Connecting solar radiance variability to the. solar dynamo using the virial theorem

Transcription

1 Connecting solar radiance variability to the solar dynamo using the virial theorem Kiepenheuer-Institut für Sonnenphysik, Freiburg i.br. steiner/

2 1. What can the virial theorem tell us? Assuming that the global solar luminosity varies approximately like solar irradiance, then the virial theorem unavoidable implies structural changes in the Sun or in partial layers of it on that same time scale. What kind of structural changes might it be? Composite of the total solar irradiance. From C. Fröhlich under more

3 What can the virial theorem tell us? (cont.) A general form of the virial theorem, including the magnetic field, is given by Chandrasekhar & Fermi (1953): 1 2 d 2 Z J dt = 2K + Ω + M + 3 P dv + S, 2 R J(t) = Z R ρr 2 dv, Z K(t) = 1 2 Ω(t) = 1 Z 2 R R ρv 2 dv, ρψdv, Z B 2 M(t) = dv, R 2µ 0 Z 3 P dv = 3(γ 1)U(t), R I S(t) = P tot ds + 1 I µ 0 R R (r B)B ds. where J is the moment of inertia, K the kinetic energy of mass motion, Ω the gravitational binding energy of the star with Ψ being the gravitational potential, M the magnetic energy, U the internal energy, and S a surface integral over the boundary R of the region R occupied by the star.

4 What can the virial theorem tell us? (cont.) 1 2 d 2 J dt 2 = 2K + Ω + M + 3 P dv + S R A variation of the total internal energy (as a consequence of luminosity variation) goes along with corresponding changes in one or several of the other virials, one of which being surely the total magnetic energy.

5 What can the virial theorem tell us? (cont.) The region R encompasses the convection zone, where the lower boundary lies beneath the overshoot region in the radiative zone with p g = const and B = 0. At the upper boundary P g = 0. R p g= const core R 1 convection zone R 2 I S(t) = R = 4πR 3 1p g (R 1 ) {z } [J] P tot ds + 1 µ 0 I + 1 µ 0 4πR 3 2fB 2 (R 2 ) {z } [J] R (r B)B ds +4πR 3 2f B2 (R 2 ) 2µ 0 {z } [J]

6 2. Estimates of the various virials The magnetic virial: M(t) = R B2 /(8π)dV. One solar cycle generates a magnetic flux of Φ Wb. If this flux resides in the overshoot layer in the form of flux tubes with a strength of 10 T, it has a total magnetic energy of M J The inertial virial: d 2 J/dt 2 J/τ 2 cyc R ρr2 dv/τ 2 cyc. d 2 J/dt M The thermal virial: The equivalent variation in thermal energy over a solar cycle due to the solar luminosity variation is δl L so that U rad J Coincidence of equal magnetic and thermal energy change (Schüssler, 1996)

7 Estimates of the various virials (cont.) The kinetic virial: The kinetic energy of convective motion in the convection zone is K con = (1/2) R ρv2 con dv = [J] = 0.16 M. In the tachocline, where magnetic intensification presumably takes place, the available convective kinetic energy reduces to K con 10 4 M. Also, because F conv v 2 conv <! 10 3, convective motion cannot be tapped in sufficient amounts. Since turbulent, convective motion is unlike to produce coherent large scale magnetic flux ropes in the overshoot layer, we assume the variation of K con with the solar cycle to be negligible, K con = 0. The kinetic energy of rotation in the convection zone is K rot = (1/2) R ρ(ω(θ,r) r)2 dv = [J] = 110 M. However, the difference to the kinetic energy of rigid rotation under the condition of equal rotational momentum, i.e., the available kinetic energy from differential rotation is K rot = [J] = 2 M.

8 Estimates of the various virials (cont.) The available kinetic energy from differential rotation in the tachocline alone amounts to K rot = [J] = M. Thus, differential rotation in the magnetic layer would have to be very efficiently replenished from layers atop for feeding the energy required for field intensification. In lack of such a mechanism we have K(t) = (1/2) R ρv2 dv = const Torsional oscillations are due to a geostrophic flow driven by temperature variations at the surface according to Spruit (2003), not to Lorentz forces.

9 Estimates of the various virials (cont.) Considering the variations over a solar cycle, only three significant virias of the full virial equation, (1/2) d2 J dt 2 = 2K + Ω + M + 3 R P dv + S, remain: Ω + M + 2U = 0. Note that for the total convection zone Ω = [J] and U = [J] A possible scenario that relates the three terms Ω, M, and U over a solar cycle is provided by the flux-tube dynamo.

10 3. Magneto-convective energy transport Magnetic flux tube at the bottom of the convection zone (dashed circle), core convection zone

11 3. Magneto-convective energy transport Magnetic flux tube at the bottom of the convection zone (dashed circle), rising through the convection zone to the core surface (solid red circle), convection zone

12 3. Magneto-convective energy transport Magnetic flux tube at the bottom of the convection zone (dashed circle), rising through the convection zone to the core surface (solid red circle), where it forms a sunspot pair convection zone - Schüssler, Caligari, Ferriz-Mas, Moreno Insertis et al. - Fisher, Fan, Longcope, Linton et al. - D Silva, Choudhuri et al.

13 Magneto-convective energy transport (cont.) 0 ln B µ zcrit p ln p 0e i e Gas pressure decrease more rapidly in the superadiabatic external atmosphere, p e, than in the adiabatic flux-tube atmosphere, p i.

14 Magneto-convective energy transport (cont.) 0 ln B µ zcrit p ln p 0e i e z exp Gas pressure decrease more rapidly in the superadiabatic external atmosphere, p e, than in the adiabatic flux-tube atmosphere, p i. At the critical height z crit = z exp, the magnetic pressure must vanish and the flux tube explodes into a cloud of weak field.

15 Magneto-convective energy transport (cont.) 0 ln B µ zcrit p ln p 0e i e z exp Gas pressure decrease more rapidly in the superadiabatic external atmosphere, p e, than in the adiabatic flux-tube atmosphere, p i. At the critical height z crit = z exp, the magnetic pressure must vanish and the flux tube explodes into a cloud of weak field. Flux intensification by flux-tube explosion : Moreno Insertis, Caligari, & Schüssler (1995), Rempel & Schüssler (2001) more

16 Magneto-convective energy transport (cont.) Simulation of the flux-tube explosion by Matthias Rempel Evolution of the entropy and the magnetic field strength in the course of a flux-tube explosion.

17 Magneto-convective energy transport (cont.) Thermal and flow structure beneath a sunspot. From the SOI/MDI home page by Kosovichev, et al. Left: The sound speed beneath a sunspot. Redder colors are faster (hotter) and bluer colors are slower (cooler). Right: Motion beneath a spot. Material flows at the depth of 0 3 Mm (a) and at 6 9 Mm (b). The outline of the sunspot umbra and penumbra is shown. The colors indicate vertical motion with red corresponding to downflow. The arrows show horizontal motion. The longest arrows correspond to 1.0 km/s.

18 4. The solar cycle in terms of the virial theorem 1 Weak toroidal field of B 1 T. Ω 1, M 1, and U 1 initial virials. 2 This field, located in a sheet near the bottom of the convection zone, gets amplified to 10 T by the process of flux-tube explosion. M = M 2 M 1 M J > 0, Ω = Ω 2 Ω 1? 0, so that, according to the virial theorem, U = U 2 U 1 = 1 2 ( Ω + M) 1032 J < 0, leaving an internal energy deficiency of another J, which must be gained by a reduction in radiation loss from the Sun in fulfillment of E tot 1 = U 1 + Ω 1 + M 1 = E tot 2 = U 2 + Ω 2 + M 2 + R ր

19 The solar cycle in terms of the virial theorem (cont.) 3 Magnetic field gets ejected or annihilated, clearing stage for the next solar cycle with opposite magnetic sign. Time of magnetic activity at the solar surface. M = M 3 M 2 M J < 0, Ω = Ω 3 Ω 2? 0, so that, according to the virial theorem, U = U 3 U 2 = 1 2 ( Ω + M) 1032 J > 0, leaving an internal energy excess of J, which must be lost by an excess of radiation from the Sun.

20 The solar cycle in terms of the virial theorem (cont.) The thermodynamic cycle: 1st stage 2nd stage extra entropy deficient downflow convection zone throttled energy flux convection zone extra radiation extra entropy rich upflow These processes work on a hydrodynamic time-scale.

21 5. Conclusion According to the present ideas - Solar radiance variability is an expression of the thermodynamic cycle associated with the magnetic solar cycle - The virial theorem provides an explanation for the hitherto coincidence of the magnetic energy generated over one solar cycle equating the solar luminosity variation integrated over the same period - Observational consequences might bear the predicted - variation of the superadiabaticity, δ, with the solar cycle, - correlation between stellar luminosity variation and the magnetic energy generated over a stellar cycle

22 Hydrodynamic vs. magnetohydrodynamic mixing-length convection s T mc mix s T mix T F tot = (1 ϕ)f c + ϕf mc δf tot = ϕ(f mc F c ) backto 3

23 The excess energy flux caused by the flux-tube explosion is according to Rempel (2001) δl = ṁ st = ṁt c p ( ad ) r = ṁ g δ r, H p {z } ad δ where ṁ is the mass per cycle period to be discharged from the initial magnetic sheet at the bottom of the convection zone: A = 8 π R 2 c d R c δl = L» ṁ = ρ 0 Ad/τ cyc, d = δ 10 6 Φ 1 πr = m. 2 cb 0» r 10 8 m backto 3

24 dweak flux sheetstrong flux sheet Abstract flux intensification process convection zone superadiabatic subadiabatic dense flux sheet core Mass in a weak magnetic layer is bottled and returned on top. Strong flux sheet buoyant below a slightly lifted atmosphere. Upward drift in subadiabatic overshoot layer reestablishes mechanical equilibrium. U 1, Ω 1, M 1 0 U 2 < U 1, Ω 2 =? Ω 1, M 2 > 0 backto 3

25 Facular regions as seen in the ecliptic plane and perpendicular to it view in ecliptic plane view perpendicular to ecliptic plane backto 1

26 Table of content 1. What can the virial theorem tell us? 2. Estimates of the various virials 3. Magneto-convective energy transport 4. The solar cycle in terms of the virial theorem 5. Conclusion 6. References

27 References Chandrasekhar, S. and Fermi, E.: 1953, ApJ 118, 116 Fröhlich C. and Lean J., 1998, in New Eyes to see inside the Sun and Stars, F.L. Deubner (ed.), IAU Symp. 185, Kluwer, Dordrecht Moreno Insertis, F., Caligari, P., and Schüssler, M.: 1995, ApJ 452, 894 Rempel, M.: 2001, Struktur und Ursprung starker Magnetfelder am Boden der solaren Konvektionszone, PhD-Thesis, Georg-August-Universität Göttingen Rempel, M. and Schüssler, M.: 2001, ApJ 552, L171 Schüssler, M.: 1996, in Solar and Astrophysical Magnetohydrodynamic Flow, NATO ASI Series Vol. 481, K.C. Tsiganos (ed), Kluwer, Dordrecht, p. 17 Steiner, O.: 2003, Astronomische Nachrichten, 324, 106 Steiner, O.: 2004, in Multi-Wavelength Investigations of Solar Activity, IAU Symp. 223, A. V. Stepanov, E.E. Benevolenskaya, & A.G. Kosovichev (eds.), in press