DIFFERENTIAL ROTATION RATE OF X-RAY BRIGHT POINTS AND SOURCE REGION OF THEIR MAGNETIC FIELDS

The solar surface rotates differentially in terms of latitude. It is reviewed in Schröter (1985). The rotation rate has been derived from the Doppler velocity (Howard & Harvey 1970; Ulrich et al. 1988; Snodgrass & Ulrich 1990) and from the motion of the tracers such as sunspots (Newton & Nunn 1951; Howard et al. 1984), magnetic fields (Wilcox & Howard 1970; Snodgrass 1983; Komm et al. 1993), and so forth. The differential rotation continues down to the bottom of the convection zone as shown in Kosovichev et al. (1997) and Schou et al. (1998) from the helioseismic approach.

The corona also shows effects of the differential rotation. Features of large-scale coronal structures including coronal holes rotate nearly rigidly (Wagner 1975; Timothy et al. 1975; Weber et al. 1999), while EUV bright points (EBPs), small-scale features in the corona, show a strong differential rotation, which has been found by Simon & Noyes (1972). The differential rotation rate of small-scale coronal features has recently been evaluated with a good accuracy by tracing EBPs in Brajša et al. (2001, 2002, 2004) and Karachik et al. (2006) from many EUV images obtained with the EUV Imaging Telescope (EIT) on the Solar Heliospheric Observatory (SOHO). They have shown that the differential rotation profile determined by EBPs roughly corresponds to the profile obtained from photospheric magnetic fields.

In the present paper, we estimate the differential rotation rate of small-scale coronal features from the motion of X-ray bright points (XBPs), X-ray counterparts of EBPs, which were observed with the Yohkoh soft X-ray telescope (SXT; Tsuneta et al. 1991). We adopt a simple statistical analysis method, which is different from methods in other previous studies. We find that there are two types of XBPs in terms of the characteristic height determined from the rotation property. We also report the variation of XBP rotation rate as a function of a parameter that is strongly associated with the lifetime of XBPs and discuss the source region of their magnetic fields.

2.1. Data

2.2. Analysis

The position of each XBP to the center of the X-ray Sun and the height of the XBP are required to calculate the rotation rate of the XBP. The position of each XBP is calculated from the center of gravity of the XBP soft X-ray intensity when extracted from soft X-ray images. Since we cannot directly measure the height of each XBP that is projected on the plane of sky and since we have interest in the statistical behavior of the XBP rotation rate, we assume that we can define a statistical or representative height H from many XBPs, which leads to a constant rotation rate over longitudes. When the height of XBPs H is known, the heliocentric longitude ϕ and latitude θ on a spherical surface, whose radius is the solar radius plus H, are calculated from the position of an XBP by adding information on the center of the X-ray Sun, B0 angle, and the angle between the solar north and the vertical direction of the image. A rotation rate in a latitude band between θ and θ + Δθ is simply calculated from two XBPs: one XBP is detected at longitude ϕ_i in an X-ray image observed at t₁ and the other is detected at longitude ϕ_j in a different X-ray image observed at t₂. The sidereal rotation rate ω_i,j that is determined from two XBPs is given as

$$\begin{equation} \omega _{i, j} (H)= \frac{\phi _{j} (t_{2}, H) - \phi _{i} (t_{1}, H)}{\Delta t} + \omega _{\rm corr}, \end{equation} \tag{ 1 }$$

where Δt ≡ t₂ − t₁ > 0 and ω_corr is a correction factor to convert the synodic rotation rate in the first term into the sidereal rotation rate by considering the orbital rotation of the Earth around the Sun.

We have made histograms f of the sidereal rotation rate from all XBP pairs that we detected in a given latitude band and in a longitude range of −50° < ϕ < 50°. If two XBPs are the same XBP observed at different times, the calculated value ω_i,j contributes to a symmetric distribution around a modal value. If, on the other hand, they are two different XBPs, the corresponding value ω_i,j contributes to a background in a random manner. The modal value of the histogram f against latitude is to be estimated as the differential rotation rate. The distribution f is actually symmetric to a modal value as shown in Figures 1(a)–(e) representing conditions of H= 12,000 km and 8 hr <Δt < 24 hr. The distribution is well fitted by Voigt profile V(ω) plus a background. Voigt function contains Lorentz function, implying the presence of a damped oscillator with a lifetime. We fit the distribution by the following equations:

$$\begin{equation} f (\omega) = V(\omega) + b_{0} + b_{1}\omega + b_{2} \omega ^{2}, \end{equation} \tag{ 2 }$$

$$\begin{equation} V(\omega) = \int _{-\infty }^{\infty } G(\omega ^{^{\prime }}) L(\omega - \omega ^{^{\prime }}) d\omega ^{^{\prime }}, \end{equation} \tag{ 3 }$$

$$\begin{equation} G(\omega) = {\rm exp}(-\omega ^{2}/2\sigma ^{2}) / (\sqrt{2\pi }\sigma), \end{equation} \tag{ 4 }$$

$$\begin{equation} L(\omega) = \frac{\gamma }{\pi }\frac{1}{ (\omega -\omega _{\rm c})^{2}+\gamma ^{2} }, \end{equation} \tag{ 5 }$$

where ω_c, σ, γ, b₀, b₁, and b₂ are fitting parameters. The model distribution is fitted very well, and we can estimate the modal value ω_c with a small error δω_c irrespective of the wide distribution. δω_c of 0.01 deg day⁻¹ is equivalent to 1.4 m s⁻¹ in the rotation velocity. In Figures 1(a)–(e), we show five latitude bands and the decrease of the modal value is clearly seen from the latitude band near the equator to a higher latitude band. The detail of the differential rotation curve is shown in the next section.

Zoom In Zoom Out Reset image size

In order to estimate H from the XBP data, similar histograms in the equatorial region were made for several longitudinal segments. The modal value ω_c in a single longitudinal segment is estimated from each histogram f. The slope dω_c/dϕ is calculated and is plotted as a function of H in Figure 1(f). In order to satisfy a constant rotation rate with respect to the longitude (dω_c/dϕ = 0), we adopt the characteristic height of XBPs H to be 12,000 km for 8 hr <Δt < 24 hr. The height is consistent with those in Simon & Noyes (1972) and Brajša et al. (2004). In Karachik et al. (2006), however, ∼80,000 km is adopted for the height of EBPs. When the height of 0 km is adopted, the equatorial rotation rate becomes larger by 0.2 deg day⁻¹ in our estimate and the rotation rate in a given latitude band becomes different in different longitudes.

The red and blue lines in Figure 2 and Table 1 show the differential rotation rates of the northern and southern hemispheres that are estimated in the present study with those in previous studies. Due to the strong visibility effect in counting XBPs from SXT images that leads to small number of XBP samples near the limb of the Sun, the rotation rate at high latitude above 60° in the heliocentric coordinates has a relatively large uncertainty. The result of the present study is consistent with Brajša et al. (2004) that is shown as diamond sign in Figure 2 and supports the validity of the method used in the present study. The differential rotation rate of XBPs has a fairly similar profile to that of the photospheric magnetic field in Komm et al. (1993). A possible reason is that XBPs are produced by the magnetic interaction of small-scale magnetic fields, magnetic reconnection, in the low part of the corona.

Zoom In Zoom Out Reset image size

The differential rotation rate is usually fitted by a function of latitude θ as

$$\begin{equation} \omega (\theta)= A + B \sin ^{2}\theta + C \sin ^{4}\theta, \end{equation} \tag{ 6 }$$

where A, B, and C are fitting parameters, though the second term is not orthogonal to the third term. The rotation profiles shown in Figure 2 are fitted by this function and the result of the fitting is tabulated in Table 2. The equatorial rotation rate of 2.919 μrad s⁻¹ or 14.41 deg day⁻¹ for a July 1996 data set in Karachik et al. (2006) appears to be coincident because a much larger value in height, ∼80,000 km, is used in their estimation.

Δt is a parameter that we can intentionally filter some types of XBP by its lifetime. In one of the ranges, Δt₁ < Δt < Δt₂, XBPs with a life time of τ_XBP < Δt₁ do not contribute to the rotation rate in the range, while the XBPs with Δt₁ < τ_XBP do contribute to the rotation rate. Figure 3(a) is a similar plot to Figure 1(f) with a different Δt in a latitude band near the equator (−10° < θ < 10°). It is clear that the characteristic height XBP H determined from the condition of zero slope (dω_c/dϕ = 0) is different in a different range of Δt. Figure 3(b) shows the characteristic height of XBP H as a function of Δt. Two characteristic heights are seen in the figure; one is at H= 4000–6000 km for Δt ≲ 8 hr and the other at H=10,000–12,000 km for Δt ≲ 8 hr. We can conclude that the short-lived XBPs whose lifetime is shorter than ∼8 hr are smaller in the characteristic height than XBPs of a longer lifetime. The lifetime of XBPs τ_XBP that has been so far measured is within about a day (Golub et al. 1974, 1976; Golub & Vaiana 1978; Zhang et al. 2001; McIntosh & Gurman 2005). Golub et al. (1974, 1976) show a histogram of XBP lifetime showing a mean lifetime of 8–9 hr. What we have found in the present study is that there are at least two types of XBP, each group has a different characteristic height H.

Zoom In Zoom Out Reset image size

Figure 4(a) shows the sidereal rotation rate ω_c in the vicinity of the equator as a function of Δt, which are derived from the fitting of histograms shown in Figure 4(b). Although Golub & Vaiana (1978) have reported that "short-lived" (1 day <τ_XBP < 2 days) XBPs rotate slower than "long-lived" (2 days <τ_XBP < 4 days) XBPs, what we have found here is the trend of a quite different timescale to what they reported. The number of short-lived XBPs will be much larger than that of long-lived XBPs as reported in the case of EBPs (McIntosh & Gurman 2005) even under a condition of the temporally insufficient sampling for short-lived EBPs. Therefore, the slow rotation rate for a small Δt is the property of short-lived XBPs. Toward Δt = 0, the rotation rate or rotation frequency is approaching to 14.1–14.2 deg day⁻¹ or 455–458 nHz, which is close to the equatorial rotation rate estimated from photospheric Doppler measurements shown in Figure 2.

Zoom In Zoom Out Reset image size

It is difficult to conclude from Figure 4(a) alone at this moment whether XBPs in X-ray images have bimodal rotation rates of 14.2 and 14.4 deg day⁻¹ each having a different lifetime or a continuous distribution of the rotation rate from 14.2 to 14.4 deg day⁻¹ with a range of lifetimes. However, there must be an XBP component with a short lifetime that nearly rotates with the photospheric Doppler rotation rate. The lifetime distribution of EBPs in McIntosh & Gurman (2005) cannot be expressed by a single exponential function, and it requires another functional form, a power law in their model, at low to intermediate lifetimes. Their study supports that there is another type of lifetime distribution at a short lifetime superposed on a distribution of ∼30 hr lifetime. We note that the sidereal rotation rate of 14.4 deg day⁻¹ is coincident with that of single, long-lived, and recurrent sunspots (see Table 2 in Schröter 1985), and that it is significantly slower than that (14.7 deg day⁻¹) of the supergranulation cells (Duvall 1980; Snodgrass & Ulrich 1990).

Since the short-lived XBPs rotate in the slightly faster rotation rate (ω_c ∼ 14.2 deg day⁻¹) than that determined from the photospheric Doppler velocity measurement, the magnetic field that is associated with the short-lived (τ_XBP≲ 8 hr) XBPs could be rooted just below the surface of the Sun at the top of the convection zone. The depth of equivalent rotation rate in the convection zone is located at ∼0.99 R_☉ for the short-lived XBPs when compared with the equatorial rotation rate in Schou et al. (1998). The origin of short-lived XBPs may explain why the number density of XBPs is nearly constant over the solar cycle as shown by Hara & Nakakubo-Morimoto (2003). On the other hand, long-lived (τ_XBP∼ 20–30 hr) XBPs or XBP-associated magnetic fields rotate in a rate of 14.4 deg day⁻¹, which is fairly faster than the photospheric plasmas, and they could be rooted at a deeper location in the convection zone. When the rotation speed of XBPs shown in Figure 2 is compared with the internal speed, the depth of equivalent rotation speed of XBPs is ∼0.97 R_☉ for all latitudes up to 60° (see Schou et al. 1998 for the internal rotation rate). The depth is still near the surface. Magnetic fields associated with XBPs will probably be created at the most upper part of the convection zone.

It is highly interesting that the rotation rate of the sunspots (14.4 deg day⁻¹) is nearly the same as that of XBPs and that the rotation rate of sunspot groups (14.5–14.6 deg day⁻¹) is larger. The depth of equivalent rotation rate for the latter is ∼0.94 R_☉ at the equator, where the equatorial rotation rate becomes the maximum all over solar radius. If the magnetic fields of sunspots are created near the base of convection zone as many solar physicists believe, the rotation speed of magnetized plasmas is increased when they emerge as a magnetic flux tube from the bottom to the depth of ∼0.94 R_☉, and they have to appear on the surface keeping the rotation rate at that depth. Since the rotation speed of sunspots is much faster than that near the base of convection zone, it is extremely difficult to understand how to launch magnetic flux tubes systematically from the bottom of convection zone to make the so-called active nests or active longitudes keeping the rotation rate of limited active areas on the surface for more than half a year.

We show that a simple analysis of a data set containing positions of many XBPs on the Sun provides a high-precision measurement of the differential rotation rates of small-scale coronal features. The simple treatment in which a new statistical estimate of the rotation rate of XBPs is performed without tracing each XBP one by one from multiple images. We assume that all XBPs in a given latitude band rotate at the same constant rate over longitudes, and it is possible to introduce a characteristic height H above the photosphere. Under this assumption, XBPs have the characteristic height to be 4000–6000 km for short-lived XBPs (τ_XBP≲ 8 hr) and 10,000–12,000 km for long-lived XBPs (τ_XBP≳ 8 hr) above the photosphere. The evaluated rotation rate of XBPs for long-lived XBPs follows the rotation rate of the photospheric magnetic fields. It is clearly shown that short-lived XBPs rotate slower than long-lived XBPs, and the lower end of the rotation rate for short-lived XBPs approaches to that estimated from the photospheric Doppler measurements. This trend suggests that the magnetic field associated with XBPs with a short lifetime is rooted just below the surface of the Sun at the top of the convection zone and that the short-lived XBPs could at least result from a different origin from active regions. The depth of the short-lived XBPs may explain why the number density of XBPs is nearly constant over the 11 year solar activity cycle.

A part of Yohkoh data is only used in the present analysis, because the calibration of the stray light sometimes failed in the preprocessing the raw data. A higher precision measurement will be expected when carefully processed data, which are calibrated as the Yohkoh Legacy Data Archive (Takeda et al. 2009), are used. When the same method is applied to the EIT images with a better count rate in the quiet Sun than Yohkoh soft X-ray images, a further high-precision measurement is possible. We expect that the EUV imager on the Solar Dynamic Observatory, which will produce data of constant cadence of 10 s with better spatial resolution, will open a much higher precision measurement of the systematic motion of small-scale magnetic components.

H.H. was supported by the Grant-in-Aid for Scientific Research (No. 16740119) from the Ministry of Education, Culture, Sports, Science and Technology of Japan. Yohkoh is a spacecraft of the Institute of Space and Astronautical Science of Japan Aerospace Exploration Agency. It observed the Sun in X-rays and γ rays from 1991 September to 2001 December.