Decay Phase Cooling and Inferred Heating of M- and X-class Solar Flares

0.68 MB·

by  Daniel F. Ryan

Draftversion January17,2014
PreprinttypesetusingLATEXstyleemulateapjv.08/22/09
DECAY PHASE COOLING AND INFERRED HEATING OF M- AND X-CLASS SOLAR FLARES
Daniel F. Ryan1,2,3, Phillip C. Chamberlin2, Ryan O. Milligan4,2,3, Peter T. Gallagher1
Draft version January 17, 2014
ABSTRACT
Inthispaper,thecoolingof72M-andX-classflaresisexaminedusingGOES/XRSandSDO/EVE.
The observed cooling rates are quantified and the observed total cooling times are compared to the
4 predictions of an analytical 0-D hydrodynamic model. It is found that the model does not fit the
1 observations well, but does provide a well defined lower limit on a flare’s total cooling time. The
0 discrepancybetweenobservationsandthemodelisthenassumedtobeprimarilyduetoheatingduring
2 the decay phase. The decay phase heating necessary to account for the discrepancy is quantified and
n found be ∼50% of the total thermally radiated energy as calculated with GOES. This decay phase
a heating is found to scale with the observed peak thermal energy. It is predicted that approximating
J the total thermal energy from the peak is minimally affected by the decay phase heating in small
6 flares. However, in the most energetic flares the decay phase heating inferred from the model can be
1 several times greater than the peak thermal energy.
Subject headings:

]
R
S 1. INTRODUCTION these models sacrifice some completeness, they are
h. Solarflaresareamongthe mostpowerfuleventsin the much faster to run allowing an easier exploration of
p solar system, releasing up to 1032 ergs in a few hours the dependence of results on different possible coronal
property values. Although these models are valuable
- or even minutes. They are believed to be powered by
o magnetic reconnection, a process whereby energy stored in increasing our understanding of flare heating and
r cooling they nonetheless suffer from drawbacks. These
in coronal magnetic fields is suddenly released. This
t
s causesarapidheating andexpansionofthe flareplasma can include arbitrary inputs of unobservable parameters
a such as heating function and number of loop strands.
which is then believed to cool by conductive, radiative
[ Previous studies focused on observations of flare cool-
and enthalpy-based processes. However,the balance be-
ing are also numerous. Culhane et al. (1970) compared
1 tween cooling and heating in solar flares and the pro-
simplecollisional,radiative,andconductivecoolingmod-
v cesseswhichdeterminethisarestillnotfullyunderstood.
elstoobservationsoffourflaresmadewiththefourthOr-
9 A greater insight into this interaction would allow us to
7 better constrain the energy release mechanisms of solar biting Solar Observatory (OSO-4). They found that col-
0 flares. lisional cooling was unphysical while conduction and ra-
4 To date there have been many studies aimed at diation were equally plausible. Although they could not
1. modeling the heating and cooling of solar flares (e.g. determine which was dominant, they did find that that
forradiativecoolingtodominate,theflaredensitywould
0 Moore & Datlowe 1975; Antiochos & Sturrock 1978; have to be high (&1011 cm−3). In contrast, conduction
4 Antiochos 1980; Fisher et al. 1985; Doschek et al.
1 1983; Cargill 1993; Klimchuk & Cargill 2001; would require low densities (∼1010 cm−3) to dominate.
: Reeves & Warren 2002; Bradshaw & Cargill 2005; In contrast, Withbroe (1978) was able to compare the
v Klimchuk 2006; Warren 2006; Warren & Winebarger relative importance of cooling mechanisms. This study
Xi 2007; Sarkar & Walsh 2008). These include full 3-D examined the differential emission measure (DEM) of a
magnetohydrodynamic (MHD) models as well as 1-D single flare using Skylab and hence determined that con-
r
a MHD models. 1-D models assume that flare loop ductive and radiative losses were comparable. This sug-
strands are magnetically isolated and therefore only gests that both processes were equally important in the
solve the MHD equations along the axis of the magnetic cooling of that flare. From discrepancies between ob-
field (e.g. Bradshaw & Cargill 2005). This is less servations and conductive and radiative cooling models,
computationally draining than the full 3-D treatment it was determined that ∼1031 ergs of additional heating
and therefore allows a higher resolution, more useful must have been deposited after the flare peak.
for detailed comparison with observation. 0-D models More recently, Jiang et al. (2006) examined loop top
have also been developed (e.g. Enthalpy-Based Thermal sourcesin6flaresusingtheReuven RamatyHigh Energy
Evolution of Loops, EBTEL; Klimchuk et al. 2008) Solar Spectroscopic Imager (RHESSI; Lin et al. 2002).
which treat field aligned average properties. Although They found that the observed cooling rate was slightly
higher than expected from radiative cooling, but signifi-
1SchoolofPhysics,TrinityCollegeDublin,Dublin2,Ireland. cantlylowerthanthatexpectedfromconduction. Toac-
2SolarPhysicsLaboratory(Code671),HeliophysicsScienceDi- count for this, they calculated that more than 1030 ergs
vision,NASAGoddardSpaceFlightCenter,Greenbelt,MD20771, of additional heating during the decay phase was neces-
U.S.A.
3CatholicUniversityofAmerica,Washington,DC20064,U.S.A. sary. This was greater than that seen during the impul-
4Queen’s University Belfast, Belfast BT7 1NN, Northern Ire- sive phase. However they concluded that much of this
land. discrepancywas more plausibly explainedby suppressed
2
conduction.
TABLE 1
Raftery et al.(2009)alsousedRHESSIalongwithsev- Wavelengthsand temperaturesof bandpassesandemission
eral other instruments to chart the thermal evolution of lines used in measuringcooling rates
a single C1.0 flare. They performed a best fit to the
Instrument Wavelength [nm] Temperature[MK]
observations using the EBTEL model and an assumed
heating function to infer radiative and conductive cool- GOES/XRS 0.05–0.4–Short >4
0.1–0.8–Long
ing profiles. Conduction was found to dominate initially
Ion Wavelength [nm] Temperature[MK]
while radiation dominated in the latter phases.
FeXXIV 19.20 15.8
A common aspect of many flare cooling studies such
FeXXII 11.71 12.6
as those mentioned above is that they focus on single or
FeXIX 10.83 10.0
small numbers of events. This means they cannot say if FeXVIII 9.39 7.9
their findings are anomalous or characteristic of flares. FeXVI 33.54 6.3
As a result, it is still unclear just how well cooling mod- FeXV 28.41 2.5
els describe ensembles of flares. In this paper we aim to FeXIV 26.47 2.0
improve upon previous studies by observing the cooling
profilesof72M-andX-classflares. Thisisdoneusingob-
Hinode/XRT is a grazing incidence X-ray telescope
servations from X-Ray Sensors onboard the Geostation-
with a spatial resolution of 1 arcsec. It provides broad-
ary Operational Environmental Satellites (GOES/XRS;
band images of the Sun in wavelengths of 0.2–20 nm. It
Hanser & Sellers1996)andtheExtremeultravioletVari-
has a maximum field of view of 34×34 arcsecs but can
abilityExperimentonboardSolarDynamicsObservatory
also focus on several smaller regions of interest simulta-
(SDO/EVE; Woods et al. 2012). The observed cooling
neously. Thetimecadencedependsontheobservingpro-
times are then compared to predictions made by the
gramused but is typically on the order of seconds. Hin-
model of Cargill et al. (1995), a simple, analytical 0-D
ode/XRT has numerous filters which can help to reduce
model. Although this model is highly simplified, it was
saturation during flares. These have quite wide temper-
chosen as a first step because it is quick and easy to
atureresponses,butallpeakaround8–13MK.Thetem-
apply to many flares. In Section 2 of this paper we de-
perature sensitivity, spatial resolution and time cadence
scribe our observations. In Section 3 we discuss the as-
ofHinode/XRT make itthe mostideal instrumentavail-
sumptions,limitationsandequationsoftheCargill et al.
able for directly measuring looplengths of hot (>5 MK)
(1995)modelanddescribehowweobservationallycalcu-
X-ray- and EUV-emitting flare plasma. It is better
lated the required inputs. In Section 4 we compare the
suited than the Atmospheric Imaging Assembly (AIA;
observed cooling times to those predicted by the model
Lemen et al. 2012) onboard SDO, which is often satu-
and quantify the discrepancy. We then infer the decay
ratedbyM-andX-classflaresandwhichhasgreatersen-
phaseheatingrequiredtoaccountforthis difference. Fi-
sitivity to cooler coronal plasma (∼1 MK). However, of
nally we outline our conclusions in Section 5.
the72flaresincludedinthisstudy,only22werewellob-
2. OBSERVATIONS&DATAANALYSIS servedbyHinode/XRT.Therefore,looplengthswerede-
terminedusingtheRTV-scalinglaw(Rosner et al.1978)
2.1. Instrumentation & Flare Sample
while Hinode/XRT was usedto quantify the uncertainty
Observations for this study were taken from three in- of this law. See Section 3.2 for further details.
struments: the XRSonboardthe GOES-14and15satel- The 72 M- and X-class flares examined in this study
lites; the Multiple Extreme ultraviolet Grating Spectro- were chosen via two criteria. Firstly, their decay phases
graph A channel (MEGS A; Hock et al. 2012a) onboard had to be temporally isolated from other flares. This
SDO/EVE; and the X-Ray Telescope onboard Hinode was determined from visual inspection of the GOES
(Hinode/XRT; Golub et al. 2007; Kano et al. 2008). lightcurves. Secondly, the flares had to be observed to
The GOES/XRS measures spatially integrated solar cool to at least 8 MK with either the GOES/XRS or
X-ray flux in two wavelength bands (long; 1–8 ˚A, and SDO/EVE MEGS-A. A complete list of the flares and
short; 0.5–4 ˚A) every two seconds. Temperature can their properties are listed in Table 2 in the Appendix.
be calculated from the ratio of these channels using the
method of White et al. (2005). In this method, coronal 2.2. Observing Flare Cooling
abundances (Feldman et al. 1992), the ionization equi- The cooling of the flares in this study was charted
libria of Mazzotta et al. (1998), and a constant density by combining the peak of the GOES temperature pro-
of1010 cm−3 wereassumed. Althoughthisfinalassump- file with the peaks of lightcurves of various tempera-
tionis probablynottrue,it wasjustifiedby White et al. ture sensitive Fe lines observed by SDO/EVE MEGS-
(2005) who used CHIANTI to compute the spectrum of A. The GOES temperature was calculated using the
an isothermal plasma at 10 MK with densities of 109, TEBBS method (Temperature and Emission measure-
1010, and 1011 cm−3. No significant differences were BasedBackgroundSubtraction; Ryan et al. 2012) which
found. performsanautomaticbackgroundsubtractionandfinds
MEGS A onboard SDO/EVE measures spatially inte- thetemperatureandemissionmeasureusingthemethod
grated solar irradiance from 6 to 37 nm with a spectral of White et al. (2005). The Fe lines used in this study
resolution of 0.1 nm. MEGS A is an 80o grazing in- alongwiththeirformationtemperaturesarelistedinTa-
cidence off-Rowland circle spectrograph and has a time ble 1. These lines were chosen because in the conditions
cadence of 10 s. Within its spectral range are a number of a solar flare, they are dominant over neighbouring
of temperature and density sensitive Fe lines useful for lines within the MEGS-A resolution and therefore min-
examining the thermodynamics of hot flares. imally blended. Before extracting these lightcurves, a
3
Fig. 2.—Histogramsshowingthenon-linear(panela)andlinear
(panel b) coefficients of the second order polynomial fits to the
observed cooling profiles of the 72 M- and X-class flares in this
study.
lightcurveswereformed. Acoolingtrackwasthengener-
atedbyplottingthepeaktimeofeachlightcurveagainst
its associated formation temperature. The cooling time
was then given by the duration of this track.
Figure 1 shows an example for an M5.5 flare which
occurred on 2010 November 06 at 15:27 UT. Figure 1a
Fig.1.— Cooling track for the 2010-Nov-06 M5.5 flare which shows the GOES temperature curve while Figures 1b–
beganat15:28UT.a)Background-subtractedGOEStemperature 1h show the lightcurves of the Fe lines measured by
profile. Peak is marked by the vertical line. b) – h) Lightcurves
SDO/EVE. The vertical lines in each panel mark the
of sequentially cooler Fe lines ranging from 15.8 MK to 2 MK
observed by SDO/EVE MEGS-A. The peak of each lightcurve is peak time of that lightcurve. The lightcurves peak in
alsomarkedbyaverticalline. i)Combinedcoolingtrackobtained order of descending temperature. This is interpreted as
by plotting the time of the peak of each profile (including GOES being due to the plasma cooling. Figure 1i shows the
temperature profile) with its associated peak temperature. The
resultantcoolingtimeisthedurationofthiscoolingtrack. resulting cooling track, with each datum point repre-
senting the peak time and temperature associated with
the lightcurves above. From this it can be seen that
this flare cooled from 17 MK to 2 MK over the course
backgroundsubtractionwasmade to eachobservedflare of 389 ± 10 seconds. The uncertainty comes from
spectrum. The background spectrum was found by av-
combining the time resolutions of the GOES/XRS and
eragingthe spectrawithinaquietperiodbeforetheflare
SDO/EVE in quadrature.
starttime. This period was determined for eachflare by
In order to parameterize the flare cooling, each flare’s
visual inspection of the GOES lightcurves. This helped
cooling profile was fit with a second order polynomial of
ensure that the behaviour of the lightcurves was mini-
the form
mallycontaminatedbyemissionfromnon-flaringplasma. T(t)=T +θt+µt2 MK (1)
The irradiance observed at the wavelength of each line 0
inTable1wasthensummedwiththatwithin±0.05nm, where t is time since the start of the cooling phase in
i.e. the spectral resolution of MEGS-A. This was done seconds, T is the temperature at the start of the cool-
0
for each spectrum taken during the flare and hence flare ing phase in megakelvin (i.e. GOES peak temperature),
4
θ is the linear cooling coefficient [MK s−1], and µ is the are given by
non-linear cooling coefficient [MK s−2]. Figure 2a shows NL2
τ =4×10−10 (3)
ahistogramofthenon-linearcoolingcoefficients,µ. The c T5/2
distribution is very narrowly peaked around zero with a
full width half max of .10−4 MK s−2 and a mean of and
-6×10−5 MK s−2. This implies that the majority of the τ =3.45×103T3/2 (4)
flarecoolingprofilesareverylinear. Thisagreesqualita- r N
tively Raftery et al. (2009), whose observedcooling pro-
respectively, where L is loop half length.
file ofa C1.0 flare wasalso quite linear. Figure 2b shows
Fromthese timescales,a totalcoolingtime canbe cal-
a histogram of the linear cooling coefficients, θ. Since
culatedifitisassumedthattheflareonlycoolsbyeither
thenon-linearcoolingcoefficientsaresosmall,thelinear
conduction or radiation at any one time. If the con-
cooling coefficients approximate the cooling rates. The
histogram ranges from -1.5–0 MK s−1 and has a mean ductive timescale is initially shorter than the radiative
of -0.035 MK s−1. This implies that the SXR-emitting timescale,the flare is assumedto coolpurely by conduc-
plasma of an average M- or X-class flare cools at a rate tion from its initial temperature, T0, until time, t∗ and
of ∼3.5×104 K s−1. It should be noted that although temperature, T∗, whenthe twotimescalesbecomeequal.
Fromthentheflareisassumedtocoolpurelyradiatively
the histogram in Figure 2b peaks at the bin centered on
tothefinaltemperature,T ,whichtakesadditionaltime,
zero, all flares have non-zero linear cooling coefficients. L
These parameterizations are used again in Section 4.2. t∗∗. Itshouldbe notedhere thatt∗ andt∗∗ anddifferent
fromτ andτ . Theformeraretheperiodswhentheflare
c r
cools by conduction and radiation respectively. The lat-
3. MODELLING ter are characteristictimescales of the cooling processes.
3.1. The Cargill Model With this in mind, the total cooling time of the flare is
given by
Tomodelthecoolingobservationsdiscussedinthepre-
vious section, we usedthe model of Cargill et al.(1995).
7/12
This model is based on conductive and radiative cooling t =τ τr0 −1 +
timescales derived from the energy transport equation tot c0 τ
which describes how a plasma’s thermal energy density "(cid:18) c0(cid:19) #
changeswithtime. Toderivethesetimescales,anumber 2τ τ 5/12 τ 1/6 T
of assumptions are made. First, the plasma is isotropic, r0 c0 1− c0 L (5)
3 τ τ T
i.e. contains no shearing motions. Second, it is isother- (cid:18) r0(cid:19) " (cid:18) r0(cid:19) (cid:18) 0(cid:19)#
mal, i.e. obeys the ideal gas law, p=Nk T. Third, the
B where subscript ‘0’ implies the value of that property
plasma β is low, implying all particle motions are along
at the start of the cooling phase. If conduction never
theaxisofthemagneticfield. Fourth,theplasmais‘col-
dominates radiation, the flare is assumed to cool purely
lisionless’. Fifth, the conductive heat flux obeys Spitzer
radiatively. This gives a total flare cooling time of
conductivity, κ T5/2∇T, where κ = 10−6. And sixth,
0 0
wtishheaedrraeeqdχuiaa=ttiev1le.y2l×oms1so0df−ue1lne9cd,tζiob=ny,2PPraranaddd,=αbe=χtwN−eζe0Tn.5α1(0eR6rogasnncmder31e0ts7−aK1l., ttot = 2τ3r0 (cid:18)1− TTL0(cid:19) (6)
1978). Byusingtheseassumptions,theenergytransport Equations 5 and 6 use the relations describing the tem-
equation can be written as poralevolutionoftemperatureduetoconductionderived
byAntiochos & Sturrock(1978)andduetoradiationde-
1 ∂p 1 ∂ ∂u rived by Antiochos (1980).
γ−1 ∂t =−γ−1∂s(pus)−p ∂ss− The Cargill model is a very simple, easy-to-use an-
alytical model. However, its simplicity gives rise to a
∂ κ T5/2∂T −χNζTα +h (2) number of limitations. It assumes that at any one time
∂s 0 ∂s cooling occurs via either conduction or radiation, with
(cid:18) (cid:20) (cid:21) (cid:19)
an instant switch between the two when their cooling
where p is pressure, s is the spatial coordinate along the timescalesareequal. Thisassumptionmaybeanaccept-
axisofthemagneticfield,u istheplasmavelocityalong ableapproximationwheneither the conductiveorradia-
s
the axis of the magnetic field, T is temperature, N is tive timescales are much longer than the other. How-
density,andh is the heatenergyaddedper unit volume, ever, it is certainly not valid when the two timescales
per unit time. are similar. In addition, this model does not accountfor
In order to derive the characteristic timescales of the enthalpy-basedcooling. This type ofcoolingis mostsig-
cooling mechanisms, Cargill et al. (1995) assumed that nificanttowardsthe endofaflarewhenthe temperature
there were no flows within the plasma (u =0) and that islowandnolongersupportstheplasmaagainstgravity.
s
there was no heating (h = 0). This implies that the Therefore these equations are not suitable for modeling
onlywaythethermalenergydensityisalteredisviacon- plasma cooling below ∼1–2 MK.
ductive and radiative heat flux. Then the characteristic Furthermore, the Cargill model treats the flare as a
conductivecoolingtimescalecanbederivedbyneglecting monolithic loop. Other 0-D models (e.g. Warren 2006;
radiative processes in Equation 2, and vice versa for the Klimchuk et al. 2008) employ the idea that flaring loops
characteristic radiative timescale. By further assuming are comprised of many smaller strands, each heated and
that the plasma is monatomic (i.e. the adiabatic con- cooled at different times. However, some recent stud-
stant, γ = 5/3) the conductive and radiative timescales ies (e.g. Aschwanden & Boerner 2011; Peter et al. 2013)
5
haveexaminedcoronalloopcross-sectionsathighresolu- uncertainty. First, the instrumental uncertainty of the
tionandfoundnodiscerniblestructure. Thisimpliesthe irradiance of the two lines. Second is the uncertainty
such strands are below a resolution of 0.2 arcsec or that due to the noise in the ratio time profile. The latter
a monolithic treatment may be justified. If the flaring wasevaluated asthe standarddeviationduring the time
loops are indeed made of sub-strands they would have period when the flare was hotter than 12 MK as deter-
the same orientation. And since the plasma is frozen mined with the GOES/XRS. This threshold was chosen
onto the magnetic field lines and diffusion across them as it ensured that the flare temperature (accounting for
is minimal, reducing the multiple strands to one spatial uncertainty)was in the valid range of the Milligan et al.
dimension along the axis of the magnetic field, as in the (2012) method. The total uncertainty in the ratio was
Cargill model, is somewhat justified via symmetry. One determined from the standard propagation of errors of
couldarguesomethingsimilarformultipleloopsinaflar- the two uncertainty sources. This uncertainty was then
ing arcade. However, this treatment implicitly assumes transformed to density by propagating the ratio’s up-
that all the loops/strands are being heated and cooled per and lower limits as per Milligan et al. (2012). It
simultaneously which results in the average behavior of should be noted that there are additional uncertainties
alltheloops/strandsbeingmodeled. Despitethisrestric- associatedmodelled relationship between FeXXI line in-
tion, such an approach can still be useful in examining tensities and density. However, these are expected to
flare hydrodynamics, especially since the additional free be much smaller than the uncertainty sources discussed
parametersintroducedbymulti-strandmodelingarevery above. For more information on the modelling of the
unwieldy when modeling a large number of flares. FeXXI lines in CHIANTI see Section 4.7.1 of Dere et al.
Finally,theCargillmodeldoesnotaccountforheating (1997), and references therein.
which has been suggested can continue well into the de- As stated in Section 2.1, loop half length, L, was de-
cayphase(e.g.Withbroe1978;Jiang et al.2006;Warren terminedusingthe RTVscalinglaw(Rosner et al.1978)
2006). Thusthisassumptionisnotverywelljustifiedand
would be expected to produce shorter predicted cooling L=(1.4×103)−3 Tm3ax (7)
times than those observed. Despite the above limita- p
(cid:18) (cid:19)
tions, the Cargill model contains much of the physics
believed to be responsible for flare cooling and quanti- where p is pressure and Tmax is the maximum tempera-
fying how well it simulates observations is important to tureintheloop. Byassumingthattheplasmaisisother-
better understand flare evolution. mal and obeys the ideal gas law, this can be rewritten
in terms of temperature, T, and density, N, which can
3.2. Observed Inputs to Cargill Model be calculated using GOES/XRS and SDO/EVE respec-
TheobservedinputsrequiredbytheCargillmodelare, tively.
initial temperature, initial density, and loop half length 1 T2
L= (8)
which is assumed to be constant. k (1.4×103)3 N
B
The initial temperature, T , is that at the beginning
0
of the observed cooling track i.e. the GOES tempera- Ideally, these loop half lengths would be directly mea-
ture peak. As stated in Section 2.2 this was calculated suredusingHinode/XRT.However,only22oftheevents
usingthe TEBBSmethod(Ryan et al.2012). This mea- in this study were well-observed by this instrument and
surement has two sources of uncertainty: one due to so it was necessary to utilize the RTV-scaling.
the instrument (Garcia 1994, Section 7) and one due In order to quantify the uncertainties of the RTV-
to the background subtraction (Ryan et al. 2012, Sec- predictedvalues,thelooplengthsofthese22eventswere
tion 3.2). The total uncertainty in the initial tempera- measuredwithHinode/XRT(Figure3). Theseareplain-
ture was found by combining these two uncertainties in of-sky measurements performed ‘by eye’. A more rigor-
quadrature. ous analysis attempting to accountfor projectionaffects
The initial density, N , was determined using the isexpectedtoalterthemeasuredlooplengthsonlyupto
0
method of Milligan et al. (2012). This method uses a factor of ∼2 which is sufficient for our purposes. The
CHIANTI(version7;Landi et al.2012)todeterminethe different XRT filters used for eachevent canbe found in
relationshipbetweendensity andthe ratiosofthreeden- Table 2 in the Appendix. These filters all peak between
sity dependentFe XXI line pairs(12.121nm/12.875nm, 8–13 MK. Their response functions show contributions
(14.214 nm + 14.228 nm)/12.875 nm, and from plasma at temperatures below 1 MK of at least
14.573 nm/12.875 nm). In this study, only the first 2 orders of magnitude lower than the peak. This sug-
ratio was used as only these lines consistently exhibited gests that the images are not significantly contaminated
increased emission due to the flares. This method is byemissionfromlowertemperatureplasmawhichmight
only valid for temperatures above 10 MK due to the otherwise affect the measured loop lengths. The one ex-
formation temperatures of these lines. It is also not ception is the Al-mesh filter. But this was only used for
sensitive outside the range 1010–1014 cm−3. However, oneeventandwasnotfoundtobeanoutlier. Wherepos-
the Cargill model only requires the initial density, i.e. sible, SDO/AIA was used to help determine the axis of
the density at the time of the peak GOES temperature. themagneticfield,alongwhichthelooplengthshouldbe
Since all the flares in this study peak above 10 MK measured. Forinstance, acombinationofAIAandXRT
and were found to have densities within this range, the moviesrevealedthatthelongaxisoftheflaresinthepan-
method of Milligan et al. (2012) is suitable. els e), o), t) of Figure 3 were flare arcades and the loop
Theuncertaintyinthedensitymeasurementwastaken lengths themselves were actually along the shorter axis.
from the uncertainty in the line intensity ratio as mea- Despiteitsusefulnessintheseinstances,AIA’ssensitivity
sured by EVE. The ratio itself has two main sources of to cooler plasma and greater tendency to saturate made
6
Fig.3.—Observedlooplengths measuredwithHinode/XRTof22flares.
7
(72%) that radiation dominated conduction for the en-
tirety of the cooling phase. Conduction initially domi-
nated radiation in only 20 flares (28%). This suggests
that flares for which radiation is the dominant cooling
mechanism(suchasthoseexaminedbyMcTiernan et al.
1993; Lo´pez Fuentes et al. 2007) are far more common
than those in which conduction initially dominates (e.g.
Moore & Datlowe 1975; Jiang et al. 2006; Raftery et al.
2009). Furthermore, Culhane et al. (1970) concluded
from examining a simple radiative cooling model that
flareplasmacoolingbyradiationinatimescaleof∼500s
would exhibit high densities (1011–1012 cm−3). The av-
erage observed cooling time of the events in Figure 5
is 653 s and their average density is 1.4×1012 cm−3,
which is very close to the conclusions of Culhane et al.
(1970). This strengthens the claimthatradiationis typ-
ically dominant overconduction throughouta flare’s de-
Fig.4.— Comparison of RTV-predicted flare loop half lengths cay phase.
withthosemeasuredwithHinode/XRT.Aloosecorrelationcanbe Tofurtherquantifythe discrepancybetweenpredicted
seenaroundthe1:1line(over-plotted). and observed cooling times (henceforth referred to as
the ‘excess cooling time’), the root-mean-square devia-
tion (RMSD) of the distribution was calculated. This
it less well suited to making the actual measurements
was found to be 961 s. Normalizing this to the mean of
than XRT. The loop half lengths implied by the XRT
theobservedcoolingtimes(653s)givesthecoefficientof
measurementswerecomparedto the RTV-predictedval-
variationoftheroot-mean-squaredeviation(CVRMSD).
ues(Figure4)andaloosecorrelationaroundthe1:1line
This quantifies the spread of the ‘excess cooling times’
was found.
relative to the mean of the observed cooling times. The
The CVRMSD (coefficient of variation of the root-
CVRMSDwasfoundtobe1.47indicatingalargespread
mean-square deviation) of the distribution in Figure 4
as is visually suggested in Figure 5.
was used to quantify the uncertainty of the RTV-
If the Cargill model is adequately describing the
predicted loop half lengths. This was found to be
cooling mechanisms of solar flares, the ‘excess cooling
1.8, implying that the loop half length and uncertainty
is given by L = L ± 1.8 · L . This com- time’ suggests that there is additional heating occur-
RTV RTV ring throughout the decay phase. Similar assumptions
pares favourably with Aschwanden & Shimizu (2013)
havebeenmade inpreviousstudies(e.g.Withbroe 1978;
who compared RTV-predicted loop lengths with the
Jiang et al. 2006; Hock et al. 2012b). In the following
length-scales of less intense flares using SDO/AIA and
found an uncertainty of ±1.6·L . section we explore just how much additional heating en-
RTV ergy is required to account for the ‘excess cooling times’
Having measured the initial temperature, initial den-
and examine the distributions of these energies.
sity and loop half length, the Cargill-predicted cooling
timeswerefoundfromEquation5ifτc0 <τr0 andEqua- 4.2. Inferring Heating During Decay Phase
tion 6 if τ > τ . The uncertainties on these cool-
c0 r0 Forradiativelydominatedflares,thedecayphaseheat-
ing times were calculatedby firstrewritting Equations 5
ing required to account for the ‘excess cooling time’ can
and 6 in terms the observed input properties (tempera-
bedeterminedfromthefollowingmodifiedversionofthe
ture, density and loop half length) and then propagat-
energy transport equation
ingtheiruncertaintiesbythestandarderrorpropagation
rules. Having done this, we then compared the model- ∂T
3k N =−χNζTα+h (9)
predicted cooling times with the observations discussed B 0 ∂t 0
in Section 2.2.
wherek isBoltzmann’sconstant,N isthe density(as-
B 0
4. RESULTS&DISCUSSION sumed to be constant and equal to the initial density
to remain consistent with the Cargill model), T is tem-
4.1. Comparing Observed and Modeled Cooling Times
perature, t is time, χ, ζ and α have the same values as
Figure5showsthecomparisonofCargill-predictedand above (1.2 × 10−19, 2, and -0.5, respectively), and h is
observedcoolingtimes for 72 M- andX-classflares. The the heating rate per unit volume. This equation is stat-
1:1 line is over-plotted for clarity. It can clearly be seen ing that the rate of change of thermal energy density
that Cargill is consistent with observations at the short- (LHS), is determined by the radiative energy losses (1st
est cooling times, but is not a good overall fit to the term,RHS) andheating (2ndterm, RHS). The totalde-
distribution. Upon closer inspection it was found that cay phase heating energy, H, can then be evaluated by
only 14 events (20%) had observed cooling times which integrating over time and multiplying by flare volume,
agreed with Cargill within experimental error. Mean- V, assumed to be constant.
while 58 (80%) disagreed. Of those, only 1 was over-
estimated by Cargill. The remaining 57 were underes- H =V×
timated. Thus these results statistically prove that the ttot ∂T(t)
Cargill model provides a lower limit to the time needed 3k N +1.2×10−19N2T(t)−1/2 dt (10)
for a flare to cool. In addition, it was found in 52 flares Z0 (cid:20) B 0 ∂t 0 (cid:21)
8
Fig.5.—ComparisonofCargill-predictedcoolingtimeswithobservedcoolingtimes. The1:1lineisoverplotted forclarity.
This analysis was performed on 38 flares within our
sample(markedinTable2intheAppendixbytheirnon-
zerovaluesinthe‘DecayPhaseHeating’column). These
flareswerechosenbecausetheCargillmodelimpliedthat
radiationwasthe dominantcoolingmechanismthrough-
outtheirdecayphases,makingEquations9and10valid.
Theseflareswerealsoseentocooldowntoatleast6MK
so the majority of their cooling could be analyzed. The
rateofchangeoftemperature,dT/dt,inEquation10was
found by differentiating the secondorderpolynomialfits
to the coolingprofiles discussedin Section2.2. The flare
volume was calculated from the density and peak emis-
sion measure using the equation,
EM
V = (11)
N2
The emission measure was calculated from ratio of the
Fig.6.— Heating during the decay phase as a function of the
GOESlongchannelfluxandtemperatureusingthesame differencebetweentheobservedandCargill-predictedcoolingtimes
assumptions and methods as described in Section 2.1 for 38 M- and X-class flares. The line over-plotted is the best fit
(White et al.2005;Ryan et al.2012,TEBBS).Thetotal tothedata(seeEquation12).
decay phase heating required to account for the ‘excess
cooling time’ was then calculated from Equation 10.
Figure 6 shows resultant energies as a function of the range from 2×1028 – 5×1030 ergs. The findings of
‘excesscoolingtime’. ThePearsoncorrelationcoefficient
Withbroe (1978) and Jiang et al. (2006) fit into the up-
of the distribution was calculated in log-log space and
per limit of this range. They inferred total decay phase
found to be 0.77, implying a statistically significant cor- heating of 1031 ergs for the 1973 September 7 flare and
relation. The following power-law was fit to the data, >1030 ergs in the 2002 September 20 flare, respectively.
H =1026.73(∆t)1.06±0.24 (12) Althoughthe energiesinFigure 7a areplausible, further
testing of the Cargill model is necessary to categorically
where H is the total heating energy during the decay prove whether they are correct. Nonetheless, from these
phase, and ∆t is the ‘excess cooling time’. The uncer- heating calculations, it is possible to work out some im-
tainty on the exponent represents one standard devia- plications of these energies in fact being correct. This
tion. This power-law quantifies, in very simple terms, provides extra ways of testing the Cargill model’s accu-
the affect of heating during the decay phase on a flare’s racy.
cooling time. Firstly, the distribution in Figure 7a was fit with an
Figure 7a shows a histogram of these energies which anexponentialusingthemethodofmaximum-likelihood,
9
(Wall & Jenkins 2003, Chapter 5: Hypothesis Testing)
implied it was best suited to the data. However energy
frequency distributions of solar flares are often found to
bepower-laws(e.g.Aschwanden2011)whichmaybedue
to self-organized criticality. In this case, we cannot rule
outthe possibility that selectioneffects may havebiased
this distribution and including more events might reveal
ittobemorepower-law-like. Withthisinmind,apower-
law was also fit to this distribution via the method of
maximum-likelihood and found to be
f(H)∝H−0.6±0.1 (14)
Next, the values in Figure 7a were compared to the
total thermally radiated energy (Figure 7b). CHIANTI
was used to determine the spectra corresponding to the
temperature and emission measure as calculated from
GOES. The total radiated energy was then found by in-
tegrating over all wavelengths and over flare duration.
The distribution in Figure 7b ranges from 0.2–0.9 and
peaksat0.5. Aspreviouslystated,theCargillmodelim-
plies that radiation is the dominant loss mechanism for
these flares. If this is true, Figure 7b suggests that the
total heating during the decay phase typically makes up
half of the flare’s total thermal energy budget.
Thesignificanceofthetotaldecayphaseheatingisfur-
ther highlighted in Figure 7c where it has been normal-
ized by the thermal energy at the flare peak, calculated
from the following equation.
E =3Nk T (15)
peak B
Thedistributionhasanegativeslopeandrangesfrom<1
to >7. This implies that the total decay phase heating
energyinferredfromthe‘excesscoolingtime’canbesev-
eral times greater than the thermal energy at the peak.
This agrees with Jiang et al. (2006) who found that the
inferred decay phase heating in the 2002 September 20
event was greater than the energy deposited during the
impulsive phase. Such a result would be significant as
previousstudies(e.g. Emslie et al.2012)haveusedpeak
thermal energy as an estimate for the total thermal en-
ergy of a flare. To quantify the relationship between de-
cay phase heating, H, and peak thermal energy, E ,
peak
a power-law was fit to the data and found to be
H =10−2.7±0.4E1.1±0.1 (16)
peak
Thisimpliesthatthetotaldecayphaseheatingasafrac-
tion of the peak thermal energy is greater for greater
valuesofthe peakthermalenergy. Inthe rangeexplored
Fig.7.— Histograms showing the required total heating during here,theaveragetotaldecayphaseheatingis∼2.5times
thedecayphaseof38M-andX-classflarestoaccountforthediffer- thepeakthermalenergy. However,thisisexpectedtobe
encebetweentheCargill-predictedobservedcoolingtimes(‘excess less for less energetic flares. Thus if the ‘excess cooling
coolingtime’). a)Log10 oftotaldecayphaseheating. b)Totalde- times’inferredfromtheCargillmodelaretobebelieved,
cay phase heating normalizedby the total energy radiatedby the
flareasmeasuredbyGOES.c)Totaldecayphaseheatingdivided estimating a flare’s total thermal energy from its peak
bythethermalenergyatthebeginningofthecoolingphase. is valid for small flares, but not for the most energetic
events.
The predictions and comparisons made here all as-
resulting in,
f(H)∝e−γ·H (13) sumethatthetotaldecayphaseheatinginferredfromthe
Cargill model is reasonable. These predictions give fur-
where f(H) is the number of events as a func- ther ways of testing the validity of the Cargillmodel via
tion of total decay phase heating energy, H, and observations or more advanced modeling of decay phase
γ = 1.7 (±0.3) × 10−30. Again, the uncer- heating.
tainty represents one standard deviation. An exponen-
tialfit waschosenbecausethe Kolmogorov-Smirnovtest 5. CONCLUSIONS
10
In this paper, the cooling phases of 72 M- and X- found that the total decay phase heating predicted from
class solar flares were examined with GOES/XRS and the Cargill model typically makes up about half of the
SDO/EVE. The cooling profiles as a function of time thermallyradiatedenergybudgetofthehotflareplasma.
were parameterized and typically found to be very Finally, it was determined that if the decay phase heat-
linear. The average cooling rate was found to be inginferredfromtheCargillmodelistobebelieved,then
∼3.5×104 K s−1. These observations were compared to peakthermalenergyisanacceptableestimatefortheto-
the predictions of the Cargill et al. (1995) model. Loop talthermalenergyofsmallflares. However,this method
halflengthsneededbythismodelwerecalculatedviathe would underestimate the thermal energy budget for the
RTV scaling law (Rosner et al. 1978). The uncertainty most energetic events.
on this law was quantified by comparing the predicted In order to confirm of refute the findings inferred us-
lengths of 22 flares within the sample with observations ing the Cargill model, comparisons with direct obser-
made by Hinode/XRT. The loop half lengths predicted vations of the decay phase heating must be made for
byRTVscalinglawweretypicallywithinafactorof3of an ensemble of flares. This would further highlight the
those seen in Hinode/XRT. strengths and weaknesses of the Cargill model. In ad-
It was found that the Cargill model provides a well dition, including more temperature sensitive lines in a
defined lower limit on flare cooling times, and the de- similar analysis to this one or performing fits of the full
viation from the model was quantified. The root-mean- EVEobservedspectrumwouldgivemorecomprehensive
squaredeviationbetweentheobservationsandthemodel observationsofthetemperatureanddensityevolutionof
was found to be 961 s and which was 1.47 of the mean theflareplasma. Studiescomparingsimilarobservations
observed cooling time. Furthermore, the Cargill model withresultsofmoreadvancedhydrodynamicsimulations
finds that radiation is the dominant loss mechanism wouldalsohelpusbetterunderstandthethermodynamic
throughout the cooling phase for 80% of flares. For the evolution and energetics of flare decay phases.
remaining 20%, Cargill finds that conduction dominates
initially, before being superseded by radiation.
Wewouldliketothanktheanonymousrefereeforpro-
Next, the ‘excess cooling time’ was assumed to be due
viding constructive feedback on this manuscript. DFR
to additional heating. The total decay phase heating
would like to thank Arthur J. White, Trevor A. Bowen,
required to account for the ‘excess cooling time’ was in-
Dr. Jim Klimchuk, Dr. Joel C. Allred and Dr. C. Alex
ferredfor 38 flareswithin the sample. The energieswere
Young for their helpful discussions. He would also like
found to be physically plausible, ranging from 2×1028 –
to thank the Fulbright Association for funding the re-
5×1030 ergs. The frequency distribution could be de-
search. PCC and DFR would like to acknowledge fund-
scribed by either an exponential with an exponent of
ing from the Living With a Star Targeted Research and
−1.7(±0.3)× 10−30 or a power-law with an exponent
TechnologyProgram. ROMisgratefultotheLeverhulme
of −0.6±0.1. These total decay phase heating energies
Trust for financial support from grant F/00203/X, and
werefound to be highly correlatedwith the ‘excess cool-
to NASA for LWS/TR&T grant NNX11AQ53G.
ing time’andwasfitwith apower-lawwith anexponent
of 1.06±0.24 and a scaling factor of 1026.73. It was also
APPENDIX
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