Observations and Modelling of Pulsed Radio Emission from CU Virginis 3

arXiv:1201.3678v1 [astro-ph.SR] 18 Jan 2012

h yoycrto msin usqety rgloe a for et expected Trigilio is Subsequently, which (1996) emission. al. gyrosynchrotron et Leone the by level 10% success- the at (2006) al. et Vir. Leto CU to and it star radio applied quiescent MCP fully the a of 1987). from modulation emission al. rotational mod et the three-dimensional (Drake explain a constructed to magnetosphere (2004) al. the of et Trigilio factor in (Lorentz trapped emit- relativistic radiation trons radio mildly gyrosynchrotron quiescent by be This to ted phase 1991). believed emis- rotational is Leone radio with emission 1993; their modulate Umana 1992; and to & 1987) observed al. (Leone al. et been et (Linsky has Drake emitters sion 1994; addition, with radio al. In vary et known its Leone 1980). to are stars, Landstreet observed stars & MCP MCP been (Borra other phase has rotational Like field stars. magnetic days) effective (MCP) 0.52 magnetic Peculiar around (polar of ically field only (period magnetic of of rotator strong field (distance fast Its the nearby a 1952). very of (Deutsch is is one and It is stars. pc) Vir) 80 Ap CU studied hereafter best (HD124224, Virginis CU INTRODUCTION 1 o.Nt .Ato.Soc. Astron. R. Not. Mon. ..Lo K.K. from emission radio Virginis pulsed CU of modelling and Observations .Ravi V. c 6 5 4 3 2 1 R eteo xelnefrAlsyAtohsc (CAASTRO Astrophysics All-sky for Excellence of Centre ARC colo hsc,Uiest fMlore akil,VIC Parkville, Melbourne, of University Physics, of School colo nomto ehoois h nvriyo Sydn of University The Technologies, Information of School colo hsc,TeUiest fAead,S 05 Aust 5005, SA Adelaide, of University The Physics, of School utai eecp ainlFclt,CIOAtooyan Astronomy CSIRO Facility, National Telescope Australia ynyIsiuefrAtooy colo hsc,TeUniv The Physics, of School Astronomy, for Institute Sydney 01RAS 2011 iclrplrsto nterdoeiso a detected was emission radio the in polarisation Circular ∼ G lcsi ntectgr fMgei Chem- Magnetic of category the in it places kG) 3 5 1 , , 2 2 ..Manchester R.N. , , 6 , ..Bray J.D. 000 –0(01 rne 1My21 M L (MN 2018 May 31 Printed (2011) 1–10 , rotation ssont aetecretplearvltm rqec depende frequency ma time the arrival in words: pulse plasma Key w correct cold consistent through the not have refracted is is to emission shown in lines, the is field model, which magnetic emission in r the maser model angles A to cyclotron perpendicular electron emission is simple various emission A consider exp ge lines. to and a field order construct Virginis, magnetic we In CU pulses, pulse. two of the second between magnetosphere the hig time in the the reverses before of arrives dependence ordering emission the frequency and lower previo sion confirm the which pulse, det period We first rotation Array. the every Compact pulses Telescope radio Australia chem polarised the magnetic larly the with of taken observations Virginis radio cm CU 20 and cm 13 present We ABSTRACT 3 , 2 , .Hobbs G. − 2 ..Keith M.J. , aito ehnss non-thermal mechanisms: radiation γ ai otnu:stars continuum: radio ≤ )elec- 2) 2 .Murphy T. , el 00 Australia 3010, l. y S 06 Australia 2006, NSW ey, pc cec,PO o 6 pig S 70 Australia 1710, NSW Epping, 76, Box P.O. Science, Space d riyo yny S 06 Australia 2006, NSW Sydney, of ersity ralia ) 2 ie,adocpe arwfeunybn ls oahar- a to close band frequency narrow hol- a a (70 occupies angle forms and large emission lines, a ECM of at 1982), model cone low cone” Dulk “loss & the directivity (Melrose In high ECM polarization. with circular radiation 100% tightl produce and be (2006) can to den- al. radiation ECM et electron the beamed. Leto requires low by that observation modelled the the as ruled of and region was emission because the waves 2008) in Langmuir sity al. to et (Trigilio due out emis- radiation radio pulsed Plasma the for sion. mechanism emission emit the as to (ECM) observed been has that star MCP pulses. CU only radio decade. 2010) the a al. is almost et observ- by Ravi Vir cm 2011; separated 20 2008, were and 2000, observations cm al. where 13 et the (Trigilio the in hav bands Vir as ing profiles CU to pulse from referred Similar observed are been pulses. profile “trailing” pulse and Th the profile”. “leading” “pulse in the components as described two hereafter is period tion cannot mechanism. which emission period gyrosynchrotron the rotation from every arise pulses circularly 100% radio two produces polarised Vir CU that discovered (2000) rgloe l 20)pooe lcrnccornmaser cyclotron electron proposed (2000) al. et Trigilio rota- one across Vir CU for flux radio observed The − A 1 tr:idvda:C igns antcfields, magnetic Virginis, CU individual: stars: T , 4 E , tl l v2.2) file style X 6 ..Gaensler B.M. , ◦ o85 to ◦ otemgei field magnetic the to ) mti oe fthe of model ometric e rqec emis- frequency her nce. 1 antefrequency the lain clypcla star peculiar ically , 6 sdtcin.In detections. us .Melrose D. , ltv othe to elative c w circu- two ect t u data. our ith gnetosphere hc the which y e e 1 , 2 K.K.Lo et al. monic of the cyclotron frequency. Although ECM is narrow every few minutes in the 2009 Dec 25 to 26 and 2010 May band, the relatively wide band of the observation can be 19 epochs. Complex visibility data in four polarisations were explained if the emission region spans a range of magnetic recorded for 15 baselines with a sampling time of 10 s. field strengths. We also made use of archival (pre-CABB) ATCA data A simple ECM model does not explain all the features of to supplement our analysis. Altogether, we analysed eight the pulsed radio emission from CU Vir. For instance, in the epochs of data at 20 cm and seven epochs at 13 cm as sum- 20 cm observing band, the leading and trailing pulses are marised in Table 1. In column order we provide the date of separated by ∼ 5 hours. In the 13 cm observing band, the observation, the observing frequencies used1, the available leading pulse was not seen in observations by Trigilio et al. bandwidth, the configuration of the ATCA2, the total inte- (2008), but was seen a decade later by Ravi et al. (2010). In gration time, the phase calibrator used and a reference for contrast to the variability of the leading pulse, the trailing the data. pulse shows remarkable stability over more than a decade We calibrated our data using the standard miriad in both observing bands. The trailing pulse at 13 cm is al- software package (Sault et al. 1995)3. The flux amplitude ways observed to arrive earlier than the corresponding 20 cm scale and the bandpass response were determined from the pulse. Trigilio et al. (2008) postulated that this has a geo- backup ATCA primary calibrator 0823−500. Observations metric explanation, which is related to where the pulse orig- of a bright compact radio source (as listed in Table 1) for 1.5 inates in the magnetosphere. The discovery from Ravi et al. minutes every 20 minutes were used to calibrate the com- (2010) that the leading 13 cm pulse arrives later than the plex gains and leakage between the orthogonal linear feeds corresponding 20 cm pulse, agrees with this interpretation in each antenna. Radio frequency interference was removed and suggests that both pulses come from the same magnetic from the visibilities using the flagging tool mirflag. pole of the star. However, until now, existing data do not To obtain light curves, we first imaged the field in the provide enough frequency coverage to study the pulse pro- standard way. Then, we took the clean components of each file in detail as a function of observing frequency. Here we source, masking the location of CU Vir, and subtracted them present new data with more complete frequency coverage from the visibility data using the miriad task uvmodel. We over the 13 cm and 20 cm observing bands and demonstrate then shifted the phase centre of the visibilities to the location with our modelling that an ECM emission model, without of CU Vir and measured the flux density directly from the propagation effects, cannot explain the observations. In a visibilities by vector averaging their real components in time recent paper, Trigilio et al. (2011) suggest the frequency de- bins of three minutes. pendence of the pulse arrival time arises from frequency dependent refraction of the radiation through the stellar magnetosphere. In this paper, we have analysed the frequency 2.1 Pulse profile dependent refraction model in more detail. The 20 and 13 cm Stokes V pulse profiles from our ob- The structure of the paper is as follows. In §2 we present servations, along with other archival epochs, are shown in the observing and analysis techniques used to detect time Figures 1 and 2 respectively. The Stokes V flux densities varying emission from CU Vir with the Australia Telescope are plotted against rotational phase using the ephemeris Compact Array (ATCA) and present our observational re- from Trigilio et al. (2011) and adjusted for the rotation of sults. In §3 we discuss our modelling and show how it relates the Earth around the Sun. We have used the rotation pe- to our observations. riod, P = 0.52071601 days from Trigilio et al. (2011) to align the light curves and set phase zero, in HJD, to be 2450966.3601 + EP, where E is the number of epochs. As 2 OBSERVATIONS AND DATA ANALYSIS can be seen in Figures 1 and 2, the rotation period is a good fit to our data as the pulses are aligned in phase. We observed CU Vir with the ATCA on six separate oc- In the May 2010 epoch, we detected both the leading casions (2009 December 23 to 26, 2010 May 19, 2010 June and trailing pulses. The time between them is 5.20±0.05 15). The observations in December 2009 were only two hours hours in the 20 cm observing band. This is similar to the each in length because they were taken during unallocated pulse separation time determined by Trigilio et al. (2000) telescope time. For these short observations, we predicted and Ravi et al. (2010) in earlier epochs. In Figure 3, we have the arrival time of the trailing pulse using the ephemeris juxtaposed measurement of the effective magnetic field from from Ravi et al. (2010) and timed our observation accord- Pyper et al. (1998) with the average Stokes V pulse profiles. ingly. We were allocated time for the May and June 2010 Assuming a dipolar field structure, and take negative effec- epochs and hence were able to observe for about nine hours tive magnetic field to mean the field lines point into the star, continuously. A summary of the observation parameters is the magnetic South pole is closest to our line of sight during given in Table 1. the shorter of the two pulse separations. All our observations were taken with the Compact Ar- ray Broadband Backend (CABB; Wilson 2011) which offers 1 a bandwidth of up to 2 GHz per intermediate frequency and We refer to the 1384 MHz pre-CABB and 1503 MHz CABB polarisation. However, the maximum bandwidth in our 20 frequencies as the 20 cm band; and the pre-CABB 2496 MHz and CABB 2335 MHz frequencies as the 13 cm band and 13 cm observing bands was limited to around 700 MHz 2 In all cases a maximum baseline of 6 km was because our observations were taken before the upgrade was used. The various labels (6A, 6C and 6D) refer fully completed. Furthermore, it was not possible to observe to different antenna spacings, which are defined at at 20 and 13 cm simultaneously. To maximise the apparent http://www.narrabri.atnf.csiro.au/operations/array_configurations/configurations.html bandwidth, we switched between the 20 and 13 cm bands 3 http://www.atnf.csiro.au/computing/software/miriad/

c 2011 RAS, MNRAS 000, 1–10 Observations and modelling of pulsed radio emission from CU Virginis 3

Table 1. Summary of ATCA observation parameters

Date Start time Frequency Bandwidth ATCA conﬁguration Integration time Phase calibrator Reference UT (MHz) (MHz) (hours)

1999 May 29 06:57 1384, 2496 128, 128 6A 9.3 B1406−076 Trigilio et al. (2008) 1999 Aug 29 01:12 1384, 2496 128, 128 6D 9.0 B1406−076 Trigilio et al. (2008) 2008 Oct 30 21:06 1384, 2368 128, 128 6A 9.0 B1416+067 Ravi et al. (2010) 2009 Dec 23 18:13 1503 742 6A 1.2 B1406−076 This work 2009 Dec 24 18:11 2335 678 6A 1.6 B1406−076 This work 2009 Dec 25 20:01 1503, 2335 742, 678 6A 1.9 B1406−076 This work 2009 Dec 26 20:13 1503, 2335 742, 678 6A 2.4 B1406−076 This work 2010 May 19 08:02 1503, 2335 742, 678 6C 8.2 B1416+067 This work 2010 Jun 15 06:41 1503 742 6C 9.2 B1402−012 This work

Table 2. Phases of the leading and trailing pulse in the 19 May 15 2010 epoch Average 12 9 Frequency Leading pulse Trailing pulse Pulse separation 6

(MHz) arrival phase arrival phase (hours) Flux (mJy) 3 0 15 29th May 1999 1.375 0.360 ± 0.002 0.782 ± 0.002 5.27 ± 0.05 12 1.675 0.366 ± 0.002 0.778 ± 0.002 5.15 ± 0.05 9 1.875 0.372 ± 0.006 0.776 ± 0.006 5.1 ± 0.1 6 2.275 - 0.76 ± 0.02 - Flux (mJy) 3 0 2.475 - 0.76 ± 0.02 - 15 29th Aug 1999 2.675 - 0.756 ± 0.006 - 12 9 6

Flux (mJy) 3 0 15 30th Oct 2008 We did not detect the 13 cm leading pulse, which had 12 9 been detected in 2008. Although the trailing pulse was de- 6 tected in all epochs for which we have data at the relevant Flux (mJy) 3 0 phase, the peak ﬂux density varied between 6-12 mJy in the 15 23rd Dec 2009 20 cm observing band, and 7-13 mJy in the 13 cm observ- 12 9 ing band. With our data we can constrain the frequency at 6 which the pulse disappears to be between 1.9 and 2.3 GHz Flux (mJy) 3 0 in the 19 May 2010 epoch. 15 25th Dec 2009 Figure 4 shows the pulse proﬁles, in 200 MHz channels, 12 9 for the 19 May 2010 epoch. The pulse arrival phase is de- 6

pendent upon the observing frequency. This dependence is Flux (mJy) 3 0 clearly observed for the trailing pulse in the 26 Dec 2009 15 26th Dec 2009 epoch (Figure 5). This frequency dependence is stable over 12 9 at least six months since we observe similar phenomenon in 6

the trailing pulse in the 26 Dec 2009 epoch. We determined Flux (mJy) 3 0 the arrival phase of the pulses by cross correlating the single 15 19th May 2010 epoch pulse and the average pulse, and ﬁnding the phase 12 of the peak of the cross correlation. The error of the pulse 9 6

arrival phase is the variation of the parameter required to Flux (mJy) 3 2 0 increase the reduced χ between the single epoch pulse and 15 15th Jun 2010 the average pulse by one unit. Table 2 summarises the arrival 12 phase for the 19 May 2010 epoch. 9 6

Flux (mJy) 3 0

0 0.2 0.4 0.6 0.8 1 Phase

3 MODELLING Figure 1. 20 cm Stokes V pulse proﬁles from all epochs listed in Table 1 aligned using the rotation period from Trigilio et al. Models for the origin of the pulsed emission from CU Vir (2011). have been proposed by Trigilio et al. (2000), Trigilio et al. (2008) and Trigilio et al. (2011). We perform numerical simulation of each of these models to produce a pulse proﬁle, and compare these with our observations.

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15 2000 Average Effective B field 12 1500 1000 9 500 6 0

Flux (mJy) 3 -500 0 -1000 15 Effective B field (G) -1500 29th May 1999 Average (13cm) 12 12 9 9 6 6

Flux (mJy) 3 Flux (mJy) 3 0 0 15 29th Aug 1999 Average (20cm) 12 12 9 9 6 6

Flux (mJy) 3 Flux (mJy) 3 0 0 15 30th Oct 2008 12 0 0.2 0.4 0.6 0.8 1 Phase 9 6 Figure 3. Top: Effective magnetic field Beff from Pyper et al.

Flux (mJy) 3 0 (1998) vs. rotation phase. A negative Beff means the field lines 15 24th Dec 2009 12 are directed into the star. Middle and bottom: Average Stokes 9 V pulse profiles at 13 cm and 20 cm. If we assume CU Vir has a 6 dipolar field structure, then the magnetic South pole is closest to

Flux (mJy) 3 our line of sight during the shorter of the two pulse separations. 0 15 25th Dec 2009 12 9 in weaker magnetic ﬁelds as the radiation escapes, and the 6

Flux (mJy) 3 growth rates for the higher harmonics are too low for the 0 15 intensity to be signiﬁcant (Melrose & Dulk 1982). 26th Dec 2009 12 The sense of the polarised emission (positive Stokes V; 9 right circular polarisation) indicates that it originates en- 6 tirely from the north magnetic pole of the star. This may

Flux (mJy) 3 0 be explained by the presence of a quadrupole component in 15 19th May 2010 12 the magnetic ﬁeld of CU Vir (Trigilio et al. 2000). Conse- 9 quently, we consider only the north magnetic hemisphere in 6 our modelling.

Flux (mJy) 3 0

0 0.2 0.4 0.6 0.8 1 Phase 3.2 Model parameters

Figure 2. 13 cm Stokes V pulse profiles from all epochs listed Table 3 lists the parameters we use for our modelling. in Table 1 aligned using the rotation period from Trigilio et al. The obliquity of the magnetic axis was calculated by (2011) Trigilio et al. (2000) from measurements of the variation in effective magnetic field strength given an inclination i. Leto et al. (2006) derived the Alfvén radius and thickness 3.1 The magnetosphere of CU Vir of the magnetosphere by fitting multifrequency radio flux densities to a three-dimensional magnetospheric model sim- The emission models described here are based on a stel- ulating the quiescent gyrosynchrotron emission. The dipole lar magnetosphere model proposed by Andre et al. (1988), moment is calculated from a value of 3 kG for the mag- shown in Figure 6. The cold torus, containing trapped ma- netic field strength at both poles from Trigilio et al. (2000). terial from the stellar wind, is an extension by Trigilio et al. Hatzes (1997) suggest an off-centre dipole model, with differ- (2004). ent surface magnetic field strengths at each pole, but these In this model, electrons are accelerated by magnetic re- values give a similar dipole moment4. We use only the cen- connection in the magnetic equatorial plane of the star, and tral value for each parameter, ignoring the uncertainty. propagate along field lines towards the magnetic poles. They are reflected through magnetic mirroring; however, electrons with sufficiently small pitch angles will instead intersect the 3.3 Emission from field lines parallel to the surface of the star. This results in a loss-cone anisotropy magnetic axis in the pitch angles of this population of electrons, allow- The first model for the pulsed emission from CU Vir is from ing them to produce radio emission through the electron Trigilio et al. (2000). The emission is assumed to be directed cyclotron maser mechanism (Trigilio et al. 2000). ◦ as a hollow cone, making an angle of 85 with the local mag- ECM emission occurs close to a harmonic of the cy- netic field. At this point, it was believed that the population clotron frequency: ν = sνcyc, where s is the harmonic number. We assume the emission to be in the second harmonic, s = 2, because the fundamental, s = 1, is generally sup- 4 Since the star is treated here simply as a rotating dipolar field, pressed by gyromagnetic absorption of the thermal plasma the actual position of its surface is not required for the modelling.

c 2011 RAS, MNRAS 000, 1–10 Observations and modelling of pulsed radio emission from CU Virginis 5

Figure 6. Cross-section of the magnetosphere of CU Vir used in our modelling (not to scale). The magnetic field is assumed to be dipolar within the Alfvén surface (heavy dashes), with equatorial radius rA. Beyond this radius, field lines (light dashes) are distorted by the stellar wind. Electrons are accelerated in equatorial current sheets beyond the Alfvén radius, and travel along field lines within the middle magnetosphere (shaded), which divides the inner and outer magnetospheres. The parameter l describes the extent of the middle magnetosphere, indicating the position of the projection (dotted) of the field line marking its outer boundary within the Alfvén surface. Cooled gas from the stellar wind accumulates in a torus (shaded) near the star.

Table 3. Adopted parameters for CU Vir. See appendix A1 for details. (iii) Determine the strength of emission in this direction. The emission pattern is a hollow cone making an angle of Parameter Symbol Value ◦ 85 with the north magnetic axis; i.e., at a magnetic latitude a Rotation axis inclination [de- i 43 ± 7 of +5◦. We assume the emission to have a narrow Gaussian grees] profile centred on this latitude. Magnetic axis obliquity [de- β 74 ± 3 a (iv) Repeat steps (i)-(iii) for different rotation phases to grees] b form a pulse profile. Stellar radius [R⊙] R∗ 2.2 c Equatorial Alfvén radius [R∗] rA 15 ± 3 c A phase of φ = 0 under the definition used in equation Thickness of middle magneto- l 1.5 1 corresponds to the point in the rotation of CU Vir when sphere [R∗] its north magnetic pole is most closely directed towards us. Dipole moment [Am2] m 5.4 × 1033 d Therefore, we shift the phase by 0.08 to match the ephemeris a Trigilio et al. (2000) as described in section 2. We do not attempt to predict the b North (1998) absolute flux density, but instead scale the peak of the sim- c Leto et al. (2006) ulated pulse profile to match our observations. d see text The result of this simulation is shown in Figure 7 (right panel). This model correctly predicts the arrival times of the of high-energy electrons would approach the star along field pulses. However, it does not fit the width of the pulses, nor lines almost parallel to the magnetic axis, as shown in Figure their frequency dependence (which had not been observed 7 (left panel). when this model was developed). Our simulation procedure is as follows: (i) Select a rotation phase φ. 3.4 Emission from dipolar magnetic field lines (ii) Determine the magnetic latitude θE of emission in the This model for the pulsed emission from CU Vir is from direction of the Earth, using: Trigilio et al. (2008). It incorporates results from Leto et al. sin θE = sin β sin i cos φ + cos β cos i (1) (2006), who determined parameters of the magnetosphere,

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Figure 7. Schematic diagram (left), and pulse profile of emission (right) for the model of ECM emission from field lines (dashed) parallel to the magnetic axis (section 3.3). The simulated emission in the 20 cm (dark arrows) and 13 cm (light arrows) bands arises from different locations, but is emitted in the same directions, hence producing identical pulse arrival times (top right), in contrast to the averaged observed pulse profile (bottom right). Dotted and solid lines represent the 13 cm and 20 cm bands respectively.

12 2.675GHz 12 2.675GHz 9 9 6 6 3 3 Flux (mJy) 0 Flux (mJy) 0 -3 -3 12 2.475GHz 12 2.475GHz 9 9 6 6 3 3 Flux (mJy) 0 Flux (mJy) 0 -3 -3 12 2.275GHz 12 2.275GHz 9 9 6 6 3 3 Flux (mJy) 0 Flux (mJy) 0 -3 -3 12 1.875GHz 12 1.875GHz 9 9 6 6 3 3 Flux (mJy) 0 Flux (mJy) 0 -3 -3 12 1.675GHz 12 1.675GHz 9 9 6 6 3 3 Flux (mJy) 0 Flux (mJy) 0 -3 -3 12 1.375GHz 12 1.375GHz 9 9 6 6 3 3 Flux (mJy) 0 Flux (mJy) 0 -3 -3 0 0.2 0.4 0.6 0.8 1 0.7 0.75 0.8 0.85 0.9 Phase Phase

Figure 4. Stokes V pulse profile from the 19 May 2010 epoch. Figure 5. Stokes V pulse profile from the 26 Dec 2009 epoch. It Each panel represents the average of 200 MHz bandwidth. The is clear that the 1.4 GHz pulse lags the 2.7 GHz pulse by ∼ 0.03 noise is higher for the three pulse profiles in the 13 cm observing phase. band compared to 20 cm observing band because more channels were removed due to RFI. (i) Select a rotation phase and convert to magnetic latitude, as in section 3.3. including the Alfvén radius rA. These values imply that the (ii) Select a point on the magnetic equator at radius rA. field lines of the middle magnetosphere near the star are (iii) From this point, trace along the field line towards the not parallel to the magnetic axis. We assume them to have north magnetic pole, following the path that would be taken a dipolar configuration, as shown in Figures 6 and 9 (left by an electron in the middle magnetosphere. panel). (iv) At each point along the field line, determine the cy- Our simulation procedure is as follows: clotron frequency. When the cyclotron frequency is equal to half of the required emission frequency (corresponding to

c 2011 RAS, MNRAS 000, 1–10 Observations and modelling of pulsed radio emission from CU Virginis 7

a distinct double-peaked structure which is absent in the real data. There is some frequency dependence, but it aﬀects the pulse width (narrower at higher frequency) rather than the arrival time.

3.5 Refracted equatorial emission This model for the pulsed emission from CU Vir is from Trigilio et al. (2011). Rather than a hollow cone centred on the local field line, the ECM emission is assumed to be directed perpendicular to the local field line, and only in the two directions which are also parallel to the magnetic equatorial plane. For each longitude in the magnetic equatorial plane, this results in emission being visible from two points in the emission region. Figure 8. Emission in a hollow cone (dotted) centred on a mag- The key point of this model is that it includes the effect of refraction as the radiation crosses the boundary netic field vector (‘B-field’) at a magnetic latitude of θB. The half-opening-angle of the cone is δ. A sample emission vector is of the cold torus region (Figure 6). We neglect the effects shown as a dashed arrow. This vector may be parameterised with of refraction from density variations within the cold torus, the angle ω, which describes its position on the cone. Equation or from the interface where the radiation exits the torus. 2 describes the relationship between ω and the magnetic latitude The refractive index outside the cold torus is assumed to of the emission, θE. be unity. Inside the torus, the refractive index nct depends on the orientation of the magnetic field relative to the ra- emission at the second harmonic), stop and note the mag- diation (Gurnett & Bhattacharjee 2005), but if we assume them to be parallel it is: netic latitude θB of the direction of the magnetic field at this point. (v) The emission is assumed to be directed as a hollow 2 νp cone centred on an axis with magnetic latitude θB. We de- nct = 1 − (4) ν(ν − νcyc) termine the parameter ω which describes the position of the s − emission vector on this cone, as shown in Figure 8: For a plasma density in the cold torus of 109 cm 3, compatible with Leto et al. (2006), the plasma frequency is sin θE − cos δ sin θB cos ω = (2) νp = 280 MHz. If we assume that the magnetic field strength sin δ cos θB at the point of refraction is approximately the same as at the (vi) If there was no solution for ω, the emission power P point of emission, we have the cyclotron frequency νcyc = in this direction is zero. Otherwise, the emission power per ν/2 (if the emission is at the second harmonic). If the angle of incidence αi with the boundary of solid angle Ω is: ◦ the torus (see Figure 10) is assumed to be 60 , as by dP 1 ∝ (3) Trigilio et al. (2011), the angle of refraction αR can be found dΩ sin ω cos θB sin δ with Snell’s law: This formula is obtained by assuming the emission power to be equally distributed around the cone. See appendix A2 for 1 details. sin αR = sin αi (5) nct (vii) Repeat steps (i)-(vi) for different rotation phases to form a pulse profile. For nct < sin αi, this implies that the radiation is en- (viii) Repeat steps (i)-(vii) for different values of δ. Sum tirely reflected at the interface, and none is transmitted. the resulting profiles, weighting them with a narrow Gaus- Under the conditions assumed here, this occurs for ν < 790 ◦ sian function centred on δ = 85 . MHz, so the pulse spectrum should be truncated below this (ix) Repeat steps (i)-(viii) for emission frequencies across frequency. However, this is below the frequency range of our the observation band. Sum the resulting profiles, weighting observations, so we cannot test this here. them equally. Our simulation procedure is as follows: (x) Repeat steps (i)-(ix) for starting points on the mag- (i) Select a rotation phase and convert to magnetic lati- netic equator with radii between rA and rA + l. Sum the tude, as in section 3.3. resulting profiles, weighting them equally. (ii) Convert from the magnetic latitude of the emission The phase and height of the pulse profile are then ad- to the angle of refraction. From our assumption about the ◦ justed as in section 3.3. The Gaussian function used in step angle of incidence, this is done by adding 60 . ◦ (viii) is very narrow (width < 0.2 ), intended only to smooth (iii) Calculate the refractive index at this frequency with over the step size in other parameters, so the width of the equation 4. resulting profile is due to other aspects of the model. (iv) Determine the angle of incidence with equation 5. Results are shown in Figure 9 (right panel). The simu- (v) Convert from the angle of incidence to the original lated pulses are too wide to fit the observations, and display magnetic latitude of the emission. From our assumption

c 2011 RAS, MNRAS 000, 1–10 8 K.K.Lo et al.

Figure 9. Schematic diagram (left), and pulse profile of emission (right) for the model of ECM emission from field lines diverging from the magnetic axis (section 3.4. The regions in the magnetosphere where emission occurs are marked with solid rings, and the dark and light arrows correspond to emission in the 20 cm and 13 cm bands. The simulated emission (top right) is spread over a very broad range of magnetic latitudes, resulting in an extended pulse profile quite distinct from our averaged observations (bottom right). Dotted and solid lines represent the 13 cm and 20 cm bands respectively. about the angle of incidence, this is done by subtracting frequency dependence and the second by its grossly dissimi- ◦ 60 . lar pulse profile. We confirm that the third model, involving (vi) Determine the strength of emission in this direction. refraction of the emission as it propagates through material We assume the emission to have a narrow Gaussian profile in the stellar magnetosphere, yields the correct frequency centred on the magnetic equator. dependence and pulse arrival times to fit our observations. (vii) Repeat steps (i)-(vi) for different rotation phases to form a pulse profile. (viii) Repeat steps (i)-(vii) for emission frequencies across ACKNOWLEDGEMENTS the observation band. Sum the resulting profiles, weighting them equally. The Australia Telescope Compact Array is part of the Aus- tralia Telescope which is funded by the Commonwealth of The phase and height of the pulse profile are then ad- Australia for operation as a National Facility managed by justed as in section 3.3. The Gaussian function used in step ◦ CSIRO. K.L. is supported by an Australian Postgraduate (vi) is very narrow (width < 0.2 ), so the width of the re- Award and a CSIRO OCE scholarship. K.L. and B.M.G. sulting profile is due to variation with frequency across the acknowledge the support of the Australian Research Coun- observation band. cil through grant DP0987072. The Centre for All-sky As- Results are shown in Figure 10 (right panel). The pulse trophysics is an Australian Research Council Centre of Ex- arrival times and their frequency dependence both fit the cellence, funded by grant CE11E0090. G.H. is the recipi- observations. The observed pulses are wider than the simu- ent of an Australian Research Council QEII Fellowship (no. lated pulses; this may be due to inherent width of the emis- DP0878388). V.R. is the recipient of a John Stocker Post- sion beam, variation of the angle of incidence, or further graduate Scholarship from the Science and Industry Endow- turbulent refraction in the cold torus. ment Fund. K.L. would like to thank Robert Braun for help- ful discussions. We are grateful to Bernie Walp for perform- ing one of our observations. 4 CONCLUSIONS We observed CU Vir with the ATCA over six epochs and, REFERENCES in the 20 cm observing band, detected 100% circularly polarised pulses twice in a rotation period. This confirms that Andre P., Montmerle T., Feigelson E. D., Stine P. C., Klein the 20 cm pulses are stable over a period of more than a K., 1988, ApJ, 335, 940 decade. We did not detect the 13 cm leading pulse that was Borra E. F., Landstreet J. D., 1980, ApJS, 42, 421 observed a year earlier by Ravi et al. (2010). For our data, Deutsch A. J., 1952, ApJ, 116, 536 the leading pulse has a spectrum which cuts off between Drake S. A., Abbott D. C., Bastian T. S., Bieging J. H., 1.9 and 2.3 GHz. We can see a clear frequency dependency Churchwell E., Dulk G., Linsky J. L., 1987, ApJ, 322, 902 in the time between leading and trailing pulses across our Gurnett D., Bhattacharjee A., 2005, Introduction to entire observing band. plasma physics. Cambridge University Press We have simulated in detail the emission models pro- Hatzes A. P., 1997, MNRAS, 288, 153 posed by Trigilio et al. (2000), Trigilio et al. (2008) and Leone F., 1991, A&A, 252, 198 Trigilio et al. (2011). We show that the first two models do Leone F., Trigilio C., Umana G., 1994, A&A, 283, 908 not fit our observations, the first being excluded by its lack of Leone F., Umana G., 1993, A&A, 268, 667

c 2011 RAS, MNRAS 000, 1–10 Observations and modelling of pulsed radio emission from CU Virginis 9

Figure 10. Schematic diagram (left), and pulse proﬁle of emission (right) for the model of refracted ECM emission in the equatorial plane (section 3.5). The regions in the magnetosphere where emission occurs are marked with solid rings. The emission in the 20 cm and 13 cm bands (dark and light arrows respectively, and solid and dotted lines on the right) is refracted as it crosses the boundary of the cold torus region (dotted). This results in frequency-dependence in the pulse arrival time, corresponding with the observations (bottom right). Note that the refraction in the left panel is exaggerated.

Leone F., Umana G., Trigilio C., 1996, A&A, 310, 271 Leto P., Trigilio C., Buemi C. S., Umana G., Leone F., ~ ˆ ˆ ˆ ˆ 2006, A&A, 458, 831 U = E − (E · S)S Linsky J. L., Drake S. A., Bastian T. S., 1992, ApJ, 393, V~ = Bˆ − (Bˆ · Sˆ)Sˆ 341 Melrose D. B., Dulk G. A., 1982, ApJ, 259, 844 with lengths: North P., 1998, A&A, 334, 181 Pyper D. M., Ryabchikova T., Malanushenko V., Kuschnig |U~ | = sin i R., Plachinda S., Savanov I., 1998, A&A, 339, 822 Ravi V., Hobbs G., Wickramasinghe D., Champion D. J., |V~ | = sin β Keith M., Manchester R. N., Norris R. P., Bray J. D., et al., 2010, MNRAS, 408, L99 The angle between these vectors is φ, which we can then Sault R. J., Teuben P. J., Wright M. C. H., 1995, in ﬁnd as: R. A. Shaw, H. E. Payne, & J. J. E. Hayes ed., Astro- nomical Data Analysis Software and Systems IV Vol. 77 U~ · V~ of Astronomical Society of the Paciﬁc Conference Series, cos φ = |U~ ||V~ | A Retrospective View of MIRIAD. p. 433 Eˆ · Bˆ − (Eˆ · Sˆ)(Sˆ · Bˆ) Trigilio C., Leto P., Leone F., Umana G., Buemi C., 2000, = (A1) A&A, 362, 281 sin i sin β Trigilio C., Leto P., Umana G., Buemi C. S., Leone F., ˆ ˆ ˆ 2008, MNRAS, 384, 1437 As E, S and B are unit vectors, we can evaluate these Trigilio C., Leto P., Umana G., Leone F., Buemi C. S., dot products as trigonometric identities: 2004, A&A, 418, 593 Trigilio C., Leto P., Umana G., Simona Buemi C., Leone sin θE − cos i cos β cos φ = (A2) F., 2011, ArXiv e-prints: 1104.3268, Submitted to ApJ sin i sin β Letters Rearranging, we obtain: Wilson W. E., Ferris R. H., Axtens P., Brown A., Davis E., Hampson G., Leach M., Roberts P., et al., 2011, MNRAS, 416, 832 sin θE = sin β sin i cos φ + cos β cos i (A3)

which is the same as equation 1.

APPENDIX A: ANGULAR RELATIONS A1 Rotation of CU Vir A2 Conical emission Equation 1 connects the rotation phase of CU Vir to the Step (v) of the procedure in section 3.4 assumes a relation, magnetic latitude of emission in the direction of the Earth. equation 2, between the magnetic latitude of emission θE The relation between these is shown, with associated angles and the position ω of that emission on a cone, as shown in and unit vectors, in Figure A1. To derive the relationship, Figure 8. To demonstrate this, we note that Figure A1 is we construct vectors: equivalent to Figure 8, with the following substitutions:

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Figure A1. Rotation of CU Vir. The magnetic axis Mˆ and the rotation axis Rˆ are inclined by β. In the reference frame of the star, the vector towards the Earth Eˆ describes a circle around point C on the rotation axis, separated from the rotation axis by i, and parameterised by the rotation phase φ. The magnetic latitude of this vector is θE . The vectors U~ and V~ are in the planes Rˆ-Mˆ and Rˆ-Eˆ respectively; for their application, see section A1.

π β → − θB 2 i → δ φ → ω Therefore the derivation of equation A2 in section A1 also demonstrates the correctness of equation 2:

sin θE − cos δ sin θB cos ω = (A4) sin δ cos θB In step (vi), we then wish to ﬁnd the emission power P per solid angle Ω in this direction. We may assume that the emission is evenly distributed around the cone (i.e. dP/dω is constant). If we diﬀerentiate equation A4 with respect to θE , we get:

dω cos θE = − (A5) dθE cos θB sin δ sin ω This allows us to obtain the distribution of emission power with magnetic latitude:

dP dP dω = dθE dω dθE cos θE dP ∝− (as is constant) (A6) cos θB sin δ sin ω dω The emission power per solid angle is related to this:

dP 1 1 dP = dΩ 2π cos θE dθE 1 ∝ (A7) cos θB sin δ sin ω This is the result used in step (vi) of section 3.4: equation 3.

c 2011 RAS, MNRAS 000, 1–10