Two-Dipole Model of the Suns Magnetic Field Bertalan Zieger Center - PowerPoint PPT Presentation

Two-Dipole Model of the Sun’s Magnetic Field Bertalan Zieger Center for Space Physics, Boston University, MA, U.S.A. & Kalevi Mursula ReSoLVE Centre of Excellence, University of Oulu, Finland

Overview Spatial Power Spectrum of the Photospheric Magnetic Field: • Eliminating the vantage point effect • Filling the polar data gaps • Normalization of the spherical harmonic coefficients Two-dipole Model: • Justifying the two-dipole approximation of the photospheric field • Derivation of the potential of two eccentric axial dipoles • Fitting the two-dipole model to observed harmonic coefficients North-South Asymmetry: • Is there a persistent north-south asymmetry during solar minima? • What is the cause of the asymmetry? • How does the photospheric asymmetry affect the coronal magnetic field and the interplanetary magnetic field at Earth?

Synoptic Map of the Photospheric Magnetic Field MWO, CR 1910, June 1996 Radial magnetic field assumption in the photosphere: assumption that the magnetic that B r = B los / sin θ , assumption was first suggested where B los is the line-of-sight magnetic field component, and ! is the colatitude.

Definition of Harmonic Coefficients Solar Physics: l 1 X X P m l (cos θ )(g m l cos m φ + h m B r ( R 0 , θ , φ ) = l sin m φ ) . l = 0 m = 0 and h m ysics, g m and are the harmonic coefficients of the spherical harmonics expansion of l l spherical harmonics spherical the radial magnetic field component in the photosphere. Geomagnetism: 1 l ◆ l + 1 ✓ R 0 X X l (cos θ )(g 0 m l cos m φ + h 0 m P m Ψ I ( r , θ , φ ) = R 0 l sin m φ ) , r l = 0 m = 0 component in the photosphere. to g 0 m and are the harmonic coefficients of the spherical harmonics expansion of and h 0 m l , l Ψ , the internal potential, also known as Gauss coefficients . Neglecting external sources in the photosphere, the two coefficients are related as: l = ( l + 1) h 0 m l = ( l + 1)g 0 m g m h m l l

Calculating the Zonal Gauss Coefficients From Latitudinal Profiles For photospheric magnetic field data with a longitude-latitude grid of N ! � N " , the zonal ( m = 0) Gauss coefficients can be expressed as N θ l = π 2 l + 1 X g 0 0 h B los i i P 0 l (cos θ i ) , 2 N θ ( l + 1) i = 1 where ⟨ B ilos ⟩ is the mean line-of-sight magnetic field at the colatitude of " i . The latitudinal profiles of the radial magnetic field can be reconstructed from the first 24 zonal Gauss coefficients as follows: 24 X ( l + 1)g 0 0 l P 0 B r ( θ ) = l (cos θ ) l = 1

Latitudinal Profiles of the Radial Magnetic Field for Three Consecutive Carrington Rotations 15 CR 1910 The decline of the polar CR 1911 10 CR 1912 field at the northern and southern ends of the profiles is an artefact due 5 to large observational B r (G) errors close to the edge 0 of the visible solar disk. -5 The change in the latitude range of data is -10 caused by the vantage point effect. -15 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90 Latitude ( ° ) Latitudinal profiles reconstructed from the first 24 harmonic coefficients are shown as solid lines.

Vantage Point (B 0 ) Effect in Solar Observations B 0 B 0 is the heliographic latitude of the central point of the solar disk due to the tilt of the ecliptic with respect to the solar equatorial plane.

Latitudinal Profile of the Radial Magnetic Field During Solar Minimum (1995-1996) 15 Two-year median profile Year 1995-1996 of the radial magnetic no polar filling 10 polar filling field after removing the erroneous observations at the highest 5º of 5 latitude. B r (G) 0 The polar data gaps are filled with zeros (red) -5 and a constant value (yellow), respectively . -10 The southern polar field -15 is significantly stronger -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90 than the northern polar Latitude ( ° ) field. Zieger et al., A&A, 2019

Zonal Harmonic Coefficients Calculated With and Without Polar Filling Harmonic Coefficients of B r a 4 2 0 (G) g l 0 -2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Gauss Coefficients of ! I Harmonic Degree b no polar filling 1 polar filling 0 (G) 0.5 g' l 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Harmonic Degree Zieger et al., A&A, 2019

Definition of Spatial Power Spectrum Z Z 1 B 2 d Ω = 1 ( �r Ψ I ) 2 d Ω = 4 π 4 π 1 l ◆ 2 l + 4 ✓ R 0 X X h (g 0 m l ) 2 + ( h 0 m l ) 2 i ( l + 1) . = r l = 1 m = 0 In the photosphere ( r = R 0 ) the power-per-degree spectrum is l X S degree h l ) 2 i (g 0 m l ) 2 + ( h 0 m = ( l + 1) l m = 0 and the zonal ( m = 0) spatial power spectrum becomes 1 = ( l + 1)(g 0 0 S zonal l ) 2 = ( l + 1)(g 0 l ) 2 , l

Zonal Spatial Power Spectrum of the Photospheric Magnetic Field (1995-1996) 10 1 no polar filling polar filling 10 0 10 -1 Power (G 2 ) 10 -2 10 -3 10 -4 10 -5 10 -6 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Harmonic Degree Zieger et al., A&A, 2019

Two-Dipole Model of the Photospheric Magnetic Field Two rings of dipoles representing the Two axial dipoles placed at the center north-south magnetic component of of each dipole ring in panel A . The decaying active regions in the magnetic potential of the dipoles in photosphere in an axisymmetric case. panel A and B are nearly identical at the solar surface.

Fitting the Two-Dipole Model to the Observed Zonal Gauss Coefficients Theoretically derived zonal Gauss coefficients of two eccentric axial dipoles ")' + %& + ( + # = %& ' ( ' ")' ! " where a 1 and a 2 are the strength of the two dipoles and z 1 and z 2 are their locations along the z-axis of symmetry. The four unknown parameters of the two-dipole model can be exactly solved using the equations for the first four Gauss coefficients: # − 6! . # + √3(27(! ( # ) . − 108! ( # + 64! ( # ) 7 + 54(! . # − # ! 4 # ! 7 # ) . (! 4 # ! . # ! 7 # ! 4 # (! 7 # ) 7 ! 4 ) ( = (9! ( @ # − 18(! . # ) . (! 7 # ) . ) # ! 7 # ) . ), 36(! . A )/(24! ( # − 3! . # − 6! ( # ) ( )/(3! . # ) ( ) , ) . = (2! 7 # − 2! ( # ) . )/(2) ( − 2) . ) , ' ( = (! . # − ' ( . ' . = ! (

Normalized Zonal Gauss Coefficients and Spatial Power Spectra for Three Solar Minima Gauss Coefficient Power Spectrum B A 10 1 1.2 1975-1976 10 0 1 1985-1986 1995-1996 0.8 10 -1 two-dipole model 0.6 10 -2 Power 0 g l 0.4 10 -3 0.2 10 -4 1975-1976 1985-1986 0 1995-1996 10 -5 two-dipole model -0.2 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 6 7 8 9 10 11 12 Harmonic Degree Harmonic Degree The two-dipole model (black) fitted to the first four zonal Gauss coefficients can reproduce the observed spatial structure of the photospheric magnetic field up to the harmonic degree 8. The low-order even zonal Gauss coefficients are significantly different from zero, indicating a persistent north-south asymmetry during solar minima.

Parameters of the Northern and Southern Dipoles A 1 Dipole Strength (G) 0.8 0.6 0.4 North 0.2 South 0 1970 1975 1980 1985 1990 1995 2000 Year B 90 80 North 70 Latitude ( ° ) South 60 50 40 30 20 10 0 1970 1975 1980 1985 1990 1995 2000 Year The southern dipole is stronger than the northern dipole during all the three solar minima. The northern and southern dipoles are located at similar northern and southern latitudes, implying that the asymmetry is caused by the different dipole strengths.

Coronal Magnetic Field Arising From the Two-Dipole Model of the Photospheric Magnetic Field The potential field source surface (PFSS), where the coronal magnetic field becomes radial, is marked by a dashed circle. The heliospheric current sheet, where the magnetic field reverses, is tilted towards the south by 4.1º.

Conclusions • The two-dipole model can reproduce the spatial structure of the photospheric magnetic field during solar minimum up to harmonic degree 8. • The north-south asymmetry is caused by the different strengths of the northern and southern dipoles rather than the difference in their heliographic latitudes. • The southern dipole was found to be stronger during all the three solar minima, indicating a persistent north-south asymmetry in the operation of the solar dynamo. • The photospheric asymmetry results in a southward tilted heliospheric current sheet (3º-5º) during solar minima, which is confirmed by heliospheric observations. • The two-dipole model could be used to fill in the polar data gaps in synoptic maps of the photospheric magnetic field.

Th Thank Yo You!