A simplified ab initio cosmic-ray modulation model: Construction and predictive capabilities Katlego Moloto Eugene Engelbrecht Adri Burger July 17, 2017 Centre for Space Research North-West University
Table of contents 1. Problem Statement 2. Modulation Model 3. Sample Solutions 2
Problem Statement
Problem Statement • Explain observed Previous solar minima spectra Differential intensity (part.m -2 .s -1 .sr -1 .MeV -1 ) Blue open symbols: A > 0 cycles modulation Red filled symbols: A < 0 cycles self-consistently 1 1977: Evenson et al. (1983) 1977: von Rosenvinge et al. (1979) 1998: Sanuki et al. (2000) 1965: Fan et al. (1966) 1965: Ormes and Webber (1968) 1965: Balasubrahmanyan et al. (1965) 1996: McDonald et al. (1998) 1987: McDonald et al. (1998) December 2009 PAMELA 0.1 0.1 1 Kinetic energy (GeV) Potgieter et al. (2013, ICRC PROC) 3
Problem Statement • Explain observed Previous solar minima spectra Differential intensity (part.m -2 .s -1 .sr -1 .MeV -1 ) Blue open symbols: A > 0 cycles modulation Red filled symbols: A < 0 cycles self-consistently • Input realistic solar minimum 1 conditions 1977: Evenson et al. (1983) 1977: von Rosenvinge et al. (1979) 1998: Sanuki et al. (2000) 1965: Fan et al. (1966) 1965: Ormes and Webber (1968) 1965: Balasubrahmanyan et al. (1965) 1996: McDonald et al. (1998) 1987: McDonald et al. (1998) December 2009 PAMELA 0.1 0.1 1 Kinetic energy (GeV) Potgieter et al. (2013, ICRC PROC) 3
Problem Statement • Explain observed Previous solar minima spectra Differential intensity (part.m -2 .s -1 .sr -1 .MeV -1 ) Blue open symbols: A > 0 cycles modulation Red filled symbols: A < 0 cycles self-consistently • Input realistic solar minimum 1 conditions 1977: Evenson et al. (1983) • Should give 1977: von Rosenvinge et al. (1979) 1998: Sanuki et al. (2000) 1965: Fan et al. (1966) predictive 1965: Ormes and Webber (1968) 1965: Balasubrahmanyan et al. (1965) capacity to 1996: McDonald et al. (1998) 1987: McDonald et al. (1998) modulation code December 2009 PAMELA 0.1 0.1 1 Kinetic energy (GeV) Potgieter et al. (2013, ICRC PROC) 3
Problem Statement • Explain observed Previous solar minima spectra Differential intensity (part.m -2 .s -1 .sr -1 .MeV -1 ) Blue open symbols: A > 0 cycles modulation Red filled symbols: A < 0 cycles self-consistently • Input realistic solar minimum 1 conditions 1977: Evenson et al. (1983) • Should give 1977: von Rosenvinge et al. (1979) 1998: Sanuki et al. (2000) 1965: Fan et al. (1966) predictive 1965: Ormes and Webber (1968) 1965: Balasubrahmanyan et al. (1965) capacity to 1996: McDonald et al. (1998) 1987: McDonald et al. (1998) modulation code December 2009 PAMELA • This can be done 0.1 0.1 1 using an ab initio Kinetic energy (GeV) approach Potgieter et al. (2013, ICRC PROC) 3
Introduction • Ab initio approach to modulation (e.g. Engelbrecht and Burger 2013a and b, APJ) requires turbulence spectra as input for diffusion tensor and not just B or alpha 4
Introduction • Ab initio approach to modulation (e.g. Engelbrecht and Burger 2013a and b, APJ) requires turbulence spectra as input for diffusion tensor and not just B or alpha • Simplified version of Engelbrecht and Burger (2015, APJ) model is used in this study; results are preliminary and qualitative rather than quantitative 4
Introduction • Ab initio approach to modulation (e.g. Engelbrecht and Burger 2013a and b, APJ) requires turbulence spectra as input for diffusion tensor and not just B or alpha • Simplified version of Engelbrecht and Burger (2015, APJ) model is used in this study; results are preliminary and qualitative rather than quantitative • Part of a project to study long-term modulation and solar-cycle dependence of turbulence parameters, thus time dependence 4
Introduction • Ab initio approach to modulation (e.g. Engelbrecht and Burger 2013a and b, APJ) requires turbulence spectra as input for diffusion tensor and not just B or alpha • Simplified version of Engelbrecht and Burger (2015, APJ) model is used in this study; results are preliminary and qualitative rather than quantitative • Part of a project to study long-term modulation and solar-cycle dependence of turbulence parameters, thus time dependence • The code used in this project is a 3D Steady-state SDE code. 4
Modulation Model
Model ∂ f ∂ t = − ( V sw + � v d � ) · ∇ f + ∇ · ( K S · ∇ f ) + 1 3( ∇ · V sw ) ∂ f ∂ ln P • The term ( V sw + � v d � ) · ∇ f describes the outward convection of cosmic rays by the solar wind and cosmic ray drift. 5
Model ∂ f ∂ t = − ( V sw + � v d � ) · ∇ f + ∇ · ( K S · ∇ f ) + 1 3( ∇ · V sw ) ∂ f ∂ ln P • The term ( V sw + � v d � ) · ∇ f describes the outward convection of cosmic rays by the solar wind and cosmic ray drift. • The term 1 3( ∇ · V sw ) ∂ f ∂ ln P describes adiabatic energy changes 5
Model ∂ f ∂ t = − ( V sw + � v d � ) · ∇ f + ∇ · ( K S · ∇ f ) + 1 3( ∇ · V sw ) ∂ f ∂ ln P • The term ( V sw + � v d � ) · ∇ f describes the outward convection of cosmic rays by the solar wind and cosmic ray drift. • The term 1 3( ∇ · V sw ) ∂ f ∂ ln P describes adiabatic energy changes • and the remaining term ∇ · ( K S · ∇ f ) describes diffusion. 5
Model • A colatitude dependent solar wind speed is used. • Parker Heliospheric magnetic field is used. • neutral sheet drift by Burger (2012, APJ) Strauss. (2010, MSC) 6
Model • A colatitude 100 dependent solar 50 0 wind speed is used. 50 • Parker Heliospheric 100 100 magnetic field is used. • neutral sheet drift by 50 Burger (2012, APJ) 0 100 50 0 50 100 Engelbrecht and Burger (2010, ASR) 7
Model • A colatitude dependent solar wind speed is used • Parker Heliospheric magnetic field is used. • neutral sheet drift by Burger (2012, APJ) 8
Steady-state 3D Model • Steady-state 3D model using “effective” values of 1 AU observations as input, taking into account outward convection by the solar wind 9
Effective Values A > 0 A < 0 A > 0 A < 0 A < 0 Classic 21 22 23 24 70 Tilt Angle [degrees] 50 30 Spot value 12Months 10 24Months 1980 1985 1990 1995 2000 2005 2010 2015 Time [Years] 10
Solar Minimum Values Year Magnetic Field Variance Tilt Angle 1987 5.8 8.3 4.1 1997 5.7 9.4 4.5 2009 4.4 5.7 11.2 Nel. (2015, MSC) 11
Variance 10 3 Zank et al. 1996 1987 Ecliptic Polar Smith et al. 2001 1997 2009 10 2 10 1 δ B 2 [nT 2 ] 10 0 10 -1 10 -2 10 -3 10 0 10 1 10 2 10 0 10 1 10 2 R [AU] R [AU] Based on Oughton et al. (2011, JGR) Based on Bavassano et al. (2000, JGR) 12
Correlation Scales 10 0 Smith et al. 2001, e-folding 2D Ecliptic Polar Smith et al. 2001, integration Slab Weygand et al. 2011 Weygand et al. 2011 10 -1 λ c [AU] 10 -2 10 -3 10 0 10 1 10 2 10 0 10 1 10 2 R [AU] R [AU] Based on Oughton et al. (2011, JGR) Based on Wicks et al. (2010, PRL) 13
Diffusion Tensor • Parallel diffusion based on QLT (Teufel & Schlickeiser 2003, A&A) � 2 � R 2 3 s � B o b 2 b � √ π ( s − 1) 4 √ π + √ π (2 − s )(4 − s ) λ � = R s bk min δ B slab , x 14
Diffusion Tensor • Parallel diffusion based on QLT (Teufel & Schlickeiser 2003, A&A) � 2 � R 2 3 s � B o b 2 b � √ π ( s − 1) 4 √ π + √ π (2 − s )(4 − s ) λ � = R s bk min δ B slab , x • Perpendicular diffusion based on NLGC (Matthaeus et al. 2003, APJL; Shalchi et al. 2004) � 2 / 3 √ 3 δ B 2 � 2 ν − 1 λ 1 / 3 a 2 F 2 ( ν ) l 2 D 2 D λ ⊥ ≈ B 2 � 4 ν o 14
Diffusion Tensor • Parallel diffusion based on QLT (Teufel & Schlickeiser 2003, A&A) � 2 � R 2 3 s � B o b 2 b � √ π ( s − 1) 4 √ π + √ π (2 − s )(4 − s ) λ � = R s bk min δ B slab , x • Perpendicular diffusion based on NLGC (Matthaeus et al. 2003, APJL; Shalchi et al. 2004) � 2 / 3 √ 3 δ B 2 � 2 ν − 1 λ 1 / 3 a 2 F 2 ( ν ) l 2 D 2 D λ ⊥ ≈ B 2 � 4 ν o • Drift coefficient derived by Engelbrecht et al (2017, APJ) � − 1 � 2 δ B T � 2 � λ ⊥ κ A = v 1 + 3 R L 2 R L B 0 14
Sample Solutions
MFP Rigidity 10 1 λ || 10 0 10 -1 λ [AU] 10 -2 λ ⊥ 10 -3 λ Α 1987 2009 1997 10 -4 0.1 1 10 Rigidity [GV] 15
MFP Radial 10 2 10 1 λ || 10 0 λ [AU] 10 -1 10 -2 λ ⊥ 10 -3 1987 λ Α 2009 1997 10 -4 1 10 100 R [AU] 16
Spectra Comparison 10 2 LIS 1987 1997 2009 IMP8, A < 0 Pamela, A < 0 10 1 IMP8, A > 0 j T [MeV.s.m 2 .sr] -1 10 0 10 -1 10 -2 10 -2 10 -1 10 0 10 1 Kinetic Energy [GeV] 17
Spectra Comparison 10 2 LIS 1987 1997 2009 2019 IMP8, A < 0 10 1 Pamela, A < 0 IMP8, A > 0 j T [MeV.s.m 2 .sr] -1 10 0 10 -1 10 -2 10 -2 10 -1 10 0 10 1 Kinetic Energy [GeV] 18
Summary and Conclusions • Simplified ab initio approach with observational inputs for large scale structures and turbulence quantities relevant to cosmic-ray modulation yields results in reasonable with all three solar minimum data sets. • Preliminary prediction for cycle 25 solar minimum is for intensity even greater than in 2009 • Model is still in the process of extension and refinement to full time dependence 19
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