A&A 481, 819-825 (2008)
M. P. McEwan - A. J. Díaz - B. Roberts
School of Mathematics and Statistics, University of St Andrews, St Andrews, Fife KY16 9SS, Scotland
Received 5 June 2007 / Accepted 30 November 2007
Context. In transverse coronal loop oscillations, two periodicities have been measured simultaneously and are interpreted as the fundamental kink mode (with period P1) and the first harmonic (with period P2). Deviations of the period ratio P1/2P2 from unity provide information about the extent of longitudinal structuring within the loop.
Aims. Here we develop an analytical approximation that describes the shift in P1/2P2 in terms of the ratio of the length 2L of a coronal loop and the density scale height .
Methods. We study the MHD wave equations in a low plasma using the thin tube approximation. Disturbances are described by a differential equation which may be solved for various equilibrium density profiles, obtaining dispersion relations in terms of Bessel functions. These dispersion relations may be used to obtain analytical approximations to the periods P1 and P2. We also present a variational approach to determining the period ratio and show how the WKB method may be used.
Results. Analytical approximations to the period ratio P1/2P2 are used to shed light on the magnitude of longitudinal structuring in a loop, leading to a determination of the density scale height. We apply our formula to the observations in Verwichte et al. (2004) and Van Doorsselaere et al. (2007), obtaining the coronal density scale height.
Conclusions. Our simple formula and approximate approaches highlight a useful analytical tool for coronal seismology. We demonstrate that P1/2P2 is linked to the density scale height, with no need for estimates of other external parameters. Given the accuracy of current observations, our formula provides a convenient means of determining density scale heights.
Key words: Sun: corona - Sun: oscillations - stars: coronae
In the past techniques in coronal seismology have rested on the accuracy of relatively uncertain input parameters. For example, Roberts et al. (1984) argued that radio observations of coronal oscillations could be used in combination with calculations of the magnetohydrodynamic (MHD) modes of oscillation of a coronal flux tube to determine in situ physical conditions in the inhomogeneous corona. More recently, Nakariakov & Ofman (2001) used the theory of the fast kink mode combined with TRACE observations to infer the magnetic field strength in a coronal loop, given reasonable estimates of plasma density. An extensive review of coronal seismology is given in Nakariakov & Verwichte (2005).
It is important in coronal seismology to reduce the need for input parameters. Observations of higher harmonics, together with the more readily determined fundamental standing harmonic of a coronal loop, promises to shed light on the longitudinal structuring in a loop (Dymova & Ruderman 2006a; Andries et al. 2005a; Dymova & Ruderman 2006b,2007; Goossens et al. 2006; Andries et al. 2005b; McEwan et al. 2006). A first observation was reported in Verwichte et al. (2004) where the fundamental kink mode and its first harmonic were observed simultaneously in an oscillating coronal loop. More recently, van Doorsselaere et al. (2007) also observed multiple harmonics oscillating in a coronal loop and obtained measurements of the fundamental period and its first harmonic with a significantly improved accuracy. Observations of multiple oscillating harmonics have also been reported in De Moortel & Brady (2007).
Andries et al. (2005b) argued that the ratio of the period of the fundamental mode of oscillation, P1, to its first harmonic, of period P2, contains information about the density stratification of the loop plasma. In a homogeneous string the ratio P1/2P2 is unity, but it deviates from this value when non-uniformity is considered; even in the standard loop model of Edwin & Roberts (1983), density non-uniformity between the inside and outside of a cylindrical magnetic flux tube creates wave dispersion which, in turn, causes the period ratio P1/2P2 to fall below unity, though the effect is generally not as large as that due to longitudinal density variation (McEwan et al. 2006). McEwan et al. (2006) studied the ratio P1/2P2 in detail and identified the various effects that contribute towards the shift of this ratio from unity. Their conclusion is that longitudinal structuring of the plasma has the greatest impact on the shift of P1/2P2 from unity; consequently, the shift contains information about the longitudinal structuring of the plasma. Other applications involving the ratio of the fundamental mode of oscillation to higher harmonics are discussed in Erdélyi & Verth (2007).
The thin tube approximation has been widely applied to the oscillations of photospheric flux tubes, stratified by gravity (see for example Ryutov & Ryutova 1976; Roberts & Webb 1979,1978; Spruit 1981; Ferriz Mas & Schüssler 1989; Stix 2004). Recently, Dymova & Ruderman (2005,2006a,b) showed its application to prominences and coronal flux tubes. Here, following Dymova & Ruderman (2006a,b) and McEwan et al. (2006), we show that the thin tube approximation applied to coronal loops leads to an analytical determination of the effect of longitudinal structuring on the kink mode of oscillation. Of particular interest is that we are able to shed light on the nature of the period ratio P1/2P2, demonstrating its departure from unity as a result of longitudinal structuring in two specific models of that structuring. The period ratio is also determined by use of a variational principle and a WKB approach.
Consider a zero-
plasma embedded in an uniform magnetic field
The plasma density
is considered to be
structured along the magnetic field, so that
naturally as a result of gravitational stratification, temperature
stratification or of non-uniform heating of the loop. Then linear perturbations
about such an equilibrium lead to the wave equations
(Díaz 2004; Díaz et al. 2002)
Dymova & Ruderman (2006a,2005) have shown from Eqs. (1) and (2) that in the thin tube limit for non-axisymmetrical kink oscillations the radial velocity
satisfies an equation of the form
We consider the case of an exponential density profile. Suppose that the plasma
outside the loop are both exponentials with the same scale height
Consider the standing modes of oscillation of a loop of length 2L. Line-tying at the photospheric/chromospheric base requires that
at z=L. Conditions at the loop apex z=0 determine the modes of oscillation. For even modes, the radial velocity
is symmetric about the loop apex so d
at z=0; the oscillation has a maximum or minimum at the loop apex. The dispersion relation for the even modes follows from Eq. (8) and these boundary conditions:
Similarly, the odd modes have a radial motion
that has a node at the loop apex z=0 as well as the loop base z=L, and these modes satisfy the dispersion relation
Suppose, then, that
is large. We may then employ the expansions for Bessel functions of
large arguments (Abramowitz & Stegun 1964):
we have neglected terms of order .
we may obtain an equation approximating the first zero of
Eq. (16), i.e. the fundamental period P1 for
In a similar way, expanding Eq. (10), we obtain for the odd mode
which, expanded as a series in
It is of interest to consider other profiles of
in addition to the exponential case. Consider the case when the kink speed squared,
varies linearly with z:
The dispersion relation for the even kink modes of oscillation in a thin coronal loop of length 2L, with a linearly varying kink speed squared (embedded in an atmosphere also with a linearly varying kink speed squared) is
The thin tube equation, Eq. (3), is amenable to an approximate treatment, as well as the exact solutions for certain specific profiles. Multiplying Eq. (3) by
and integrating along the loop from the apex to the base gives
These trial functions are chosen to give the appropriate behaviour for that accords with the fundamental mode of period P1 and its first harmonic of period P2. Equation (35) determines the frequency (and so the periods P1 and P2) for any choice of profile , by evaluating the integrals I and J.
The variational approach leading to Eq. (34) requires trial functions which may either be chosen for simplicity (as in Sect. 2.3) or by methods that lead to more accurate representations of the actual eigenfunctions. The WKB method provides a powerful and convenient means of generating such functions (Bender & Orszag 1978).
We write Eq. (3) in the form
Obtaining an explicit expression for the first eigenvalue
is more difficult. For the even modes,
this gives a relation between A and B in Eq. (42). The line-tying boundary condition
then implies a transcendental equation for :
Expressions (43) and (44) may be used to approximate the period ratio for the exponential and linear profiles discussed earlier, and the agreement they give is excellent. These formulae may also used to approximate the period for different density profiles, even if analytical or numerical solutions are difficult to obtain, needing only the integrals in Eq. (41) to be evaluated for s=1 and then the transcendental Eq. (44) to be solved to obtain an approximation for the period ratio.
We have pointed out a number of ways in which the period ratio P1/2P2 may be determined, using either analytical solutions and their approximations for specific profiles of or by considering a variational or WKB formulation for these profiles or other choices. We now return to the exponential profile, with the aim of using our analytical results to deduce the scale of variation in from observations. Other profiles give similar results to the exponential case.
Consider then the periods P1 and P2 in a loop that is exponentially structured (with density profiles given by Eq. (6)). Equations (19) and (23) determine these periods in terms of
the period of a kink mode in a loop that has no longitudinal structuring;
depends upon the loop length 2L, the internal and external Alfvén speeds, and the internal and external densities. However, by combining P1 and P2 to form their ratio we may eliminate
obtaining a period ratio
P1/2P2 which depends upon the ratio
of loop half length L to density scale height
Figure 1 shows the variation of P1/2P2 with as determined for the exponential profile; we show the approximate formula (46) together with the exact solution of the full dispersion relations (9) and (10). The agreement is excellent (and incidentally also agrees with a full numerical solution of Eqs. (1) and (2) carried out by Díaz et al. 2007).
|Figure 1: The period ratio P1/2P2 for a thin tube with exponential longitudinal structuring. The dashed curve corresponds to the numerically determined solution of the dispersion relations (9) and (10). The solid curve corresponds to the analytical approximation Eq. (46).|
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Table 1: Coronal seismology using P1/2P2: scale heights.
Table 2: Coronal seismology using P1/2P2: density ratio .
|Figure 2: The period ratio P1/2P2 as a function of the density ratio of internal density at the loop apex to density at the loop base. The solid curve corresponds to the exponentially structured density profile, numerically determined by the solution of dispersion relations (9) and (10). Also shown for comparison are the results for a selection of other profiles, considering the linear profile (dashed line), the profile discussed by Dymova & Ruderman (2006a), Eq. (48) (dotted line), the profile (Eq. (50)) with kL=1 (dot-dashed line) and the profile discussed by Andries et al. (2005b) (Eq. (49), dot-dot-dashed line); for these profiles we have used the variational formulation.|
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is a measure of the scale of longitudinal variation
in the square of the kink speed,
In the exponential
is the scale height of
at all points
along the loop. The assumption of similar profiles in internal plasma density
and external plasma density
to the density scale height. We can express these
quantities in terms of the ratio
at the loop apex to
at the loop base:
The main conclusion to be drawn from Fig. 2 is that all profiles give similar period ratios when is not greatly different from , but for much smaller than there is a significant variation in the period ratio with choice of profile. Consequently, other considerations are needed before any confident deduction of may be made from observational knowledge of the period ratio. Moreover, notice that different profiles give a range of values for the density ratio, once a value for P1/2P2 has been fixed. Therefore, it should be taken into account when using Eq. (46) that other density profiles might give different values, although it provides a handy approximation for a simple estimation of which roughly agrees with similar profiles. Notice also that the profile in Eq. (49) satisfies the condition in Eq. (5), showing that the profiles which include such cusps at the apex behave similarly, and thus this cusp is a minor feature regarding the period ratio.
With this in mind, we consider the application of the results for the exponential profile to observations. Consider Eq. (46) applied to the data observed in Verwichte et al. (2004). van Doorsselaere et al. (2007) reanalysed this data obtaining a higher accuracy in determining periods and found
P1/2P2 for two cases (labelled C and D in Verwichte et al. 2004; and van Doorsselaere et al. 2007) to be
Calculating the associated value of
from Eq. (46), for the exponential case, gives
Finally, using Fig. 2 we may calculate the density ratio for the three cases assuming an exponential profile. For case C we find and for case D we find . For case E we determine . If other profiles are assumed we obtain the results in Table 2, where we deduce the density ratio in the loops for different equilibrium profiles using the results in Fig. 2 (a ``-'' sign means that this ratio cannot be achieved with that particular equilibrium profile). These values lay in the error margin given by the uncertainty in the measurement of the periods. We conclude that in the current available observations, the choice of model is a minor source for the error in Eq. (46) compared with observational uncertainties.
McEwan et al. (2006) showed that the dominant cause of a shift in P1/2P2from unity is the longitudinal structuring of the plasma, exceeding shifts caused by other effects such as dispersion. The observational determination of P1/2P2 thus yields information about the longitudinal structure. In the thin tube limit, the ratio P1/2P2 depends only upon the ratio of the loop half-length L to the scale height . It is independent of other parameters, such as the propagation speed . Thus by using Eq. (46) one can determine the coronal density scale height without relying on other input parameters. In the three currently known observational determinations of the periods P1 and P2, significant error bars arise in the determination of in each case. However, instruments and techniques for measuring periods are more advanced than those for determining number density or magnetic field strength, so we have here a method for coronal seismology that relies on relatively well known input parameters. In conclusion, the period ratio P1/2P2 is potentially a powerful tool for atmospheric seismology.
Potentially, formula (46) has applications for stellar observations also. The determination of input parameters for stellar seismology is difficult, mainly due to the great distances involved. However, observations of stellar coronal oscillations have been reported (Mathioudakis et al. 2006,2003; Mitra-Kraev et al. 2005), although only single harmonics have so far been detected. With observations of higher harmonics it should prove possible to determine stellar density scale heights.
The authors acknowledge financial support from the Particle Physics and Astronomy Research Council and thank Dr. Gavin Donnelly for useful discussions. Tom van Doorsselaere and Michael Ruderman are also acknowledged for their accurate remarks and help in the discussion of the results.