. Therefore, to cal-culate the parameters, in addition to the value Гper, webring in the ratio of asymptotic Mach numbers in thedirections y (Mas(0)) and z (Mas(90)):

(10)

21 aεΜ

2 3norm _ norm per per

2 3_ norm per per

1 2

_ norm par2

( * , , , ),

( * , , , )

[ (0) (90)] ,( ( , , (90)) )

[ (0) (90)] .

s o o o

sy s o o

as as

sz s o as

as as

r r R b

R R R b

M Mb b b M

M M

−

−

−

Δ = − = Γ Δ ε Γ γ

= Γ ε Γ γ

×= γ + β

×

GNETISM AND AERONOMY Vol. 60 No. 2 2020

ANALYTICAL DESCRIPTION OF THE NEAR PLANETARY BOW SHOCK 167

Fig. 6. Comparison of the parameters of bow shock model (2) fitting the MGD calculations with parameters calculated in theGD model (Verigin et al., 2001a) for an incident solar wind f lux with a magnetic field parallel to the f low velocity. The solid linesin panels (a, b, c) show the approximating dependences.

0 0.1 0.2 0.3 0.4

0.1

0.2

0.3

0.4

0.8 1.2 1.6 2.0 2.40.8

1.2

1.6

2.0

–0.8 –0.4 0 0.4 0.8 1.2–0.8

–0.4

0

0.4

0.8

1.2rs – ro = 1.003Δnorm

bs = bs_norm + 0.2Rs = 1.009Rs_norm

Rs_norm bs_normΔnorm

rs – ro Rs bs(a) (b) (c)

Fig. 7. Examples of the position and shape of a bow shock generated near two different obstacles in an incoming solar wind f lowwith a magnetic field parallel to the velocity. The points show the MGD calculations, and the solid lines represent approximationwith GD calculations and expressions (9).

0

1

2

3

4

5

–3 –2 –1 0 1 2–3 –2 –1 0 1 20

1

2

3

4

5

X X

� = 2Ms = 6Ma = 5

� = 5/3Ms = 8Ma = 5

b0 = 0b0 = –1

The renormalized parameters (10) Δnorm (a) andRsy_norm (b) in Fig. 8 are compared with parametersrs – ro and Rsy obtained via approximation of the MGDcalculations of the bow shock position and shape. Inaddition, the curvature radii of the subsolar region ofthe bow shock (c) were approximated in planes (x, y)and (x, z).

Figure 8 shows that an additional parameter(χ(ϑbv) ~ 1.4) must be introduced in order to deter-mine the standoff distance of the bow shock in a solar-wind flow with a magnetic field that is perpendicularto the direction of the plasma-flow velocity. Thisparameter obviously depends on the direction of themagnetic field and is equal to unity for an anglebetween the directions of the solar-wind velocity andthe magnetic field of ϑbv = 0 (9). Magnitude Rsу cor-

GEOMAGNETISM AND AERONOMY Vol. 60 No. 2

relates well with Rsy_norm (Fig. 8b). The interdepen-dence of the curvature radii in planes (x, y) and (x, z)Rsz = (Fig. 8c) makes it possible to calculateRsz_norm if Rsy_norm is known.

The bluntness in plane (x, z) bsz (Fig. 8d) is approx-imated well by bsz_norm + 0.21, which is calculated withthe corresponding expression (10). The bluntness inplane (x, y) bsy (Fig. 8e) can be calculated with bsz_norm.The GD parameter ds_norm depends only on the shapeof the obstacle bo, but parameter ds in the approxima-tion of MGD calculations (Fig. 8f), on average, is sig-nificantly lower than the GD ds = 0.4ds_norm.

Figure 9 shows two examples of the bow shockposition and shape as determined with renormalizedGD formulas for a solar-wind flux with a magnetic

1 2per syRΓ

2020

168

GEOMAGNETISM AND AERONOMY Vol. 60 No. 2 2020

KOTOVA et al.

Fig. 8. Comparison of the parameters of the bow shock model (2) fitting the MGD calculations with parameters calculated in theGD model (Verigin et al., 2001a) for an incident solar wind flux with a magnetic field perpendicular to the f low velocity. The solidlines on the panels show the approximating dependences.

0 0.2 0.4 0.6

0.2

0.4

0.6

1.6 1.8 2.0 2.2

1.6

1.8

2.0

2.2

1.6 1.8 2.0 2.2 2.4

1.6

1.8

2.0

2.2

2.4

–0.4 0 0.4 0.8

–0.4

0

0.4

0.8

0 0.4 0.8 1.2 1.6

0.4

0.8

1.2

1.6

–0.4 0 0.4

–0.4

0

0.4

rs – ro = 1.37Δnorm

bsz = bsz_norm + 0.21 bsz = bsy + 0.35 ds = 0.4ds_norm

Rsy = 0.99Rsy_norm

Rsz = Rsy√(Gрег)

bsz_norm bsy ds_norm

Δnorm Rsy_norm Rsy

RszRsyrs – ro

bsz bsz ds

(a) (b) (c)

(d) (e) (f)

Fig. 9. Examples of the position and shape of the bow shock in the plane (x, y) (lower curves (a)) and the plane (x, z) (upper curves(b)) that formed around two different obstacles in an oncoming solar wind flow with magnetic field perpendicular to the velocity.The points show the MGD calculation, and the solid lines represent approximation with GD calculations, expressions (10), andcorrections in accordance with Fig. 8.

0

1

2

3

4

5

–3 –2 –1 0 1 2–3 –2 –1 0 1 20

1

2

3

4

5

X X

ba

b

aY/Z Y/Z

γ = 5/3Ms = 6Ma = 5

γ = 5/3Ms = 8Ma = 5

b0 = 0

b0 = –1

ANALYTICAL DESCRIPTION OF THE NEAR PLANETARY BOW SHOCK 169

field that is perpendicular to the velocity vector of theflow around different obstacles.

7. CONCLUSIONSThis paper considered cases of the formation of the

bow shock in front of obstacles of various shapes whenthe interplanetary magnetic field was directed parallelto the solar-wind flow and when it was perpendicularto the f low.

It was shown that the rational function that wasused earlier in the GD approximation can be used toapproximate the surface of the planetary bow shock.

For cases with a magnetic field vector that is paral-lel to the f low-velocity vector, it is enough to deter-mine four free parameters: the distance to the nosepoint, the radius of the curvature, the bluntness at thenose point, and the transition parameter to the asymp-totic angle of inclination. These parameters can beapproximated with GD calculations.

It was shown for cases with a magnetic field vectorthat is perpendicular to the plasma flow-velocity vec-tor that the surface of the planetary bow shock can beapproximated with six free parameters: the distance tothe nose point, the radii of the curvature, the blunt-ness at the nose point in planes (x, y) and (x, z), andthe transition parameter to the asymptotic angle ofinclination. Relations for the recalculation of theseparameters with GD calculations were obtained.

In the future, it is supposed to determine theparameters that allow describing the planetary bowshock by an analytical rational function for an arbi-trary direction of the interplanetary magnetic field,and to approximate these parameters also based ongas-dynamic calculations.

FUNDING

The work was done in accordance with the plan of scien-tific research works at IKI of Russian Academy of Scienceswith the partial support of the Program no. 12 of the Presid-ium of the Russian Academy of Sciences, “Issues of the ori-gin and evolution of the Universe based on ground-basedobservations and space research.”

REFERENCESBieber, J.W. and Stone, E.C., Energetic electron bursts in

the magnetopause electron layer and in interplanetaryspace, in Magnetospheric Boundary Layers, A SydneyChapman Conference, 1979,ESA SP-131.

Chao, J.K., Wu, D.J., Lin, C.-H., Yang, Y.-H., Wang, X.Y.,Kessel, M., Chen, S.H., and Lepping, R.P., Models forthe size and shape of the Earth’s magnetopause andbow shock,in Space Weather Study Using MultipointTechniques, Lyu, L.-H., Ed., Pergamon, 2002,pp. 360–368.

Chapman, J.F. and Cairns, I.H., Three-dimensional mod-eling of Earth’s bow shock: Shock shape as a function of

GEOMAGNETISM AND AERONOMY Vol. 60 No. 2

AlfvénMach number, J. Geophys. Res., 2003, vol. 108,A051174.https://doi.org/10.1029/2002JA009569

Chen, Y., Tóth, G., Cassak, P., Jia, X., Gombosi, T.I.,Slavin, J.A., Markidis, S., Peng, I.B., Jordanova, V.K.,and Henderson, M.G., Global three-dimensional sim-ulation of Earth’s dayside reconnection using a two-way coupled magnetohydrodynamics with embeddedparticle-in-cell model: Initial results, J. Geophys. Res.,2017, vol. 122, pp. 10318–10335.https://doi.org/10.1002/2017JA024186

Fairfield, D.H., Average and unusual locations of theEarth’s magnetopause and bow shock, J. Geophys. Res.,1971, vol. 76, no. 28, pp. 6700–6716.

Fairfield, D.H., Cairns, I.H., Desch, M.D., Szabo, A.,Lazarus, A.J., and Aellig, M.R., The location of lowMach number bow shocks at Earth, J. Geophys. Res.,2001, vol. 106, no. A11, pp. 25361–23376.

Formisano, V., Orientation and shape of the Earth’s bowshock in three dimensions, Planet. Space Sci., 1979,vol. 27, p. 1151.

Jelínek, K., Němeček, Z., and Šafránková, J., A new ap-proach to magnetopause and bow shock modelingbased on automated region identification, J. Geophys.Res., 2012, vol. 117, A05208.https://doi.org/10.1029/2011JA017252

Kotova, G., Verigin, M., Zastenker, G., Nikolaeva, N.,Smolkin, B., Slavin, J., Szabo, A., Merka, J.,Němeček, Z., and Šafránková, J., Bow shock observa-tions by Prognoz–Prognoz 11 data: Analysis and modelcomparison, Adv. Space Res., 2005, vol. 36, pp. 1958–1963.

Ledvina, S.A., Ma, Y.-J., and Kallio, E., Modeling andsimulating f lowing plasmas and related phenomena,Space Sci. Rev., 2008, vol. 139, pp. 143–189.https://doi.org/10.1007/s11214-008-9384-6

Merka, J., Szabo, A., Narock, T.W., King, J.H., Paula-rena, K.I., and Richardson, J.D., A comparison ofIMP 8 observed bow shock positions with model pre-dictions, J. Geophys. Res., 2003, vol. 108, A021077.https://doi.org/10.1029/2002JA009384

Meziane, K., Alrefay, T.Y., and Hamza, A., On the shapeand motion of the Earth’s bow shock, Planet. SpaceSci., 2014, vols. 93–94, pp. 1–9.

Němeček, Z., and Šafránková, J., The Earth’s bow shockand magnetopause position as a result of solar wind–magnetosphere interaction, J. Atmos. Terr. Phys., 1991,vol. 53, pp. 1049–1054.

Peredo, M., Slavin, J.A., Mazur, E., and Curtis, S.A.,Three-dimensional position and shape of the bowshock and their variation with Alfvénic, sonic and mag-netosonicMach numbers and interplanetary magneticfield orientation, J. Geophys. Res., 1995, vol. 100,no. A5, pp. 7907–7916.

Petrinec, S.M. and Russell, C.T., Hydrodynamic andMHD equations across the bow shock and along thesurfaces of planetary obstacles, Space Sci. Rev., 1997,vol. 79, pp. 757–791.

Slavin, J.A. and Holzer, R.E., Solar wind flow about theterrestrial planets. 1.Modeling bow shock position andshape, J. Geophys. Res., 1981, vol. 86, no. A13,pp. 11401–11418.

2020

170 KOTOVA et al.

Verigin, M.I.,Location and shape of near-planetary bowshocks: gas-dynamical andMHD aspects, Solnechno-zem-nye svyazi i elektromagnitnye predvestniki zemletryasenii.Sb. dokladov III Mezhdunarodnoi konferentsii 16–21 avgusta 2004 g. (Solar–Terrestrial and Electromag-netic Precursors of Earthquakes. Proceedings of the In-ternational Conference of 16–21 August 2004), Pet-ropavlovsk-Kamchatskii: IKIR DVO RAN, 2004,pp. 49–69.

Verigin, M.I., Kotova, G.A., Remizov, A.P., et al., Studiesof the Martian bow shock response to the variation of themagnetosphere dimensions according to TAUS andMAGMA measurements aboard the PHOBOS 2 orbiter,Adv. Space Res., 1997, vol. 20, no. 2, pp. 155–158.

Verigin, M.I., Kotova, G.A., Remizov, A.P., et al.,Shapeand location of planetary bow shocks, Cosmic Res.,1999, vol. 37, no. 1, pp. 34–39.

Verigin, M., Kotova, G., Szabo, A., Slavin, J., Gombosi, T.,Kabin, K., Shugaev, F., and Kalinchenko, A., WINDobservations of the terrestrial bow shock 3-D shape and

motion, Earth Planets Space, 2001a, vol. 53, no. 10,pp. 1001–1009.

Verigin, M.I., Kotova, G.A., Slavin, J., et al., Analysis ofthe 3-D shape of the terrestrial bow shock by Inter-ball/Magion 4 observations, Adv. Space Res., 2001b,vol. 28, no. 6, pp. 857–862.

Verigin, M., Slavin, J., Szabo, A., Kotova, G., and Gombo-si, T., Planetary bow shocks: Asymptotic MHDMachcones, Earth Planets Space, 2003a, vol. 55, pp. 33–38.

Verigin, M., Slavin, J., Szabo, A., Gombosi, T., Kotova, G.,Plochova, O., Szegö, K., Tatrallyay, M., Kabin, K.,and Shugaev, F., Planetary bow shocks: Gasdynamicanalytic approach, J. Geophys. Res., 2003b, vol. 108,A081323.https://doi.org/10.1029/2002JA009711

Wang, M., Lu, J.Y., Yuan, H.Z., Kabin, K., Liu, Z.Q.,Zhao, M.X., and Li, G.,The dipole tilt angle depen-dence of the bow shock for southward IMF: MHD re-sults, Planet. Space Sci., 2015, vol. 106, pp. 99–107.

GEOMAGNETISM AND AERONOMY Vol. 60 No. 2 2020