Non-Ideal MHD Structure of Pulsar Magnetospheres

C. Kalapotharakos1,2, D. Kazanas1, A. Harding1, I. Contopoulos3

1Astrophysics Science Division, NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA2University of Maryland, College Park UMDCP/CRESST), College Park, MD 20742, USA3Research Center for Astronomy, Academy of Athens, Athens 11527, Greece

We study pulsar magnetospheres that deviate from the Ideal Force-Free MHD condition employed in their modeling over the past decade. Non-ideal MHD allows for electric field components parallel to the pulsar magnetic field, providing thus the possibility for particle acceleration and radiation production, something that was by default absent in IMHD treatments. We modify our current 3D force-free IMHD numerical code by implementing a variety of prescriptions for the parallel electric field. The ultimate goal is to find realistic magnetospheric solutions that can explain the observational phenomenology but are also compatible with the underlying microphysics.

AbstractAbstract

The Code

• We use a time dependent Finite Difference Time Domain (FDTD) code in order to evolve our systems

• We have implemented the Perfectly Matched Layer (PML) technique which is an absorbing and non reflecting outer boundary. This allows us to follow the magnetosphere for many rotations (Kalapotharakos & Contopoulos 2009).

Ideal Force-Free Solutions

B 0

E B 0 eE J B 0

E 1

4e

I-MHD Force-Free

Et

B 4 J

Bt

E

We solve the time-dependent Maxwell equations

considering the Ideal Force-Free conditions

Ideal Force-Free Solutions

The above expression for the current density is equivalent to

||2e

E BJ E

B

where Ε|| is the electric fieldcomponent parallel to the magnetic field

for

Ideal Force-Free Solutions

• In Ideal Force-Free MHD the conductivity is infinite while the parallel electric field and dissipation is (by definition) 0. (Kalapotharakos & Contopoulos 2009, Spitkovsky 2006, Timokhin 2006)

• For α=0 the ratio J/ρe exceeds the value c (space-like current) only along the zero charge lines and along the separatrix that separates the closed from the open field lines while it is less than c (time-like current) near the star.

• For higher values of α there are larger regions corresponding to space-like currents.

Ideal Force-Free Solutions

0

Color: Poloidal current (Purple means 0 red means high value)Stream lines: Poloidal current

Color: Charge density (blue means negative red means positive)

Stream lines: Poloidal magnetic field lines

Color: ConductivityStream lines: Poloidal magnetic field lines

Color: Ε|| (green means 0 red means parallel B blue means antiparallel to B)Stream lines: Poloidal magnetic field lines

Color: J/ρe (green means ≈c red means >c blue means <c)Stream lines: Poloidal magnetic field lines

Color: J E dissipation∙ (Purple means 0 red means high value)Stream lines: Poloidal magnetic field lines

α is the angle between the magnetic axis and the rotational axis

Min

MaxThe length unit is theThe length unit is thelight cylinder Rlight cylinder Rlclc

Non-Ideal Force-Free Solutions (I)We present the results

corresponding to σ=25.0

• We observe that the general structure of the magnetic field lines does not change dramatically

• Nevertheless, we see that the previously (in the ideal case) open field lines close on the equatorial current sheet due to the dissipation we have there.

• The ratio J/ρe has been decreased. There are areas where the space-like current has been transformed to time-like one.

• For α=0 we have parallel electric fields along the separatrices and above the magnetic poles.

• For α>0 there are regions inside the closed field lines area with E||.

Non-Ideal Force-Free Solutions (II)

0 • The areas that develop E|| are similar to the solutions with the constant σ.

• Nevertheless, in these cases the E|| is stronger along the separatrices and inside the closed field lines area while it is weaker above the poles.

• The ratio J/ρe is equal to c except some regions where it is less than c.

Ideal Force-Free Solutions

30

90

Non-Ideal Force-Free Solutions (I)

30

90

Non-Ideal Force-Free Solutions (II)

30

90

Non-Ideal Force-Free Solutions• The real pulsar magnetosphere cannot be the

vacuum (there are parallel electric fields but there are not particles to be accelerated) and cannot be Ideal Force-Free (there are particles but there are not parallel electric fields to accelerate them).

• The real solution should be somewhere in between the vacuum and the Ideal Force-Free.

• Another issue is that the Ideal Force-Free MHD solutions consider abundant charges in order to satisfy the MHD solution. This means that the constraints that may be implied by the underlying microphysics are overridden. The Ideal Force-Free solutions do not ensure by themselves that there is the necessary microphysics being able to reproduce the corresponding charge density.

Non-Ideal Force-Free Solutions (I)The firstfirst prescriptionprescription we used in order

to search the spectrum of the Non-Ideal Force-Free solutions was the variation of the conductivity in the expression (see also page 6)

||2e

E BJ E

B

As σ varies from 0 to infinity the corresponding solutions lie between the vacuum and the Ideal Force-Free.

Non-Ideal Force-Free Solutions (II)

In some cases the microphysics indicates that the ratio J/ρe should be near c.

We managed to construct solutions where the ratio J/ρe is as close as possible to c (and never above c).

||2e

E BJ E

B

In this case the conductivity σ varies in order to achieve the ratio J/ρe=c

Non-Ideal Force-Free Solutions (III)Gruzinov (2007, 2008) proposed a covariant

dissipative ohmic law and named it Strong Field Electrodynamics (SFE) 2 2 2 2

0 0 02 2

0e

E B E B B E EJ

B E

where2 2

2 2 2 2 2 00 0 0 0 0 2 2

0 0

, , 0,B E

B E B E B E B E EB E

• SFE introduces dissipation only at the space-like currents (J/ρe≥c) regions where flows of two different kind of charges are needed.

• The ratio J/ρe is always ≥c (space-like).

• In SFE singularities appear in the regions that is supposed to be time-like. However, the parallel electric field oscillates around zero, giving a dissipationless time-like time-averaged current.

0 0,B E

Non-Ideal Force-Free Solutions (III)

0 30

We present the results corresponding to constant σ=30.

Non-Ideal Force-Free Solutions (III) 90

• The developed E|| is again along the separatrices, along the currents sheet outside the light cylinder and above the poles.

• Nevertheless, the E|| is restricted only to the regions with space-like currents.

• We have everywhere J/ρe≥c.

• Wherever the solution demands time-like currents we observe a noisy behavior (see the bottom-right panels) corresponding to dissipation-less time-like time-averaged current.

Conclusions• The realistic pulsar magnetosphere should be

dissipative since we observe particle radiation. This means that the real pulsar magnetosphere will be somewhere in between the vacuum and the Ideal Force-Free solutions.

• The realistic pulsar magnetosphere should also obey rules implied by micro-physics.

• We tried 3 different prescriptions for the introduction of dissipation.

• The general trend is that we have parallel electric fields along the separatrices and above the magnetic poles. However, the relative strength of these fields depends on the considered prescription.Reference

sKalapotharakos & Contopoulos, A&A, 496, 495 (2009)

Gruzinov, arXiv:astro-ph/9902288 (1999)Spitkovsky, A. 2006, ApJ, 648, 51.Timokhin, A. 2006, MNRAS, 368, 1055.