coronal seismology using transverse oscillations of non-planar coronal loops

Solar Phys (2012) 278:177–185DOI 10.1007/s11207-011-9926-z

Coronal Seismology Using Transverse Oscillationsof Non-planar Coronal Loops

A. Scott · M.S. Ruderman

Received: 14 October 2011 / Accepted: 22 December 2011 / Published online: 31 January 2012© Springer Science+Business Media B.V. 2012

Abstract We continue studying the robustness of coronal seismology. We concentrate ontwo seismological applications: the estimate of coronal scale height using the ratio of periodsof the fundamental harmonic and first overtone of kink oscillations, and the estimate ofmagnetic-field magnitude using the fundamental harmonic. Our analysis is based on themodel of non-planar coronal loops suggested by Ruderman and Scott (Astron. Astrophys.529, A33, 2011), which was formulated using the linearized MHD equations. We show thatthe loop non-planarity does not affect the ratio of periods of the fundamental harmonic andfirst overtone, and thus it is unimportant for the estimates of the coronal scale height. We alsoshow that the density variation along the loop and the loop non-planarity only weakly affectthe estimates of the magnetic-field magnitude. Hence, using the simplest model of coronalloops, which is a straight homogeneous magnetic cylinder, provides sufficiently accurateestimates for the magnetic-field magnitude.

Keywords Coronal seismology · Magnetic fields · Oscillations

1. Introduction

Transverse oscillations of coronal loops were first observed by the Transition Region andCoronal Explorer (TRACE) in 1998. These observations were reported by Aschwanden etal. (1999) and Nakariakov et al. (1999), who interpreted them as fast kink oscillations ofmagnetic-flux tubes. Although these oscillations deserve attention on their own, their mainimportance is related to the fact that they are one of the principal tools of a new and fastemerging branch of solar physics called coronal seismology.

Coronal seismology was first suggested by Uchida (1970) and Roberts, Edwin, and Benz(1984). Its main idea is to estimate the parameters of the coronal plasma and magnetic field

A. Scott · M.S. Ruderman (�)Solar Physics and Space Plasma Research Centre (SP²RC), School of Mathematics and Statistics,University of Sheffield, Hicks Building, Hounsfield Road, Sheffield, S3 7RH, UKe-mail: [email protected]

mailto:[email protected]

178 A. Scott, M.S. Ruderman

using observations of waves propagating in the corona. After the kink oscillations of coro-nal magnetic loops were observed, Nakariakov and Ofman (2001) used them to estimatethe magnetic-field magnitude in coronal loops. Later, Verwichte et al. (2004) reported si-multaneous observations of the fundamental harmonic and first overtone of the coronal-loopkink oscillations. An important property of these oscillations was that the ratio of periodsof the fundamental harmonic and first overtone was smaller than two. Andries, Arregui, andGoossens (2005) suggested that this deviation of the period ratio from two is related to thedensity variation along the loop due to the gravitational stratification of the solar atmosphere,and used this period ratio to estimate the density scale height in the corona.

To estimate the magnetic-field magnitude using the observations of coronal-loop kinkoscillations, Nakariakov and Ofman (2001) adopted the simplest model of a coronal loop,which is a straight, homogeneous, magnetic cylinder (e.g. Edwin and Roberts, 1983). An-dries, Arregui, and Goossens (2005) modelled a coronal loop by a semicircular magnetictube with the constant circular cross section situated in a vertical plane. A very importantquestion related to coronal seismology is how robust the results are. What happens if we usemore sophisticated models of coronal loops to estimate the plasma and magnetic-field pa-rameters? This question has been partly answered in relation to the atmospheric scale-heightestimates. Ruderman, Verth, and Erdélyi (2008) and Verth, Erdélyi, and Jess (2008) showedthat accounting for the loop expansion can strongly affect the estimates of the density scaleheight. On the other hand, Dymova and Ruderman (2006) and Morton and Erdélyi (2009)showed that the shape of a coronal loop has only a moderate effect on the estimate of thedensity scale height. McEwan et al. (2006) concluded that the wave dispersion related to thefinite thickness of coronal loops is very weak and can be neglected. Morton and Ruderman(2011) studied the kink oscillations of a semicircular coronal loop with a constant ellipticcross section and showed that the estimate of the density scale height is independent of theratio of the ellipse axes. Robertson, Ruderman, and Taroyan (2010) considered the kink os-cillations of a coronal loop consisting of two parallel threads with a constant circular crosssection. They found that the estimate of the density scale height obtained using this model isexactly the same as the estimate obtained using the model of a monolithic coronal loop. Asfor the estimate of the magnetic field magnitude, to our knowledge, no models of coronalloops except the simplest one were used in this study.

Recently, Aschwanden et al. (2008) and Aschwanden (2009) used STEREO data to re-construct the three-dimensional geometry of several coronal loops. These studies showedthat there are loops that do not lie in a plane. Hence, it is interesting to study how the loopnon-planarity can affect the results of coronal seismology, which we address here. We alsostudy how the density variation along the loop can affect the estimate of the magnetic-fieldmagnitude. The article is organized as follows: In the next section, we briefly describe themodel of a non-planar coronal loop developed by Ruderman and Scott (2011). In Section 3we consider the estimates of the density scale height using kink oscillations of non-planarcoronal loops. In Section 4 we investigate how the density variation along the loop and theloop non-planarity affect the estimate of the magnetic-field magnitude. Section 5 containsthe summary of the obtained results and our conclusions.

2. The Model of a Non-planar Coronal Loop

Our analysis is based on a model of a non-planar coronal loop derived using the linearizedmagnetohydrodynamics (MHD) equations by Ruderman and Scott (2011). In this sectionwe briefly describe this model. We use Cartesian coordinates [x, y, z] with the z-axis in the

Seismology with Non-planar Coronal Loops 179

Figure 1 Helical loop geometry.Footpoints are anchored in thephotosphere at the xy-plane andz represents the height in theatmosphere; s is the distancealong the loop, and R is theradius of the semicircular loopprojection on the yz-plane. 2πq

is the helix pitch.

vertical direction, and cylindrical coordinates [� , ϕ, x]. These two coordinate systems arerelated by y = � cosϕ and z = � sinϕ. In cylindrical coordinates, the force-free (but notpotential) equilibrium magnetic field is defined by

B� = 0, Bϕ = q�B0

q2 + � 2, Bx = q2B0

q2 + � 2, (1)

where B0 and q are constants. We consider this magnetic field in the half-space z > 0 andassume that the magnetic-field lines are frozen in the dense photospheric plasma at z = 0.Note that every magnetic-field line is a part of a helix. The coronal-loop boundary is formedby the magnetic-field lines crossing the circle of radius a in the xy-plane. This circle iscentred on the y-axis a distance R from the origin, a � R. The magnetic-field line crossingthe xy-plane at the centre of this circle is the tube axis. It is defined by the equations

x = qϕ, y = R cosϕ, z = R sinϕ, (2)

where ϕ varies from 0 to π . Similar to all other magnetic-field lines, it is a part of a helix.The pitch of this helix is equal to 2πq . The projection of the tube axis on the yz-plane is ahalf-circle of radius R. It is shown by Ruderman and Scott (2011) that the loop cross sectionis a circle of radius a everywhere. The condition a � R implies that the loop is thin. Thevariation of the magnetic field-magnitude inside the loop is of the order of a/R, so we canconsider this magnitude as approximately constant. It follows from Equation (1) that thismagnitude is given by

B∗ = qB0√q2 + R2

. (3)

A sketch of the magnetic loop is shown in Figure 1.In what follows, we use the length s along the loop axis related to ϕ by

s = Rϕ√

1 + γ 2, γ = q/R. (4)

The parameter γ measures the loop non-planarity. When γ = 0 the loop axis is a semicircleof radius R lying in the yz-plane, and the projections of loops with other values of γ on the


Figure 2 The loop projection onthe xy-plane. The solid, dashed,and dashed-dotted linescorrespond to γ = 1.0, 0.6, and0.2, respectively. The distancesare given in units of thehalf-distance between thefootpoints R̄ = R

√1 + π2γ 2/4.

horizontal plane are shown in Figure 2. Note that, to obtain a non-zero magnetic field in aplanar loop, we have to take B0 → ∞ simultaneously with γ → 0 in such a way that γB0 isconstant.

It is assumed that the densities inside the loop [ρi] and outside the loop in the loop vicinity[ρe] can vary along the loop but do not vary in the directions orthogonal to the loop axis.Hence, ρi = ρi(s) and ρe = ρe(s). In addition, it is assumed that the ratio of the densities isconstant: ρi(s)/ρe(s) = ζ > 1.

Ruderman and Scott (2011) show that the kink-oscillation frequencies of a thin non-planar loop are determined by the eigenvalue problem

d2�

ds2+ ω2

C2k (s)

� = 0, �(0) = �(L) = 0, (5)

where

C2k (s) = 2B2∗

μ0[ρi(s) + ρe(s)] , (6)

μ0 is the magnetic permeability of free space, and L is the loop length given by

L = π√

q2 + R2. (7)

The function � determines the displacement of the loop axis ξ as a function of s. We do notgive the relation between � and ξ because it is not used in what follows.

Equation (5) is exactly the same as the equation describing the kink oscillations of athin straight magnetic tube with the density varying along the tube derived by Dymova andRuderman (2005). Hence, we see that neither the loop curvature nor the loop torsion relatedto its non-planarity affects the frequencies of the loop kink oscillations directly. The maineffect related to the loop non-planarity is the variation of the oscillation polarization alongthe loop, which we do not consider.

The typical values of γ corresponding to realistic coronal loops can be found using the re-construction of coronal loop geometry given by Aschwanden et al. (2008) and Aschwanden(2009). Aschwanden et al. (2008) introduced the coplanarity parameter P∗ to characterizethe loop non-planarity. This parameter is equal to the maximum distance of a point on the


loop axis from the loop plane divided by the mean curvature radius of the loop. Let us con-nect this parameter with γ . By definition the loop plane is the plane that contains the loopapex and footpoints. It is straightforward to obtain that its equation is

2Rx + πqy = πqR. (8)

Then, using Equation (2), we obtain that the distance of a point on the loop axis from theloop plane is

d(ϕ) = γR2ϕ − π(1 − cosϕ)

√4 + π2γ 2

. (9)

The maximum distance is equal to the maximum value of function d(ϕ) in the interval[0,π ]. We easily find that this maximum value is taken at ϕ = ϕM = arcsin(2/π), and it isequal to

dM = γR

√π2 − 4 − 2 arccos(2/π)

√4 + π2γ 2

. (10)

The curvature radius of the loop is R(1 + γ 2), so we obtain

P∗ = dM

R(1 + γ 2)= γ

√π2 − 4 − 2 arccos(2/π)

(1 + γ 2)√

4 + π2γ 2. (11)

P∗ takes its maximum value when γ = γM , where

γM =√√

1 + 2π2 − 1

π≈ 0.6, (12)

and this maximum value is equal to

PM = π√√

1 + 2π2 − 1[√π2 − 4 − 2 arccos(2/π)](π2 + √

1 + 2π2 − 1)√

3 + √1 + 2π2

≈ 0.106. (13)

The dependence of P∗ on γ is shown in Figure 3. We see that P∗ is a non-monotonicfunction of γ . It increases monotonically from 0 to PM in the interval (0, γM), and decreasesmonotonically from PM to 0 in the interval (γM,∞). Hence, for any P∗ < PM there are twocorresponding values of γ , γ− and γ+. To choose one of them we note that the ratio of thedistance between the loop footpoints and the loop height is

√4 + π2γ 2. We have

√4 + π2γ 2− ≤

√3 +

√1 + 2π2 ≈ 2.75,

√4 + π2γ 2+ ≥

√3 +

√1 + 2π2 ≈ 2.75.

(14)

In all coronal loops examined by Aschwanden et al. (2008) and Aschwanden (2009) theratio of the distance between the loop footpoints and the loop height was smaller than 2.75.Hence, we have to choose γ−. P∗ varies from 0.03 to 0.11 for the loops examined by As-chwanden et al. (2008). The latter value is approximately equal to PM . Then, using Figure 3,we obtain that the corresponding variation of γ− is from 0.09 to 0.6. Aschwanden (2009)examined a loop with very strong non-planarity, P∗ = 0.21. Since the maximum value of P∗in our model is approximately 0.11, we conclude that our model loop cannot reproduce this


Figure 3 The dependence ofcoplanarity parameter P∗ on γ .

loop. This conclusion is supported by the fact that the loop examined by Aschwanden (2009)is characterized by strong variation of the curvature along the loop, while in our model thecurvature is constant.

Summarizing, we conclude that our model can reproduce non-planar coronal loops withsmall or moderate non-planarity. We expect that, typically, γ is smaller than or equal to 0.6,and it almost definitely does not exceed one.

3. Estimation of the Atmospheric Scale Height

In this section we consider the estimation of atmospheric scale height using kink oscillationsof non-planar coronal loops. In what follows, we assume that the loop is immersed in anisothermal atmosphere with atmospheric scale height H , and that the plasma temperatureinside and outside the loop is the same. Using the relation z = R sinϕ and Equation (4) weobtain

ρi(s) = ρf exp

(− R

Hsin

(s

R√

1 + γ 2

)), (15)

where ρf is the plasma density inside the loop at the loop footpoints. Using this equationand introducing the dimensionless frequency and distance S,

2 = π2ω2R2(1 + γ 2)

C2kf

, S = s

L, C2

kf = 2ζB2∗μ0ρf(1 + ζ )

, (16)

we rewrite Equation (5) as

d2�

dS2+ 2 exp

(− R

Hsin(πS)

)� = 0, �(0) = �(1) = 0. (17)

Let 1 and 2 be the dimensionless frequencies of the fundamental mode and first overtone,respectively. Since Equation (17) does not contain the non-planarity parameter γ , it followsthat both 1 and 2 are independent of γ . Since 2/1 is a monotonically decreasingfunction of R/H , we conclude that the loop non-planarity does not affect the estimationof the atmospheric scale height. Note that the dimensional frequencies of the fundamental


Figure 4 The magnitude of thedimensionless magnetic field B̃∗inside the coronal loop as afunction of R/H for variousvalues of the non-planarityparameter γ . The solid, dashed,dashed-dotted, and dotted linescorrespond to γ = 0, 0.2, 0.6,

and 1.

mode and first overtone, ω1 and ω2 , do depend on γ . However, since ω2/ω1 = 2/1, theirratio is independent of γ . It is important to remember that these results apply for linearizedMHD, since the loop model we have used was derived using the linearized MHD equations.

4. Estimation of the Magnetic Field Magnitude

In this section we study how the density variation along the loop and the loop non-planaritycan affect the estimation of the magnetic-field magnitude inside the loop. Observationally itis easier to obtain not R, but the distance between the footpoints [2R̄], which is related to R

by

R̄ = R√

1 + π2γ 2/4. (18)

Then, using Equation (16), we obtain for the magnetic-field magnitude inside the loop,

B∗ = 2π2R̄

P

√2μ0ρf(1 + ζ )(1 + γ 2)

ζ(4 + π2γ 2), (19)

where now is the dimensionless frequency of the fundamental mode of kink oscillationsfound using Equation (17), and P = 2π/ω is the observed period of the fundamental mode.Note that this equation gives the value of B∗ in teslas. Figure 4 shows the dependence of thedimensionless magnetic-field magnitude inside the loop [B̃∗], defined by

B̃∗ = B∗PR̄

√μ0ρa

= 2π2eR/2H

√2(1 + ζ )(1 + γ 2)

ζ(4 + π2γ 2), (20)

on the ratio of the loop height to the atmospheric scale height [R/H ] for various values ofthe non-planarity parameter γ . In Equation (20) ρa = ρf exp(−R/H) is the plasma densityat the loop apex.

As an example, we consider the same event that was used by Nakariakov and Ofman(2001). In this event the half-distance between the loop footpoints was R̄ = 6.0 × 107 m,and the fundamental-mode period was P = 360 seconds. Following Nakariakov and Ofman(2001), we take ζ = 10. Nakariakov and Ofman (2001) also estimated that the plasma den-sity at the loop apex was 3.3×10−12 kg m−3. If we assume that the loop is planar (γ = 0) and


the atmosphere is not stratified (R/H = 0), then, using Equation (19), we obtain B∗ ≈ 16 G,which agrees very well with the estimate of B∗ = 13 ± 9 G given by Nakariakov and Ofman(2001). Let us now take R/H = 1, which is a reasonable value for a loop that has a distanceof 1.2 × 108 m between its footpoints and a plasma temperature of about 106 K. The nu-merical solution of Equation (17) with R = H gives ≈ 4.8. Substituting these values andζ = 10 into Equation (19) yields

B∗ ≈ 1.7 × 10−3

√1 + γ 2

1 + π2γ 2/4, (21)

where, once again, the value of B∗ is measured in teslas. Taking γ = 0, we obtain fromthis equation B∗ ≈ 17 G, while for γ = 1 we obtain B∗ ≈ 13 G. Hence, the variation ofthe estimate of the magnetic-field magnitude caused by accounting for the stratification andnon-planarity is less than 20% for these typical loop parameters.

We see from the analysis in this section that the effects of density variation along theloop and loop non-planarity are weak, and the approach used by Nakariakov and Ofman(2001) for estimating the magnetic-field magnitude inside the loop is quite robust in spite ofits extreme simplicity.

5. Summary and Conclusions

In this article we addressed the problem of the robustness of coronal seismology. We consid-ered two applications of coronal seismology: the estimation of the atmospheric scale heightusing the ratio of periods of the fundamental mode and first overtone of kink oscillations,and the estimation of the magnetic-field magnitude using the fundamental mode of kink os-cillations. Our analysis was based on the model of non-planar coronal loops suggested byRuderman and Scott (2011). They, in particular, derived the equation governing the kinkoscillations of non-planar loops in the thin-tube approximation. We assumed that the loop isimmersed in an isothermal atmosphere with equal plasma temperatures inside and outsidethe loop. Then, using the governing equation written in dimensionless form, we showed thatthe ratio of periods of the fundamental mode and first overtone of kink oscillations is inde-pendent of the loop non-planarity. Hence, the loop non-planarity can be added to the loopproperties that are unimportant for the estimate of the coronal scale height, at least if we usethe model of non-planar coronal loops suggested by Ruderman and Scott (2011).

We also investigated how density variation along the loop and the loop non-planarity af-fect the estimate of the magnetic-field magnitude. We found that, for typical coronal loopparameters, the variation of the estimate of the magnetic field magnitude caused by account-ing for the stratification and non-planarity is less than 20%. We confirmed this conclusion byconsidering the event of a coronal-loop kink oscillation previously used by Nakariakov andOfman (2001) to estimate the magnetic-field magnitude in the corona. Hence, using the sim-plest model of a coronal loop in the form of a homogeneous straight cylindrical tube similarto the method of Nakariakov and Ofman (2001) provides a sufficiently accurate estimate ofthe coronal magnetic field.

Acknowledgements The authors are grateful to the STFC for their financial support. M.S.R. acknowledgesthe support of a Royal Society Leverhulme Trust Senior Research Fellowship.


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coronal seismology using transverse oscillations of non-planar coronal loops

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