## 1. Introduction and Motivation

[2] Interstellar pickup ions, i.e., interstellar matter that penetrates into the solar system and is ionized in the inner heliosphere and picked up by the outward moving solar wind, are currently under intensive observational and theoretical investigation. This is partly due to the fact that these ions are most likely the source of the so-called anomalous cosmic ray component [*Fisk et al.*, 1974]. Acceleration of these ions to high energies is most probably at the termination shock of the solar wind [*Pesses et al.*, 1981]. Furthermore, pickup ions have been observed to be efficiently injected into an acceleration process in corotating interaction regions [*Gloeckler et al.*, 1994], which are bounded by a forward and reverse shock pair. Both, the termination shock as well as corotating shocks are basically quasi-perpendicular shocks, i.e., the angle Θ_{Bn} between the shock normal and the upstream magnetic field is larger than 45°. There exist a number of competing acceleration mechanisms for pickup ions at quasi-perpendicular shocks. In the classical first-order Fermi acceleration mechanism, the particle must have a sufficiently high energy that it can scatter diffusively across the microstructure and macrostructure of the shock and experience the compression between the converging upstream and downstream flows. One-dimensional (1-D) hybrid simulations of quasi-perpendicular shocks by *Liewer et al.* [1993] and *Kucharek and Scholer* [1995] where pickup ions had been included have shown that these ions are only reflected with reasonable rates when the shock normal angle is smaller than ∼60°. *Kucharek and Scholer* [1995] suggested that these backscattered pickup ions contribute to the injection pool for a diffusive shock acceleration process, although the termination shock finds itself with a Θ_{Bn} < 60° for only ∼10% of the time. A somewhat different approach was taken by *Giacalone et al.* [1994]. They also use a 1-D hybrid simulation of a quasi-perpendicular shock but include an ad hoc scattering perpendicular to the magnetic field. This enables pickup ions to diffuse across field lines. *Giacalone et al.* [1994] found that when reasonable scattering times are considered pickup ions are injected and accelerated to fairly high energies, whereas solar wind ions are not. *Giacalone et al.* [1997] have pointed out that stationary shocks, as the termination shock and planetary bow shocks, might have different pickup reflection properties than interplanetary travelling shocks. Pickup ions have in the inertial frame a speed between zero and twice the solar wind speed. Because propagating shocks move only about 100 km/s faster than the 400–700 km/s solar wind, nonaccelerated interstellar pickup ions have already many times the shock ramming velocity when they encounter interplanetary travelling shocks and constitute thus a suprathermal particle population. *Giacalone et al.* [1997] proposed that in a first step pickup ions are accelerated in the inner heliosphere. These ions are then further accelerated at the termination shock to become the anomalous cosmic rays.

[3] A possible injection and acceleration mechanism at quasi-perpendicular shocks, which strongly favors the injection of pickup ions over solar wind ions, is shock surfing. Based on an idea by *Sagdeev* [1966], *Zank et al.* [1996] and *Lee et al.* [1996] have investigated the possibility that pickup ions are trapped at the shock between the electrostatic potential of the shock and the upstream Lorentz force: part of the pickup population incident on the shock has so little energy in the shock frame that it cannot overcome the electrostatic barrier and is reflected at the shock ramp. Upstream it is again turned around by the Lorentz force. The time spent upstream of the shock determines the maximum energy gain for the trapped pickup ions. This time is limited by two conditions: (1) the particles' incident normal velocity *v*_{x} must satisfy the condition (*m*/2)*v*_{x}^{2} ≤ *e*Φ, where *m* is the particles' mass, Φ the cross-shock potential, and *e* the electron charge and (2) the Lorentz force is smaller than the force exerted by the electrostatic potential *e*Φ, i.e., *ev*_{y}*B* ≤ *e*Φ/*L*_{s}, where *B* is the magnetic field magnitude and *L*_{s} the length scale of the cross-shock potential [*Zank et al.*, 1996]. Here, the electrostatic potential gradient has been approximated by *e*Φ/*L*_{s}. Thus, the maximum energy a particle can reach by shock surfing is inverse proportional to the square of the cross-shock potential length scale. Assuming that the cross-shock potential is of the order of the upstream bulk energy and that *L*_{s} is of the order of the electron inertial length λ_{e} = *c*/ω_{pe} (*c* = velocity of light, ω_{pe} = electron plasma frequency) the maximum pickup ion energy is close to about 1 MeV/nucleon. A somewhat different analysis by *Lee et al.* [1996] results in maximum energies of 12 MeV/nucleon. If, however, the scale length is of the order of the ion inertial length, λ_{i}, the maximum energy is by the ratio *m*_{i}/*m*_{e} smaller. Thus, the length scale of the cross-shock electrostatic potential determines the effectiveness of shock surfing acceleration. Furthermore, intrinsic nonstationarities of the shock may also compromise the acceleration efficiency.

[4] Direct observations of shock potential length scales at interplanetary shocks and the Earth's bow shock are rare, the reason being that it is not possible with single spacecraft observations to distinguish unambiguously between spatial and temporal structures. There are two reports on thin shock ramps: based on ISEE-1 and ISEE-2 data, *Scudder et al.* [1986] reported a high β (β = magnetic field to particle pressure) shock observation with a length scale of the magnetic field ramp of ∼4λ_{e}, and *Newbury and Russell* [1996] observed one almost perpendicular bow shock with a magnetic field ramp scale given by ∼2λ_{e}. But is not clear whether these observations are rare exceptions or a common occurrence. A determination of the magnetic field ramp of 11 quasi-perpendicular shocks measured by OGO-5 and ISEE-1/2 was made by *Balikhin et al.* [1995]. Four of the shocks reported had magnetic ramp scales smaller than 0.15∼4λ_{i}. In addition it is not clear how the magnetic ramp scale compares with the scale of the shock potential. *Scudder* [1995] derived empirically the potential profile through a quasi-perpendicular bow shock. The total potential jump occurs over a scale, which is an order of magnitude larger than that of the magnetic ramp. According to the result by *Scudder* [1995], the electric field within the ramp is considerably smaller than one would get by putting the whole potential drop across the magnetic ramp.

[5] In the present paper, we will determine the length scale of the potential by computer simulations of collisionless shocks. Computer simulations of shocks have been performed in the past by the hybrid method and by the particle-in-cell (PIC) method. In the hybrid method, ions are treated as macroparticles and electrons as a fluid. With a massless electron fluid, it is not possible to resolve the electron scale. Application of the hybrid method with a finite electron mass to shocks can produce serious problems: on one hand the electron inertial length has to be resolved, on the other hand the resistive length scale should not be resolved. Otherwise, the electron fluid is artificially heated in the shock ramp. Since the hybrid method has artificial and intrinsic resistivities, one should take refuge to the PIC method, where both ions and electrons are treated as particles. This will be done in the present paper.

[6] Beginning with the study of *Biskamp and Welter* [1972], collisionless shocks have been investigated by PIC simulations for more than three decades. It has been shown that perpendicular and quasi-perpendicular shocks are intrinsically nonstationary and considerable attention has been paid to the cyclic self-reformation [e.g., *Biskamp and Welter*, 1972; *Lembège and Dawson*, 1987; *Lembège and Savoini*, 1992]. This process is due to the fact that part of the upstream ions are reflected at the shock front and are responsible for the formation of the foot. The accumulation of these ions in time is responsible for the cyclic self-reformation with timescales of the order of the inverse ion gyrofrequency Ω_{ci}. We will concentrate in this paper on quasi-perpendicular shocks with Θ_{Bn} close to 90°, i.e., on shocks where the deviation of Θ_{Bn} from 90° is small so that dispersive effects are negligible. At shocks with Θ_{Bn} > Θ_{tr} where cos Θ_{tr} = (*m*_{e}/*m*_{i})^{1/2}*M*_{A} (*M*_{A} = upstream Alfvén Mach number). a whistler wave train will trail the shock, whereas for Θ_{Bn} < Θ_{tr} a whistler precursor is emitted from the shock [e.g., *Balikhin et al.*, 1995]. For example in the case of a shock with Alfvén Mach number *M*_{A} = 5 one obtains Θ_{tr} ≈ 83° (assuming *m*_{i}/*m*_{e} = 1840). In the case Θ_{Bn} < Θ_{tr} upstream whistlers can interact with reflected ions and steepen up, which also results eventually in shock reformation [e.g., *Krasnoselskikh et al.*, 2002]. We will concentrate in the following on quasi-perpendicular shocks with Θ_{Bn} > Θ_{tr}. The length scales of the magnetic ramp in more oblique shocks (Θ_{Bn} = 55°) as obtained from PIC simulations has been compared on a statistical basis with the length scales of the electric field gradient by *Lembège et al.* [1999]. These authors found that the spatial scales over which the magnetic field changes are of the same order as those over which the potential acts, and that these are of the order of a few *c*/ω_{pe}.

[7] There is a further motivation to perform additional PIC simulations of perpendicular and quasi-perpendicular shocks, and to determine length scales in these shocks. Two important parameters enter a PIC simulation: (1) the mass ratio *m*_{i}/*m*_{e} and (2) the ratio of electron plasma frequency to gyrofrequency ω_{pe}/Ω_{ce}. The problem of performing PIC simulations with a realistic value of *m*_{i}/*m*_{e} is well known; not so well known is the problem arising from the second quantity. As far as *m*_{i}/*m*_{e} is concerned there seems to exist in the literature only one quasi-perpendicular shock simulation with a realistic value. This is the work by *Liewer et al.* [1991], where the electron dynamics in low Mach number (*M*_{A} ≈ 2.8) shocks is investigated by a 1-D PIC simulation. These subcritical shock simulations were only run to a fraction of an inverse ion gyroperiod. In most PIC simulations the second quantity, ω_{pe}/Ω_{ce}, is assumed to be of the order of 1, i.e., simulations are done for shocks in the strongly magnetized condition. Note that ω_{pe}/Ω_{ce} can be written in terms of the ratio of velocity of light *c* and Alfvén velocity *v*_{A}:

[8] In the solar wind at the Earth's orbit this quantity is 100–200. In contrast, because of computational constraints of PIC codes, *Biskamp and Welter* [1972] used a value of ω_{pe}/Ω_{ce} = 5, *Lembège and Dawson* [1987] used 1, and *Liewer et al.* [1991] used 1–4. *Quest* [1986] has pointed out that the use small values of ω_{pe}/Ω_{ce} overemphasizes charge separation effects. He argued that separation electric fields in thin shock structures possibly destabilize the shock and could lead to artificial reformation. Recently, a 1-D simulation in the weakly magnetized condition with a value of ω_{pe}/Ω_{ce} = 20 has been presented by *Shimada and Hoshino* [2000]. However, because of computational constraints, these authors chose a value of *m*_{i}/*m*_{e} = 20, which makes it difficult to differentiate between ion and electron length scales.

[9] In this paper we will present a number of 1-D shock simulation runs with different mass ratios and different ratios of the electron plasma to gyrofrequency. At present, computer resources prohibit a simulation with a realistic mass ratio and at the same time a value of ω_{pe}/Ω_{ce} appropriate for the solar wind. However, we can vary both parameters independently and are then able to infer the structure of a bow shock/termination shock or an interplanetary traveling shock. Among the runs, we will present one run with *m*_{i}/*m*_{e} = 1840, τ = (ω_{pe}/Ω_{ce})^{2} = 8, and one run with *m*_{i}/*m*_{e} = 400, τ = 60. We will also vary the second important parameter, the upstream ion β, between β_{i} = 0.1 and β_{i} = 0.4. Low β values are expected to be characteristic for the heliospheric termination shock. We will restrict ourselves to nondispersive quasi-perpendicular shocks, and chose Θ_{Bn} = 87°. Less oblique collisionless shocks, which have dispersive upstream whistlers, have recently been investigated by *Krasnoselskikh et al.* [2002]. A compilation of the parameters used in the various runs is given in Table 1.

Run | M_{A} | Θ_{Bn} | m_{i}/m_{e} | (ω_{pe}/Ω_{ce})^{2} | β_{i} | β_{e} |
---|---|---|---|---|---|---|

1 | 4.2 | 87 | 400 | 10 | 0.1 | 0.2 |

2 | 4.2 | 87 | 1840 | 4 | 0.1 | 0.2 |

3 | 6.2 | 87 | 1840 | 4 | 0.05 | 0.1 |

4 | 6.2 | 90 | 1840 | 4 | 0.05 | 0.1 |

5 | 4.2 | 87 | 400 | 60 | 0.1 | 0.2 |

6 | 4.2 | 87 | 400 | 60 | 0.4 | 0.4 |

7 | 4.2 | 87 | 400 | 10 | 0.4 | 0.2 |

8 | 4.2 | 87 | 1840 | 8 | 0.4 | 0.4 |