5.6 Pitfalls of low-degree splitting measurements

Unfortunately, all the measurements described in Section 5.5 suffer from similar problems, as summarized below:

  1. The two components of the l = 1 mode are so close together (probably less than one microhertz apart) that they are resolved only for modes below about 2.2 mHz. This has implications for the measurements:

    1. Estimates of the splittings of unresolved components are highly prone to systematic error (Appourchaux et al., 2000a).
    2. The components that can be resolved have small amplitudes (Figure 15View Image) and therefore require both observations over extended periods and high signal-to-noise ratios.
    3. On the other hand, these low-frequency modes have the advantage that they show very little frequency shift with the solar cycle, which simplifies the analysis of long time series.
  2. Even though the low-degree modes penetrate deep into the solar interior, they spend most of their time in the outer layers of the Sun and are not very sensitive to the core; conversely, estimates of the core rotation are very sensitive to small errors in the splitting measurements.
  3. In order to properly estimate the rotation profile in the deep interior it is necessary to combine the low-degree splittings with medium-degree ones in an inversion. However, because the low-degree modes are so few – a few dozen at most, compared to a couple of thousand medium-degree multiplets with tens of thousands of individual frequencies or coefficients – the need for extremely precise measurements is even more pressing. Also, combining data from different instruments with different systematic errors may cause problems, particularly if the observations were made at different epochs of the solar cycle.

Point 1 above was noted by Loudagh et al. (1993Jump To The Next Citation Point) and Elsworth et al. (1995), and point 2 by Loudagh et al. (1993) and Lazrek et al. (1996), who point out that “An accuracy of about 30 nHz, or (1 year)–1 on the measurement of the l = 1 rotation splitting does not really permit, then, to discriminate between a solar core rotating twice as fast as the rest or not rotating at all!”. An approach to addressing point 3 was made by Tomczyk et al. (1995) with the newly-built LOWL instrument, an imaging instrument optimized for lower degrees. They obtained splittings for 1 ≤ l ≤ 100, and inferred a rotational profile down to 0.2 R⊙, finding a rotation rate that barely varied with radius between 0.2 R⊙ and 0.6R ⊙, apart from a low-significance bump around 0.4R ⊙.

Eff-Darwich and Korzennik (1998Jump To The Next Citation Point) further addressed point 3 when they combined results from several different instruments, including GONG, BiSON, MDI, and GOLF. They give a nice illustration of the tendency of higher-frequency low-degree mode splittings to be biased upward by the mode width, a point that was further illustrated by Chaplin et al. (2001), and conclude that with the then-available data it is not possible to rule out fast rotation in the core below 0.18 R⊙.

Charbonneau et al. (1998) used a genetic forward-modeling approach to analyze the LOWL data, with results favoring a rigidly-rotating core.

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