1. The heliospheric magnetic flux density through several solar cycles Géza Erdős (1) and André Balogh (2) (1) MTA Wigner FK RMI, Budapest, Hungary (2) Imperial College London, UK

2. Heliospheric magnetic field Importance: Heliospheric physics:Modulation of cosmic rays Energy deposited to magnetospheres Solar physics: evolution of the magnetic field of the Sun during sunspot cycles Definition of the open magnetic flux of the Sun: the number of field lines crossing the source surface of the solar wind. The magnetic flux density in the interplanetary space is characterized by B R The total magnetic flux of the Sun is zero. In order to characterize the magnetic state of the Sun, it is customary to use abs(B R ) However, the magnetic flux is a signed quantity (inward or outward crossings of the field lines with a surface), which makes problems!

3. Fluctuations of the heliospheric magnetic field Effect of the fluctuations: on the distribution of B R “excess flux” by increasing heliospheric range Method for correction of magnetic field fluctuations: Separation of data by magnetic sectors Neglect of fluctuations perpendicular to Parker line Comparison of Ulysses magnetic flux with 1 AU data: Latitudinal independence Radial independence (excess flux problem) Solar cycle dependence: Possible north-south displacement of current sheet Comparison of interplanetary magnetic flux with source surface field

4. Distribution of the open flux Different open flux distribution in the slow and fast solar wind (Ulysses) Different open flux distribution during solar minimum and maximum (OMNI) Distribution of the open magnetic flux density, B R R 2 is a complex function of heliospheric location, solar wind velocity and phase of solar cycle. Two-humped distributions (negative and positive magnetic polarities), or one-humped distributions? Solar origin, or evolution in solar wind?

5. Helios 1-2 Two-humped B R distributions close to Sun Two-humped 2D distributions everywhere Projection of 2D distribution to horizontal line: positive and negative polarities overlap at larger distances from the Sun, because Fluctuations increase Parker angle increases Erdős and Balogh, ApJ, 753, 130 (2012 )

6. Methods of corrections for fluctuations Uncorrected open flux: Probability density function of B R (black line) OMNI data from 1965 to present 6 hour averages Slow solar wind Solar minimum

7. Methods of corrections for fluctuations Parker field line, determined from solar wind velocity Probability density function of B P OMNI data from 1965 to present 6 hour averages Slow solar wind Solar minimum

8. Separation of data by magnetic sectors Identification of magnetic sector from the sign of B P Separation of B R distribution according to polarity (blue and red lines) Possible improvement of sector identification: Ulysses high latitude observations during solar minimum (unipolar sector) Electron heat flux Direction of propagation of Alfvén waves OMNI data from 1965 to present 6 hour averages Slow solar wind Solar minimum

9. Neglect fluctuations perpendicular to Parker line Parker field line, determined from solar wind velocity Probability density function of B P Re-scaling by cos(Parker angle) Justification of approximation: symmetric distribution perpendicular to Parker line OMNI data from 1965 to present 6 hour averages Slow solar wind Solar minimum

10. Ulysses/OMNI comparison, uncorrected flux OMNI magnetic flux density (green) Ulysses magnetic flux density (red) Ulysses magnetic flux is far exceeding the OMNI data at aphelion of Ulysses Heliospheric range and latitude of Ulysses

11. Ulysses/OMNI comparison, corrected flux OMNI magnetic flux density (green) Ulysses magnetic flux density (red) The discrepancy between OMNI and Ulysses magnetic fluxes disappeared when correction is made for fluctuations. Heliospheric range and latitude of Ulysses

12. Magnetic flux, as a function of heliospheric range and latitude If the magnetic flux density is calculated from the average of abs(B R ), then the flux seems to increase by heliocentric distance (“flux excess”, Lockwood et al., 2009) The “flux excess” disappears, if correction is made for fluctuations No variation of the magnetic flux by heliolatitude (Forsyth et al., 1996, Owens et al., 2008)

13. Explanation of latitudinal independence of the flux Inside a sphere with a radius of about 10 R S the magnetic field pressure is larger than that of the plasma. If the magnetic field is larger in a place of the source surface (for instance, at the poles), then the larger magnetic pressure diverts the flow from the radial expansion until equilibrium is reached (Smith, 2008). Question: Is there an equilibrium between the positive and negative magnetic flux density? If yes, no north-south displacement of the heliospheric current sheet is expected.

14. North-South (a)symmetry Difference between the magnetic flux averages in positive and negative sectors: 22 year wave fit results in an amplitude of 5%, corresponding to 1.5  offset of current sheet. Not convincing result, large errors. (Mursula and Kalevi, 2004)

15. Source surface and heliospheric magnetic flux Good agreement between solar and heliospheric magnetic flux, except rasing phase of solar cycle Source surface magnetic field: courtesy of Yi-Ming Wang

16. Source surface and heliospheric magnetic flux Good agreement between solar and heliospheric magnetic flux, except rasing phase of solar cycle Source surface magnetic field: courtesy of Yi-Ming Wang

17. Conclusions The distribution of the open magnetic flux density, B R R 2 depends in a complex way on the heliospheric location, type of solar wind (slow or fast) and solar activity. Variations in B R R 2 are largely caused by fluctuations of the magnetic field around the average Parker field lines. The effects of the fluctuations can be reduced (2 methods were presented) Results show that there is no flux excess in the outer heliosphere The magnetic flux is uniformly distributed in the heliosphere (no latitudinal dependence) A small difference between the magnetic flux of opposite polarities may be real, but there is a large variability in the observations Good agreement between heliospheric and source surface magnetic fluxes, except during the rising phase of solar cycles