Direct Detection of Solar Angular Momentum Loss with the Wind Spacecraft

The measurements required to accurately constrain the AM content in the solar wind particles are challenging to make (see the discussion in Section 3a of Pizzo et al. 1983). Not only are the fluctuations in the AM flux comparable to the average value, but from an instrument standpoint, small errors in determining the wind velocity translate to large errors in the AM flux (because the radial wind speed is 2–3 orders of magnitude larger than the typical tangential speed of 1–10 km s⁻¹ at 1 au). The latter problem appears to be the main reason why data from most spacecraft have not been used to measure AM (see Figure 6 of Sauty et al. 2005, which shows data from the Ulysses spacecraft; there is an approximately 1 yr periodicty in the observations that is likely due to spacecraft pointing). The magnetic field direction is generally more accurately determined because it is not as radial as the flow, and the instruments used are less sensitive to spacecraft pointing than the particle detectors (which get different exposures as the spacecraft pointing changes). Therefore, the magnetic stress component of the AM flux is typically better constrained.

While the Advanced Composition Explorer spacecraft's nonradial solar wind speed measurements show the expected behaviors during periods of high variability (Owens & Cargill 2004), they appear to suffer from the same spacecraft-pointing-related issues as Ulysses over longer time averages, in this case showing a strong ∼6 month periodicity. The Wind and Interplanetary Monitoring Platform 8 (IMP8) spacecraft do not obviously show such features. Furthermore, during the period of overlap between Wind and IMP8, there is good agreement in tangential wind speed, both in terms of the distributions and time series (linear regression of r = 0.81 at the hourly timescale), suggesting limited instrumental effects.

In this work we focus on the high time cadence Wind observations. Wind was designed to be a comprehensive solar wind laboratory for long-term solar wind measurements, and has certainly stood the test of time; currently approaching its 25th yr since launch (1994 November 1st). During its mission lifetime the Wind spacecraft completed multiple orbits of the Earth–Moon system, before relocating to a halo orbit about the L₁ Lagrangian point (on the Sun–Earth line) in 2004 May. All the while collecting plasma and magnetic field measurements of the solar wind and Earth's magnetosphere with the Solar Wind Experiment (Ogilvie et al. 1995; Kasper et al. 2006) and Magnetic Field Investigation instruments (Lepping et al. 1995).

We analyze data recorded by the Wind spacecraft⁴ from 1994 November to 2019 June. Using data taken when the spacecraft was immersed in the solar wind, i.e., outside the Earth's magnetosphere. Additionally, we remove times when the spacecraft encountered interplanetary coronal mass ejections (ICMEs) using the catalogs⁵ of Cane & Richardson (2003) and Richardson & Cane (2010) because ICMEs can produce large, nonradial, local flows that are not likely representative of global AM loss (Owens & Cargill 2004). For times not covered by the ICME catalog (1994 November–1996 June), we remove data with properties that are indicative of ICMEs, specifically data with a proton density greater than 70 cm⁻³ or field strengths greater than 30nT (a similar method was used by Cohen 2011 on Ulysses data).

Measurements of the solar wind magnetic field vector, proton density, and velocity are available throughout the entire Wind mission at ∼2 minute cadence. These parameters have a small number of entries flagged by the instrument team as containing unusable data, which we simply remove. Similarly, measurements of the alpha particle density and velocity are available; however, the number of unusable data entries (where the proton and alpha particle populations cannot be deconvolved by the detector) is far greater. Therefore, when the alpha particles are flagged as unusable, we assume that the alpha particle density is 4% of the proton density (a representative value taken from Borrini et al. 1983) and that the alphas' velocities are identical to the protons'. We transform the vector quantities of velocity and magnetic field from GSE coordinates to RTN coordinates, where R points from the Sun to the spacecraft, T points perpendicular to the Sun's rotation axis in the direction of rotation, and N completes the right-handed triad (further details are available in Fränz & Harper 2002).

For each quantity derived using Wind data in this work, we calculate values at the smallest available cadence (∼2 minutes) and then average them over each Carrington rotation (CR, ∼27 days) in our data set. This helps to remove longitudinal variability caused by the rotation of features on the solar surface and smooths local fluctuations that occur on a range of shorter timescales. Finally, we require that each CR-average has more than 50% of the data from that time period (after our cuts have been made). Otherwise, that CR is removed. In the top panel of Figure 1, we plot the tangential wind speed of the protons and alpha particles as observed by Wind. For the tangential speeds shown in Figure 1, we have weighted the CR averages by density, in order to reduce the obscuring effect of wind stream-interactions (see the discussion in Section 3.3). Figure 1 shows typical tangential flow speeds of a few km s⁻¹, with variability that appears genuine and not to suffer from the errors present in data from other spacecraft (as discussed in Section 2.1).

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The solar wind removes AM from the Sun at a rate proportional to the mass flux (ρv_r) multiplied by the specific AM per unit mass (Λ). Using data from the Wind spacecraft, we plot the mass flux in the protons, alpha particles, and their total in the middle panel of Figure 1. We multiply each by 4πr² for an estimate of the global mass-loss rate,

$$\begin{eqnarray}&&\dot{M}\approx \langle 4\pi {r}^{2}({\rho }_{p}{v}_{r,p}+{\rho }_{\alpha }{v}_{r,\alpha }){\rangle }_{\mathrm{CR}},\end{eqnarray} \tag{ 1 }$$

where the spacecraft's radial distance from the Sun is r, the radial wind speed is v_r, the solar wind density is ρ, the subscripts p and α denote the proton and alpha particle components, and $\langle {\rangle }_{\mathrm{CR}}$ denotes an average over a (∼27 day) CR. The total mass flux is dominated by the proton component of the wind and varies in a way that does not precisely correlate with the Sun's activity cycle (see also Phillips et al. 1995; McComas et al. 2000; Finley et al. 2018; Mishra et al. 2019). By contrast, the alpha particle mass flux appears to be more strongly correlated with solar activity throughout the Wind data set (which is not surprising as the relative abundance of helium in the equatorial solar wind is strongly correlated with solar activity, see Kasper et al. 2007).

We define the specific AM as the AM flux divided by the proton mass flux (i.e., the specific AM per proton in the solar wind), which is given by,

$$\begin{eqnarray}&&{\rm{\Lambda }}={\left\langle r\sin \theta \left({v}_{t,p}+{v}_{t,\alpha }\displaystyle \frac{{\rho }_{\alpha }{v}_{r,\alpha }}{{\rho }_{p}{v}_{r,p}}-\displaystyle \frac{{B}_{t}{B}_{r}}{4\pi {\rho }_{p}{v}_{r,p}}\right)\right\rangle }_{\mathrm{CR}},\end{eqnarray} \tag{ 2 }$$

where θ is the heliographic latitude of the spacecraft, v_t is the tangential wind velocity, B_r is the radial magnetic field strength and B_t is the tangential magnetic field strength. The first term in Equation (2) is the mechanical AM carried by the protons, the second term relates to the relative contribution of the alpha particles, and the final term describes the AM content of the magnetic field stresses. Equation (2) does not include the correction factor for the magnetic stresses which accounts for thermal pressure anisotropies, as it is expected to be negligible (see Marsch & Richter 1984b). In the bottom panel of Figure 1, we plot the total specific AM along with the individual proton, alpha particle, and magnetic field components. We use density-weighted tangential velocities, as in the top panel of Figure 1, to reduce the effect of wind stream-interactions (see the discussion in Section 3.3). Figure 1 shows the protons to dominate the specific AM of the solar wind, with the magnetic field stresses and alpha particles carrying much less specific AM (per proton).

The total AM flux in the protons, alpha particles, and magnetic field stresses is given by multiplying the specific AM by the proton mass flux,

$$\begin{eqnarray}\begin{array}{rcl}{F}_{\mathrm{AM}} & = & \langle {\rho }_{p}{v}_{r,p}{\rm{\Lambda }}{\rangle }_{\mathrm{CR}}=\left\langle \Space{0ex}{2.8ex}{0ex}r\sin \theta \left(\Space{0ex}{2.7ex}{0ex}{\rho }_{p}{v}_{r,p}{v}_{t,p}\right.\right.\\ & & +\,{\left.\left.{\rho }_{\alpha }{v}_{r,\alpha }{v}_{t,\alpha }-\displaystyle \frac{{B}_{t}{B}_{r}}{4\pi }\right)\right\rangle }_{\mathrm{CR}}.\end{array}\end{eqnarray} \tag{ 3 }$$

We plot the AM fluxes (multiplied by radial distance squared) in the protons, alphas, magnetic field, and their total in the top panel of Figure 2. There is a large scatter/variability in the AM flux, despite averaging over whole CRs. The variability is mainly due to the varying specific AM (i.e., in the tangential wind speed), rather than changes in the mass flux (see Figure 1), and which is likely affected by local fluctuations in the solar wind, caused by transients (Roberts et al. 1987; Tokumaru et al. 2012). The solid black line in Figure 2 shows a 13-CR (i.e., ∼1 yr) moving average on the total AM flux, which more clearly describes the longer-term variability of the AM flux. Our data set contains sunspot cycles 23 and 24 (left and right halves of the figures, respectively), which have notable differences in their AM fluxes. Generally, during times of increased solar activity the specific AM of the protons and magnetic field stresses increase together, such that F_AM,p/F_AM,B does not vary with solar activity. We find cycle 24, which is currently in its declining phase, has a much lower average AM flux than cycle 23 (∼40% of cycle 23).

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The average value for the AM flux, and that of each constituent, is listed and compared to previous estimates in Table 1. The Wind total is primarily composed of the proton and magnetic field components, with the alpha particles contributing a small and mostly negative AM flux contribution. In comparison with the work of Pizzo et al. (1983) and Marsch & Richter (1984a), the Wind data show a much stronger AM flux in the protons and a large reduction (in amplitude) to the AM flux carried by the alpha particles. These differences could be related to long-term change in the solar wind. For example, the solar wind appears denser in the last decade compared to the Helios era (see McComas et al. 2013). Or alternatively, due to the exchange of momentum between protons and alphas as the wind propagates into the heliosphere (for which there is some evidence in Sanchez-Diaz et al. 2016).

Table 1.  Mean of the CR-averaged Solar Angular Momentum Fluxes

Component	$\langle {r}^{2}{F}_{\mathrm{AM}}\rangle$	Source	Citation
	(×1030erg/ster)
Protons	0.29	Wind	This work
	0.17	Helios	Pizzo et al. (1983)
	∼1	Mariner 5	Lazarus & Goldstein (1971)
Alpha Particles	−0.02	Wind	This work
	−0.13	Helios	Pizzo et al. (1983)
Magnetic Field	0.12	Wind	This work
	0.15	Helios	Pizzo et al. (1983)
	0.23	Mariner 5	Lazarus & Goldstein (1971)
Total	0.39	Wind	This work
	0.20	Helios	Pizzo et al. (1983)
	0.26	Theory	This work, Equations (4) and (7)

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The AM flux in the magnetic field stress in the Wind data is similar to that determined by Pizzo et al. (1983) and Marsch & Richter (1984a) but is smaller than that determined by Lazarus & Goldstein (1971). Interestingly, the dominant contribution to the Wind-measured AM flux comes from the protons, with the magnetic field of secondary importance. In simplified MHD simulations of the solar wind (such as those of Finley & Matt 2017), the ratio F_AM,p/F_AM,B depends on parameters such that the larger the Alfvén radius (R_A) the larger the contribution of the magnetic field. The average ratio measured by Wind is F_AM,p/F_AM,B = 2.6, which is significantly different from the ratio of ∼1 found by Pizzo et al. (1983). Marsch & Richter (1984a) showed that Helios data from smaller heliocentric distances gives larger ratios, which might account for the difference. The proton-dominated regime shown by the Wind data is consistent with MHD simulations that have cylindrically averaged R_A smaller than 15R_⊙.

To show our result in the context of current theoretical predictions, we compare to the AM loss rate of Finley et al. (2018), which was derived using MHD simulations. In their work, the AM loss rate is given by,

$$\begin{eqnarray}\begin{array}{rcl}{\dot{J}}_{\mathrm{FM}18} & = & (2.3\times {10}^{30}[\mathrm{erg}]){\left(\displaystyle \frac{\dot{M}}{1.1\times {10}^{12}[{\rm{g}}{{\rm{s}}}^{-1}]}\right)}^{0.26}\\ & & \times {\left(\displaystyle \frac{{\phi }_{\mathrm{open}}}{8.0\times {10}^{22}[\mathrm{Mx}]}\right)}^{1.48},\end{array}\end{eqnarray} \tag{ 4 }$$

where the AM loss rate of the Sun is parameterized in terms of the mass-loss rate, $\dot{M}$, and the open magnetic flux, ${\phi }_{\mathrm{open}}$. The open magnetic flux in the solar wind is estimated by,

$$\begin{eqnarray}&&{\phi }_{\mathrm{open}}=\langle 4\pi {r}^{2}| {B}_{r}{| }_{1{hr}}{\rangle }_{\mathrm{CR}},\end{eqnarray} \tag{ 5 }$$

where the average value of the radial magnetic field is assumed to be representative of the global open magnetic flux in the solar wind. This assumption has been discussed by many previous authors (Wang & Sheeley 1995; Lockwood et al. 2004; Pinto & Rouillard 2017) and has observational support (Smith & Balogh 1995; Owens et al. 2008). Using Equation (5) we plot the open magnetic flux using data from the Wind spacecraft in the middle panel of Figure 1 with a solid green line.

Using Equation (4) we calculate the predicted AM loss rate of the solar wind, where the mass-loss rate and open magnetic flux (Equations (1) and (5)) are calculated using data from the Wind spacecraft. We then relate the AM loss rate and AM flux using,

$$\begin{eqnarray}&&\dot{J}={\oint }_{A}{{\boldsymbol{F}}}_{\mathrm{AM}}\cdot d{\boldsymbol{A}}={F}_{\mathrm{AM},\mathrm{eq}}{\int }_{0}^{2\pi }{\int }_{0}^{\pi }{r}^{2}{\left(\sin \theta \right)}^{3}d\theta d\phi ,\end{eqnarray} \tag{ 6 }$$

where A represents a closed surface in the heliosphere (we adopt a sphere of radius r), ϕ is heliographic longitude, and F_AM,eq is the AM flux in the solar equatorial plane, assumed to be equivalent to that measured by CR averages of data taken in the ecliptic. As the AM flux in the solar wind is expected to vary with latitude, we have assumed a physically motivated functional form,⁶ ${{\boldsymbol{F}}}_{\mathrm{AM}}(\theta )\approx {F}_{\mathrm{AM},\mathrm{eq}}{\left(\sin \theta \right)}^{2}\hat{{\boldsymbol{r}}}$. By rearranging Equation (6) we produce a relation for the equatorial AM flux,

$$\begin{eqnarray}&&{F}_{\mathrm{AM},\mathrm{eq}}=\displaystyle \frac{\dot{J}}{2\pi {r}^{2}\int {\left(\sin \theta \right)}^{3}d\theta }\approx \displaystyle \frac{\dot{J}}{2.7\pi {r}^{2}}.\end{eqnarray} \tag{ 7 }$$

The AM flux from Equation (7), using the AM loss rate from Equation (4), is plotted with a solid purple line in the top panel of Figure 2. Strikingly, this result matches well during solar minimum wind conditions. However, it consistently underestimates the AM flux during solar maxima. The Finley et al. (2018) AM loss rates were derived from simulations with only one wind acceleration profile, but differing wind acceleration profiles have been shown to affect the predicted AM loss rates (Pantolmos & Matt 2017). Therefore changes in the balance of fast and slow wind in the heliosphere are not taken into account by this model. It is known that the proportion of slow wind changes significantly from solar minimum to maximum, while the ecliptic remains essentially dominated by the slow wind the whole time (Wind encountered slow wind streams, with v_r < 500 km s⁻¹, 82% of the time). Importantly, this implies that the Wind observations may be more representative of global conditions at solar maximum than solar minimum. Uncertainties in our assumed latitudinal distribution of AM flux prevent us from producing a more conclusive estimate of the global AM loss rate. For us to better constrain this, there is a need for simultaneous observations at higher latitude (e.g., combined measurements with both the Wind spacecraft and the upcoming Solar Orbiter) but at present, the current approach is the best we can do without introducing further uncertainty.

Rearranging Equation (7) produces an estimate of the global AM loss rate based on the average AM flux detected by the Wind spacecraft, ${\dot{J}}_{{Wind}}=2.7\pi \langle {r}^{2}{F}_{\mathrm{AM}}\rangle =3.3\times {10}^{30}$ erg. This AM loss rate is approximately half that required by the empirical Skumanich relationship, where rotation period evolves proportional to the square root of stellar age (Skumanich 1972). Specifically, for the Sun's rotation to follow the Skumanich relationship, the present-day AM loss rate must be ≈6.2 × 10³⁰ erg (Finley et al. 2018). The torque-averaged Alfvén radius, ${R}_{{\rm{A}}}=\sqrt{\dot{J}/(\dot{M}{\rm{\Omega }})}$, implied by the Wind result is R_A ≈ 15R_⊙, in contrast to R_A ≈ 20R_⊙ using the AM loss rate required for Skumanich-like rotation. We note the value of R_A from Wind is in better agreement with MHD simulations that reproduce the observed ratio of F_AM,p/F_AM,B (see Section 2.4).

The (unknown) systematic uncertainties in our result could be large enough to resolve this discrepancy. However, taken at face value, and assuming the Sun is not special, our result could be evidence that stars deviate significantly from the Skumanich relationship at around the solar age (or Rossby number, for example, as suggested by van Saders et al. 2016). Alternatively, our result could mean that the present-day solar wind is in some kind of "low state," such that the AM loss rate averaged over timescales of ≫25 yr is significantly larger (see Finley et al. 2018, 2019a for a discussion and other caveats).

Detecting the AM flux is complicated by the myriad of transients and fluctuations in the solar wind. With sufficient spatial averaging of the heliosphere (or sufficient temporal averaging at a fixed location), the contribution of transients to the AM flux is likely to be small. However, with the available observations, large transient structures can bias estimates of the AM flux. In this work we have attempted to remove times when ICMEs interacted with the Wind spacecraft. We show the number of near-Earth ICMEs per CR as a color gradient in the bottom panel of Figure 2, which is well correlated with solar activity. The plasma properties of ICMEs are often very different to the ambient wind, typically having stronger magnetic fields and increased mass fluxes. Surprisingly, if we include these events in our calculation, the computed equatorial AM flux decreases by 4%. Although we have been careful to remove such events, ICME catalogs are not perfect, and therefore errors due to ICMEs are more likely to be introduced in times of high solar activity, or times where no ICME catalogs are available (i.e., 1994 November–1996 June).

Additionally, as noted by previous authors (Lazarus & Goldstein 1971; Pizzo et al. 1983; Marsch & Richter 1984a), our results contain evidence for fast–slow wind interactions. The net effect of these interactions is expected to be zero, given sufficient averaging. We plot the average AM flux in the solar wind particles with radial wind speeds greater and less than 500 km s⁻¹ separately in the lower panel of Figure 2. The slower component of the wind, when compared with the total particle AM flux plotted in black, is shown to carry the bulk of the AM flux in the particles. The faster component is shown to have a mostly small or negative AM flux. However, this component does not strongly contribute to the total AM flux during each CR because of the small fraction (on average 18%) of the time Wind encountered this flow, but also because fast wind streams tend to carry smaller mass flux, further reducing their contribution to the total AM flux.

This dichotomy between faster and slower wind streams occurs because of interactions within the solar wind as it propagates into interplanetary space. When fast and slow wind streams "collide," the slow wind undergoes an acceleration in the direction of corotation and the fast component is deflected oppositely (see Figure 1 in Pizzo 1978). Though most of this acceleration occurs in the radial direction, some is directed tangentially. The impact this has on our fluxes is far more pronounced in the faster component because it is typically less dense than the slower component. This effect makes the tenuous AM flux signal harder to distinguish when simply looking at the raw tangential wind speeds, and has been shown to become increasingly important with increasing heliocentric distances (see Figure 2 in Marsch & Richter 1984a). Since Wind data are taken at ∼1 au, and in the equatorial plane (where stream-interactions are expected to be more pronounced), we chose to present the tangential wind speeds and specific AM in Section 2 weighted by density. Doing so produces values that are more representative of their contribution to the AM flux.