104. Comparing Different Methods for Computing Solar Angular Momentum Loss

Author: Adam J. Finley, Sean P. Matt and Victor See, University of Exeter.

Introduction

There have only been a few of attempts to quantify the rate at which angular momentum is lost from the Sun due to the solar wind. In this nugget, we bring to light two methods, one using in-situ observations of the solar wind, coupled with recent stellar wind modelling, and the other based on studying the rotation rates of other Sun-like stars.

The rotation rates of Sun-like stars are observed to decrease with age due to angular momentum loss through their magnetised stellar winds. There are models that describe this rotation period evolution (e.g. [1]), which show us how stellar wind torques evolve over 100Myr to Gyr timescales, independently of our understating of the physical mechanism governing the wind torques. When applied to the Sun, these models predict a near-single value of 6.2×10³⁰erg for the solar wind torque, which is insensitive to the Sun’s previous rotation history (see Fig. 1). This is due to the observed convergence of rotation periods and the dependence of wind torques on rotation rates [2]. This is our first method, which we will compare to our recent magnetohydrodynamic (MHD) simulations results for the current solar angular momentum loss rate [3].

**Figure 1.** – Evolution of the stellar wind torque for a solar mass star (using the model of [1]). Tracks for three different initial rotation rates, fast, intermediate and slow are shown. The rotation rates and thus their wind torques converge over time, producing a single value at the age of the Sun of 6.2×10³⁰erg.

Previous Measurements

Here we draw attention to two previous estimates for the solar wind torque which utilise in-situ observations (see Fig. 2, upper panel); using the Helios spacecraft [4], and data from Ulysses [5]. The direct measurement of the solar angular momentum flow by [4] should be the most accurate method, however this work required assuming very significant spacecraft pointing corrections. Therefore, it is not clear how robust this measurement was. Both measurements are displayed in Fig. 2 with coloured dots.

**Figure 2.** Top: Solar angular momentum loss rate (solar wind torque) calculated using data from the ACE spacecraft and the open flux torque formulation from [7] vs. time (coloured line). Data from Cycle 23 is coloured green and the current Cycle 24 is blue. Previous estimates for the solar angular momentum loss rate are also plotted. Bottom: Sunspot number vs. time. We show cycle 24 to be weaker in both activity and the predicted torque. This could explain the larger value from [4] , which was measured during a stronger magnetic cycle.

Magnetohydrodynamic Modelling of Stellar Winds

We have considered two semi-analytic approaches to calculating the solar wind angular momentum loss. These semi-analytic prescriptions for the angular momentum loss rate are available [6,7], and are based on over 160 stellar wind simulations using the PLUTO MHD code [8]. The MHD simulations are performed using axisymmetric magnetic geometries combined with polytropic Parker-like wind solutions, which are relaxed to a steady state. The first semi-analytic description of the braking process is formulated using the surface dipole, quadrupole and octupole magnetic field components along with the mass loss rate of the wind. It is further shown that the large scale magnetic field (i.e. the dipole) is the most significant in controlling the angular momentum loss rate of stellar winds. A second semi-analytic formula is available which parameterises the solar wind torque using the open magnetic flux in the wind. The simplicity of the semi-analytic derivation for the open-flux torque formulation (see [9]) suggests that this method produces the most reliable torque. This method is shown to be insensitive to surface geometry and any details of how the field is opened [10].

The Sun’s Variable Angular Momentum Loss Rate

Both semi-analytic formulae are applied to the solar wind in [3]. We use remote observations of the surface magnetic field from SOHO/MDI and SDO/HMI (pySHTOOLS is used to decompose the field into components), and estimates of the mass loss rate and open magnetic flux from in-situ measurements taken by the ACE satellite. The solar wind torque is recovered over the last 20 years, and is shown to vary with the solar cycle using both formulae. However, the surface field formula produces a much smaller value for the solar wind torque than the open flux formula. The reason for this discrepancy may relate (in part) to the “open flux problem” where models of the solar wind often fail to predict the correct amount of open magnetic flux at 1au (based on solar magnetogram observations). Here we present the result from the open flux formulation in Fig. 2, compared with previous estimates [4,5]. The sunspot cycle is shown in the lower panel of Fig. 2, which appears correlated with the angular momentum loss rate. The average value of the solar wind torque is 2.3×10³⁰erg, which is smaller than required by considering the observed rotation period evolution of Sun-like stars (6.2×10³⁰erg). It remains unclear why the MHD wind torques do not agree with rotation evolution torques.

Conclusions

Using semi-analytic formulae (based on MHD simulations), we show the angular momentum loss rate of the Sun varies in phase with the solar spot cycle. The average value of the solar wind torque is discrepant depending on whether we use observations of the surface magnetic field, or the open magnetic flux in the wind as our input variables. Both of our MHD-based torques are found to be systematically lower than torques derived from rotation evolution models. This also appears to be true for other Sun-like stars [11], and even if we consider millennial-scale variability of the Sun by using indirect proxies of solar activity [12]. This implies that there are some fundamental issues with current stellar wind models and/or observations, which have yet to be understood.

References

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