Ian Cunnyngham1, Marcelo Emilio2, Jeff Kuhn1, Isabelle Scholl1, Rock Bush3
1 Institute for Astronomy, University of Hawaii, 34 Ohia Ku St., Pukalani, Maui, HI, 96790, USA
2 Ponta Grossa State University, Ponta Grossa Parana, Brazil, 84030-900.
3 Stanford University, Stanford, CA 94305, USA
A new analysis using about four years of HMI 5-min solar p-mode limb oscillations as a rotation “tracer” finds a large velocity gradient in a thin region at the top of the photosphere1. This shear occurs where the solar atmosphere radiates energy and angular momentum. We suggest that the net effect of the photospheric angular momentum loss is similar to Poynting-Robertson “photon braking” on, for example, Sun-orbiting dust. The resultant photospheric torque is readily computed and, over the Sun’s lifetime, is found to be comparable with the apparent angular momentum deficit in the outer 5% of the Sun known as the near-surface shear layer.
From space, limb solar oscillations can be measured to microarcsecond positional and 10-6 relative brightness accuracy with the HMI2. This instrument obtains 4K × 4K pixel full-disk images in narrow wavelength passbands over a range of linear and circular optical polarization states every 45 seconds [3]. We use six 7.6 pm-wide passbands that are spaced in wavelength by 7.0 pm steps across an Iron Fraunhofer line at a central wavelength of 617.334 nm. We analyzed several polarization states in each of the six filtergram timeseries to obtain 12 independent measurements of the limb brightness αi (θ,t) and position, βi (θ,t) (i=0 … 11) following the techniques described in ref. 2. For these measurements the different polarization states provide independent datasets with no apparent polarization dependence. The mean limb position (solar radius) is derived from the average of βi over θ. Fig. 1 shows the apparent solar radius in each filter over the 3.5 year duration of these data. This limb displacement is largest (biggest solar radius) at the line core and varies with observation time because the solar-frame filter wavelengths vary with the orbital Doppler shift of the satellite. This radius variation is also consistent with the apparent radius variations derived from HMI Venus transit timing data using a different analysis3. We compute the effective height of each dataset, i, from the Fig. 1 data.
Figure 1 | Measured solar radius variation versus filter wavelength during a 3.5 years duration observing window. Colors indicate HMI filtergrams and their nominal central wavelength steps in Angstroms are indicated on the right axis. The HMI continuum data are indicated with their observing campaign labels ‘mercury’ and ‘venus.’
As this surface structure rotates over the limb onto or off the visible solar disk the projection in the plane of the sky causes this oscillating limb structure to appear to rotate clockwise or counterclockwise along the limb at a rate Ψ, depending on whether the north rotation axis is pointed into or out of the plane. We use the fact that the Sun’s axis is inclined by B=7.2 degrees from the normal to the ecliptic to determine the mean solar rotation at the atmospheric depth of the limb structure.
Acoustic p-modes with 5-min periods brighten and displace the photosphere with cyclic frequencies νnlm where n, l, and m are the radial, angular and azimuthal spherical harmonic mode indices. For example, the solar surface displacement due to an (nlm) p-mode has the spatial form δr(θ,ϕ,t)=Real(anlm Ylm (θ,ϕ) exp(iωnlm t)) with Ylm a spherical harmonic. The mode displacement amplitudes are of order 10 microarcseconds, which corresponds to modal relative brightness amplitudes of about 10-6. Fig. 2 shows an example limb k-ν power spectrum. Full-disk p-mode frequencies for l=k and m=0 are over-plotted to show the agreement with conventional Doppler helioseismology4. Each ridge structure that runs from the lower left to upper right here corresponds to a fixed radial p-mode node number, n. Only the highest amplitude, and highest k=l data point in each “parabolic” power distribution corresponds to the frequency of the global p-mode. Spatial mode “leakage” accounts for the spurious parabolic structure.
Figure 2 | Limb power spectrum, |α ̃(k,ν)|2, for line core filter data. Overplotted open circles show observed global m=0 mode frequencies4. Blue cross symbols show selected mode leakage calculations of some m≠0 full-disk modes for the indicated angular (l) and radial (n) modes into the limb harmonics, k, indicated on the vertical axis. The color scale legend is indicated on the right in units of fractional brightness fluctuation (squared) per frequency bin.
The circular average mean solar rotation is derived from the time variation of the frequency shift of the power ridges in the k-v spectra. The total vertical range of all the limb measurements is about 120 km in the photosphere and the 12 measurements with six filter wavelengths and several polarization states cluster into three different heights. Fig. 3 shows these near-limb rotation results compared to Doppler and GONG full-disk p-mode inversion rotation results5 on logarithmic vertical and horizontal scales. This photospheric shear is apparently larger than has been measured anywhere in the interior.
Figure 3 | Solar angular rotation rate versus depth and colatitude on a log-log scale derived from ref. 7. ‘Limb’ points (cross symbols) show our observed mean solar rotation shear; ‘Mean Doppler’ (star symbols) indicate the angle-averaged Doppler rotation over the indicated indeterminate depth range; ‘Inversion’ shows the p-mode inversion rotation rates at three latitudes through the interior, with the near-surface and tachocline shear zones annotated. The Sun’s temperature minimum defines the outer reference radius (Rsun) for the horizontal scale.
The torque, or angular momentum loss rate, implied by the Sun’s luminous power, P, and rotation is
dL/dt= (2PΩR2)⁄(3c2) which, when integrated over the lifetime of the Sun, is within an order of magnitude of the angular momentum deficit implied by the helioseismic near surface shear rotation. Ref. 1 further discusses the radial form of the surface rotation and how this deficit can be consistent with the Sun’s surface diffusion rate and estimates of the surface viscosity.
References
[1] Cunnyngham, I., Emilio, M., Kuhn, J. R., Scholl, I, Bush, R., 2017, Phys. Rev. Letts. , 118 , 051102
[2] Kuhn,J. R., Bush, R., Emilio, M., & Scholl, I. F. 2012, Science , 337, 1638
[3] Emilio, E., Couvidat, S., Bush, R.I. Kuhn, J. R., Scholl, I. F. 2015, ApJ, 798, 48
[4] Korzennik, S. G. 2005, ApJ , 626, 585
[5] Schou, I., et al. 1998, ApJ, 505, 390