A method to measure the transverse magnetic field and orient the rotational axis of stars
Francesco Leone, Cesare Scalia, Manuele Gangi, Marina Giarrusso, Matteo Munari, Salvatore Scuderi, Corrado Trigilio, Martin Stift
DD raft version O ctober
1, 2018
Preprint typeset using L A TEX style AASTeX6 v. 1.0
A METHOD TO MEASURE THE TRANSVERSE MAGNETIC FIELD ANDORIENT THE ROTATIONAL AXIS OF STARS F rancesco L eone , C esare S calia , M anuele G angi , M arina G iarrusso , Universit`a di Catania, Dipartimento di Fisica e Astronomia, Sezione Astrofisica,Via S. Sofia 78, I–95123 Catania, Italy M atteo M unari , S alvatore S cuderi , C orrado T rigilio INAF - Osservatorio Astrofisico di Catania, Via S. Sofia 78, I–95123 Catania, Italy M artin S tift Armagh Observatory, College Hill, Armagh BT61 9DG. Northern Ireland
ABSTRACTDirect measurements of the stellar magnetic fields are based on the splitting of spectral lines into polarizedZeeman components. With few exceptions, Zeeman signatures are hidden in data noise and a number ofmethods have been developed to measure the average, over the visible stellar disk, of longitudinal componentsof the magnetic field. As to faint stars, at present observable only with low resolution spectropolarimetry, amethod is based on the regression of the Stokes V signal against the first derivative of Stokes I . Here we presentan extension of this method to obtain a direct measurement of the transverse component of stellar magneticfields by the regression of high resolution Stokes Q and U as a function of the second derivative of Stokes I .We also show that it is possible to determine the orientation in the sky of the rotation axis of a star on thebasis of the periodic variability of the transverse component due to its rotation. The method is applied to data,obtained with the Catania Astrophysical Observatory Spectropolarimeter , along the rotational period of thewell known magnetic star β CrB.
Keywords:
Stars: magnetic fields – Physical data and processes: polarization – Star individual: β CrB – Tech-niques: polarimetric INTRODUCTIONIn stellar astrophysics, magnetic fields are measured bymeans of the Zeeman e ff ect, whereby the (2 J + σ − - and σ + -components ( ∆ M = ±
1) are circularly polarized,the π -components ( ∆ M =
0) linearly. For weak to moderatefields, the displacements in wavelength of the σ -componentsfrom the unsplit line position λ (in Å) due to a magnetic field (cid:126) B (in G) is given by ∆ λ = .
67 10 − ¯ g λ | (cid:126) B | (1)where ¯ g is the so called “e ff ective Land´e factor”, related tothe Land´e factors g and g of the involved energy levels by¯ g = . g + g ) + .
25 ( g − g ) d (2)with d = [ J ( J + − J ( J + σ − and the σ + components of a simple Zee-man triplet (¯ g = .
0) at λ = v e sin i ≈ . − or an instrumental resolution of R =
200 000. In order to establish the presence of a stellarmagnetic field, it rather makes sense to measure the distancebetween the respective centers of gravity of a spectral line inleft-hand (lcp) and right-hand (rcp) circularly polarized light.The distance in wavelength between the lcp and rcp centersof gravity is proportional to the disk-averaged line-of-sightcomponent B z of the magnetic field vector, called “e ff ectivemagnetic field” by Babcock (1947). B || = W F I c (cid:90) π d φ (cid:90) π/ B z cos θ sin θ d θ × (cid:90) [ I c − I λ ] d λ (3) a r X i v : . [ a s t r o - ph . S R ] S e p where W is the equivalent width of the line, F I c denotes thecontinuum flux at the wavelength of the line; φ and θ are po-lar coordinates. I c and I λ represent the respective continuumand line intensities at the coordinate ( θ , φ ). B || is commonlyobtained from the relation given by Mathys (1994): R (1) V = W (cid:90) V c − V λ I c ( λ − λ ) d λ = .
67 10 − ¯ g λ B || (4)It is a fact that with increasing instrumental smearing,Stokes polarization profiles rapidly become unobservable(Leone & Catanzaro 2001; Leone et al. 2003); on the otherhand, high resolution spectropolarimetry is at present lim-ited to bright (V (cid:46)
10) stars. To overcome these limitations,Angel & Landstreet (1970) introduced a method based onnarrow-band ( ∼
30 Å) circular photopolarimetry in the wingsof Balmer lines for the measurement of magnetic fields ofstars that could not be observed with high-resolution spec-tropolarimetry. The di ff erence between the opposite cir-cularly polarized photometric intensities is converted to awavelength shift and subsequently to the e ff ective longitu-dinal field B || . Another method, suggested by Bagnulo et al.(2002b), is based on the relation between Stokes V and I forspectral lines whose intrinsic width is larger than the mag-netic splitting (Mathys 1989): V λ I λ = − .
67 10 − ¯ g λ B || I λ ∂ I λ ∂λ (5)This linear fitting of Stokes V against the gradient of Stokes I (Eq. 5) to measure the e ff ective magnetic field of faint tar-gets on the basis of low resolution spectropolarimetry with-out wasting any circular polarized signal has opened a newwindow. A method to measure the magnetic fields of previ-ously inscrutable objects has indeed been largely used. Thereader can refer to Bagnulo et al. (2015) for a review on thismethod and its results.The problem of measuring the magnetic field of faint starsrepresents a special case of the more general problem of howto recover Stokes profiles “hidden” in photon noise. Withreference to the very weak magnetic fields of late-type stars,the solution introduced by a lamented colleague and friend,Meir Semel, consisted in adding the Stokes V profiles of alllines present in a spectrum, obtaining a pseudo profile of avery high signal to noise (S / N) ratio (Semel & Li 1996). Thisidea has been further developed by Donati et al. (1997) whointroduced the
Least Squares Deconvolution (LSD) method.Later, Semel et al. (2006) initiated yet another approach tothe add-up of Stokes profiles from noisy spectra, based on
Principal Component Analysis .The measurement of the B || component is important to as-sign a lower limit to the strength of a magnetic field. Butin order to constrain the magnetic topology, the transversecomponent B ⊥ is necessary too. To our knowledge, no di-rect measurements of the transverse component of a stellarmagnetic field have yet been obtained. No relations simi-lar to Eqs. 4 and 5 have yet been implemented. According to Landi Degl’Innocenti & Landolfi (2004), Stokes Q and U arerelated to the second derivative of Stokes I by Q λ I λ = − .
45 10 − ¯ G λ B ⊥ cos 2 χ I λ ∂ I ∂λ (6) U λ I λ = − .
45 10 − ¯ G λ B ⊥ sin 2 χ I λ ∂ I ∂λ (7)where ¯ G = ¯ g − δ (8)is the second order e ff ective Land´e factor, with δ = ( g − g ) (16 s − d − / s = [ J ( J + + J ( J + J and J the angular momenta of the involved energylevels.Stokes Q and U signals across the line profiles are weakerthan the V signal and instrumental smearing is more destruc-tive for Stokes Q and U profiles than for Stokes V (Leoneet al. 2003) because their variations are more complex andoccur on shorter wavelength scales. As a result, Stokes Q and U have rarely been detected – being hidden in the noiseeven in stars characterized by very strong Stokes V signals– but it is worth mentioning that Wade et al. (2000) havesuccessfully applied the LSD method also to Stokes Q and U profiles. When observed, Stokes Q and U profiles repre-sent a strong constraint to the magnetic geometry (Bagnuloet al. 2001). Following Landi Degl’Innocenti et al. (1981)who showed that broadband linear polarization arises fromsaturation e ff ects in spectral lines formed in a magnetic field(Calamai et al. 1975), Bagnulo et al. (1995) have used phase-resolved broadband linear photopolarimetry to constrain stel-lar magnetic geometries.In Section 3, we show that application of the linear regres-sion method to high resolution Stokes V spectra results inhighly accurate measurements of the stellar e ff ective mag-netic field (hereafter longitudinal field ). An extension of thisregression method to high resolution Stokes Q and U spec-tra on the other hand results in a direct measure of the meantransverse component of the field (hereafter transverse field ).For this purpose, we have obtained a series of full Stokes IQUV spectra of β CrB (Section 2) over its rotational pe-riod with the
Catania Astrophysical Observatory Spectropo-larimeter (Leone et al. 2016).In Section 4.2, we will show that, as a consequence of thestellar rotation, the transverse component of the magneticfield describes a closed loop in the sky, o ff ering the possi-bility to determine the orientation of the rotational axis. β CRB OBSERVATIONS AND DATA REDUCTIONEver since Babcock (1949b), β CrB has been one of themost studied magnetic chemically peculiar main sequencestar. Distinctive characteristics of this class of stars are a )a very strong magnetic field as inferred from the integrated Figure 1 . Observed Stokes I , Q , U and V spectra of the magnetic star β CrB at rotational phase φ = .
66 (top block of six panels) and φ = . Q and U as a function ofthe second derivative of Stokes I , and Stokes V as a function of the first derivative of Stokes I . Noise spectra (Eq. 11) are shown to quantify thephoton and extraction errors.
Noise constant with wavelength validates the correctness of the data reduction. Slope with errors are reported.
Zeeman e ff ect. Typical fields are 1 - 10 kG, the strongestknown reaching 35 kG; b ) Variability of the magnetic field,spectral lines and luminosity with the same period ; the lon-gitudinal magnetic field often reverses its sign. So far, theoblique rotator is the only model that provides an acceptableinterpretation of the above-mentioned phenomena (Babcock1949a; Stibbs 1950). It is essentially based on two hypothe-ses: The magnetic field is largely dipolar with the dipole Periods typically measure 2 −
10 d, however much shorter and longerperiods have been found, see Catalano et al. (1993) and references therein. axis inclined with respect to the the rotational axis, and )Over- and under-abundances of chemical elements are dis-tributed non-homogeneously over the stellar surface. All ob-served variations are a direct consequence of stellar rotation.For comparison with results on β CrB found in the litera-ture we adopted the measurements of the longitudinal fieldby Mathys (1994), the measurements of the “surface” field(the integrated field modulus) B s = W F I c (cid:90) π d φ (cid:90) π/ | B | cos θ sin θ d θ × (cid:90) [ I c − I λ ] d λ (9)by Mathys et al. (1997) and the ephemeris by Bagnulo et al.(2001): JD ( B max || ) = . + . β CrB has been measured 32times over its rotational period. These data have been ob-tained with the
Catania Astrophysical Observatory Spec-tropolarimeter (CAOS) from June to July 2014 in the 370-860 nm range with resolution R =
55 000 (Leone et al. 2016),the minimum signal-to-noise ratio was S / N = V by setting the fast axis of the quarter wave-plateretarder to α = + ◦ and − ◦ respectively. The fast axisof the half wave-plate retarder has been rotated by α = ◦ and 45 ◦ to measure Stokes Q , and by α = . ◦ and 67 . ◦ tomeasure Stokes U .There are several methods to measure the degree of po-larization from o -rdinary and e -xtraordinary beams from thepolarizer. As to the dual beam spectropolarimetry, the ratiomethod was introduced by Tinbergen & Rutten (1992). It isassumed that there is a time independent (instrumental) sen-sitivity G , for example due to pixel-by-pixel e ffi ciency varia-tions – together with a time dependent sensitivity F of spectra– for example due to variations in the transparency of the sky.So a photon noise dominated Stokes parameter (generically P = V , Q or U ) can be obtained from the recorded o -rdinaryand e -xtraordinary spectra, S α, o and S α, e respectively, at ro-tations α and α by: S α , o = . I + P ) G o F α S α , e = . I − P ) G e F α S α , o = . I − P ) G o F α S α , e = . I + P ) G e F α Hence: PI = R P − R P + R P = S α , o / S α , e S α , o / S α , e In addition we compute the noise polarization spectrum: NI = R N − R N + R N = S α , o / S α , e S α , o / S α , e (11)to check any possible error in Stokes P / I . Without errors, the noise polarization spectrum is expected to present no depen-dence on the Stokes I derivatives (Leone 2007; Leone et al.2011).The preferred use of Eqs. 5, 6 and 7 over the original re-lations given by Landi Degl’Innocenti & Landolfi (2004) isdue to the higher accuracy that can be achieved in measuring Q / I , U / I and V / I as compared to Q , U and V . MEASURING MAGNETIC FIELD COMPONENTSAs stated in the introduction, the linear fitting of Stokes V versus the first derivative of Stokes I of Balmer line profileshas opened a new way to measure B || of stars on the basisof low resolution spectra. Introducing this method, Bag-nulo et al. (2002b) quoted a series of papers based on pho-topolarimetry of Balmer line wings to justify the validity of Figure 2 . Examples of C ossam simulations for a magnetic dipole,B p =
10 kG, orthogonal to the rotational axis and along the E-W di-rection. Left panel: star rotating at 3 km s − , right panel: 18 km s − . Table 1 . Ratio between the measured transverse field and the ex-pected value, derived by applying the derivative method. In thisparticular case the equatorial velocity is equal to the projected ve-locity. v eq [km s − ]0 3 6 12 18B p [G] 10 1.93 1.18 1.17 0.79 3.51100 0.96 1.13 1.13 1.13 3.371000 0.93 1.06 1.08 1.15 3.3210000 0.81 0.84 1.02 1.07 1.21 Eq. 5 also for the whole visible disk of a star with a complexmagnetic field and despite the limb darkening (Mathys et al.2000).Mart´ınez Gonz´alez & Asensio Ramos (2012) have shownthat Eqs. 5, 6 and 7 are valid for disk-integrated line profilesof rotating stars with a magnetic dipolar field, provided therotational velocity is not larger than eight times the Dopplerwidth of the local absorption profiles. We have performednumerical tests with C ossam (Stift et al. 2012) to find outhow far the derivative of the Stokes I profile reflects Zeemanbroadening before being dominated by the rotational broad-ening. As a limiting case, we have assumed the dipole axisorthogonal to the rotation axis, both being tangent to thecelestial sphere. Two cases are shown in Figure 2 and re-sults are summarized in Table 1 for the spectral resolution ofCAOS. These numerical simulations show that by applyingthe slope method, the transverse field of a star observed withCAOS is estimated correctly to within 20% for rotational ve-locities up to 12 km s − . We ascribe the anomalous value fora non-rotating star with a weak (10 G polar) field to the factthat the spectral line profiles are dominated by the 5.5 km s − instrumental smearing.We have also addressed the capability to measure the trans-verse component of fields that are not purely dipolar. As abenchmark, we have extended the previous numerical testswith C ossam for a star rotating at 3 km s − and B p =
10 kG.The dipole, whose axis is still going through the center ofthe star, has been displaced in the direction of the positivepole. As a function of the decentering in units of the stellarradius a , the ratio between the mesured transverse field andthe expected value is r ( a = . = . r ( a = . = . r ( a = . = . r ( a = . = .
15, and r ( a = . = . β CrB, which displays a rotational velocity of3 km s − (Ryabchikova et al. 2004).3.1. The longitudinal field component of β CrB
We have applied the method to our high resolution spec-tra and found a very high precision of the measurements.Figure 1 shows Stokes I and V of β CrB at rotational phases φ = .
66 and 0.85 in a 30 Å interval centered on theFe ii V as a func-tion of the first derivative of Stokes I and its linear fit. If ¯ g =
1, the slope gives an error in the measured B || of about 40 G.It is worthwhile noting that the same procedure, as applied tothe noise spectra, gives a much smaller error of less than 4 G.We ascribe the 40 G error to the line-by-line di ff erences inthe ¯ g Land´e factors, resulting in the superposition of straightlines with di ff erent slopes. The observed Stokes I and V pro-files of a generic spectral line k , with e ff ective Land´e factor g k e ff , define a straight line in the − . × − λ I λ ∂ I λ ∂λ vs V λ I λ plane whose slope is c k = g k e ff B || . Using a set of N spectrallines we measure an average value for the longitudinal field < B || > = (cid:80) c k N = < g e ff > B || . The relative error in the longitu-dinal field measure is given by the dispersion of the e ff ectiveLand´e factors.Even though the precision is very high, the accuracy ofthe longitudinal field measurements depends on the adopted¯ g value; usually this is assumed equal to unity. In Leone(2007), we have numerically shown that the average value ofthe Land´e factors of the spectral lines of the magnetic star γ Equ, observed in the 3780-4480 Å interval and weightedby their intensity, is about 1.1. As to β CrB, adoptingthe e ff ective temperature, gravity and abundances given byRyabchikova et al. (2004), we have extracted from VALDthe list of expected spectral lines and found an average valueof ¯ g = . ± .
4. We conclude that the linear regressionmethod measures the longitudinal field of a star with a preci-sion equal to the standard distribution of the e ff ective Land´efactors of the spectral lines involved.3.2. The transverse field component
As an extension to the method described above to mea-sure the longitudinal field, we have plotted the Stokes Q and U signals as a function of the second derivative of Stokes I (Eqs. 6 and 7). Figure 1 shows the expected linear dependen-cies for β CrB at two di ff erent rotational phases. The conversion of the slopes to transverse field measuresis less straightforward than in the longitudinal case. Line-by-line di ff erences in the second order Land´e factors are largerthan di ff erences in the e ff ective Land´e factors (Eq. 2). Thesecond order Land´e factors can become negative (Eq. 8), ef-fective Land´e factors only very exceptionally. In a list ofsolar Fe i lines given by Landi Degl’Innocenti & Landolfi(2004) some 8% of ¯ G values are negative.Table 2 reports the transverse field of β CrB by applyingEqs. 6 and 7 to 50 Å blocks of CAOS spectra in the 5000to 6000 Å interval. As for ¯ g , the adopted ¯ G of a block rep-resents the average of the G value of the predicted spectrallines. In order to check the reliability of our quantitativemeasurements of the transverse field, we have applied themethod also to the Fe ii T´elescope H´eliographique pour l’Etudedu Magn´etisme et des Instabilit´es Solaires (THEMIS).As applied to our collected spectra and on the basis of theephemeris given in Eq. 10, β CrB presents a transverse fieldthat varies with the rotation period (Figure 3). The averagevalue is about 1 kG and the amplitude as large as 0.25 kG.A comparison (Table 2) with results from the Fe ii χ is variable too with the rotation period, seeFigure 3. Since by definition, χ is limited to the range 0 − ◦ , it exhibits a saw-tooth behavior. THE ADDED-VALUE OF THE TRANSVERSE FIELDLarge e ff orts have gone into the study of stellar magneticfields (Mestel 1999) but it is still not possible to predict themagnetic field geometry of an ApBp star. As mentioned inthe introduction, the magnetic variability of early-type up-per main sequence stars is thought to be due to a mainlydipolar field, with the dipole axis inclined with respect tothe rotational axis. Once the mean field modulus could bedetermined in addition to the longitudinal field it becameclear that the magnetic configurations went beyond simpledipoles (Preston 1967). Deutsch (1970) was the first tomodel the field with a series of spherical harmonics, Land-street (1970) introduced the decentred dipole, Landstreet &Mathys (2000) adopted a field characterized by a co-lineardipole, quadrupole and octupole geometry and Bagnulo et al.(2002a) modeled the field by a superposition of a dipole anda quadrupole field, arbitrarily oriented.It has been known for quite some time that the surface fieldof β CrB cannot be represented by a simple dipole (Wol ff &Wol ff Figure 3 . Longitudinal and transverse magnetic field of β CrB as well as the angle χ of the transverse component with respect to the North-South meridian (measured counterclockwise) are plotted as a function of the rotational phase. The left panels show the expected variations fora dipole, whereas the right panels pertain to a field resulting from the superposition of a dipole, a quadrupole and an octupole. The vertical linemarks the positive extremum of the longitudinal field, i.e. the rotational phase when the line of sight, the rotation axis and the dipole axis all liein the same plane. At this phase, the transverse field is also aligned with the rotation axis; the angle χ gives the orientation of the rotation axiswith respect to the North-South direction in the sky. Table 2 . Measured transverse magnetic field of β CrB. Eqs. 6 and7 have been applied to CAOS spectra in the range 5000 to 6000 Åand to a well known single iron line.5000 - 6000 Å Fe ii B ⊥ ± σ χ ± σ B ⊥ ± σ χ ± σ ◦ kG ◦ ± ± ± ± ± ± ± ± ± ±
12 0.841 ± ± ± ±
11 0.564 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±
16 0.794 ± ± ± ±
11 0.608 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±
24 0.703 ± ± ± ± ± ± ± ±
17 1.000 ± ± ± ± ± ± ± ± ± ± ± ±
20 0.691 ± ± ± ± ± ± ± ±
17 0.483 ± ± ± ± ± ± of a pure dipole (Stibbs 1950) B d || ( t ) = + u − u B d (cid:32) cos i cos β d + sin i sin β d cos 2 π P (cid:33) (12)– where u is the limb coe ffi cient, i the angle between the lineof sight and the rotation axis, β d the angle between dipoleand rotation axes, B d the magnetic field strength at the polesand P the rotation period. Hence, the Schwarzschild (1950)relation B d || ( min , max ) = + u − u B d β d ± i ) (13)and Preston (1971) relationtan i tan β d = − r || + r || (14)– where r || is the ratio between minimum and maximumlongitudinal field values – one can establish combinations of i , β d and B d which match an observed sinusoidal B || variabil-ity. We note that the combination i = ◦ , β d = ◦ and Figure 4 . Two examples of magnetic dipoles indistinguishable fromthe respective longitudinal field variations that present very di ff erent χ variations. Field values are given in units of the polar strength. B d = . i = ◦ , β d = ◦ and B d = . β CrB, it is obvi-ously necessary to assume a magnetic field geometry with-out cylindrical symmetry (Mathys 1993). We have thus de-cided to model the magnetic variability by taking a dipole,a quadrupole and an octupole with symmetry axes pointingin di ff erent directions with respect to the rotation axis andwith respect to each other. As the reference plane we adoptthe plane defined by the rotation axis and the line of sight;the rotation phase φ is zero when the dipole axis lies in thisplane. The right panel of Figure 3 shows the result of our bestfit with i = ◦ , B d = + . β d = ◦ , B q = + . β q = ◦ , φ q = ◦ , B o = − . β o = ◦ , φ o = ◦ . φ q and φ o represent the azimuth of quadrupole and octupolerespectively.The problem of the uniqueness of this particular magneticconfiguration is outside the scope of this paper. At present wefocus exclusively on the added value of knowing the trans-verse component in relation to the orientation of the rota-tional axis, the radius and the equatorial velocity of magneticstars. 4.1. Degeneracy between i and β d Fig. 3 shows that the angle χ is dominated by the dipolarcomponent with only a negligible dependence on the higherorder components of the magnetic field. This doesn’t re-ally come as a surprise: Schwarzschild (1950) has shownthat the maximum value of the longitudinal field is equal to ∼
30% of the polar value for a dipole and equal to ∼
5% fora quadrupole. Numerical integration over the visible stellardisk shows that the same holds true for the transverse field: B max || = B max ⊥ ∼ . B d ∼ . B q ∼ . B o , consideringalso the octupole. This is an intuitive result since the longi-tudinal field for an observer simply is the transverse field ofanother observer located at 90 ◦ . For example, the longitu-dinal component as measured by an observer located abovethe north pole of a dipole is the transverse component foran observer lying in the magnetic equator. The latter cansee half of the southern hemisphere that presents exactly themagnetic field configuration of the invisible half of the northhemisphere.It is straight to show that the previous relations 12, 13 and14 together with an equal set of relations where i is replacedby i + ◦ , that are valid for the transverse field, break thedegeneracy between i and β d . We conclude that the knowl-edge of the transverse field component removes the indeter-minacy in the Schwarzschild relation (Eq. 13) between theangles formed by the rotation axis with the line-of-sight ( i )and the magnetic axis ( β ).We note that it is not necessary to solve these equations tosolve the degeneracy between i and β d when the χ variationwith the stellar rotation is available. It happens that, if i islarger than β d the χ variation is not a sawtooth (Figure 4).4.2. Orientation of the stellar rotational axis
The longitudinal and transverse components of a dipolarfield are projected along the dipole axis. This, within theframework of the oblique rotator model, describes a conearound the rotation pole. It happens that when we observethe extrema of the longitudinal field, the transverse field isprojected onto the rotation axis. This means that, when weobserve the extrema of the longitudinal field, the measuredangle χ represents the angle between the rotation axis andthe North-South direction in the sky. This simple considera-tion gives us the possibility to determine the absolute orien-tation of the rotation axis of a star hosting a dominant dipolarmagnetic field. From our data we conclude that the rotationaxis of β CrB is tilted by about 110 ◦ with respect to the N-Sdirection.4.3. Equatorial velocity and Stellar radius
Once the degeneracy between i and β d removed, the stel-lar radius can be inferred from the relation valid for a rigidspherical rotatorv e sin i[km s − ] P[days] = . (cid:12) ] sin i (15) where P is the rotational period. As to β CrB, Kurtz et al.(2007) report a v e sin i in the range 3 . − . − . The inde-terminacy ( i = ◦ , β d = ◦ ) or ( i = ◦ , β d = ◦ ) fromthe Schwarzschild relation would thus result in the followingvalues of the stellar radius: 1 . ± . R (cid:12) or 2 . ± . R (cid:12) .Our determination of the angle i = ◦ (implying 2 . R (cid:12) )agrees with the interferometric value of 2 . R (cid:12) for the radiusof β CrB obtained by Bruntt et al. (2010). The equatorial ve-locity lies between 6.6 and 8.4 km s − . CONCLUSIONSThe linear regression between Stokes V and the first deriva-tive of Stokes I in low resolution spectroscopy was intro-duced by Bagnulo et al. (2002b) as a method for estimatingthe longitudinal magnetic fields of faint stars.We have carried out phase-resolved and high-resolutionfull Stokes spectropolarimetry of the magnetic chemicallypeculiar star β CrB with the
Catania Astrophysical Obser-vatory Spectropolarimeter (Leone et al. 2016). On the basisof these data, we have shown that it is possible to extendthe previous method to the high resolution spectropolarime-try with the more general aim of recovering the Stokes pro-files hidden in the photon noise. A condition of faint starsas observed at low resolution but also of very weak stellarmagnetic fields. The precision appears to be limited by ourknowledge of Land´e factors and by the non homogeneousdistribution of chemical elements on the visible disk. Leone& Catanzaro (2004) found that measuring the longitudinalfield, element by element, di ff erent values are obtained moni-toring the equivalent width variations with the rotation periodof HD 24712.We have also shown that a regression of Stokes Q and U with respect to the second derivative of Stokes I provides adirect measure of the transverse component of a stellar mag-netic field and its orientation in the sky. If the magnetic fieldis not symmetric with respect to the rotation axis, the trans-verse field vector rotates in the sky. Having found that thedipolar component of the field is mainly responsible for thetransverse component, we conclude that it is possible to de-termine the orientation of the rotation axis with respect tothe sky: the value of the angle between the rotation axis andthe North-South direction corresponds to the value of χ atthe rotational phase where the longitudinal field reaches anextremum, viz. Θ = χ ( B extrem . || )To our knowledge, the transverse component has never be-fore been measured directly. The interpretation of broad-band linear photopolarimetry by Landi Degl’Innocenti et al.(1981), based on the linear polarization properties of spectrallines formed in the presence of a magnetic field and its appli-cation to phase-resolved data by Bagnulo et al. (1995) to con-strain the magnetic field geometries of chemically peculiarstars represent an approach somewhat similar to ours. It isworthwhile noting that β CrB has been modeled from phase-resolved broadband linear photopolarimetry by Leroy et al.(1995) and by Bagnulo et al. (2000) who found
Θ = ◦ and Θ = ◦ respectively. These values have to be com-pared with our result of Θ = ◦ .In view of the improving capability to obtain high resolu-tion spatial observations via optical and radio interferometry, it becomes increasingly important to know the orientation ofthe rotation axis in the sky. The determination of the trans-verse field is thus fundamental in multi-parametric problemssuch as the 3D mapping of the magnetospheres of early-typeradio stars (Trigilio et al. 2004; Leone et al. 2010; Trigilioet al. 2011).REFERENCES Angel, J. R. P., & Landstreet, J. D. 1970, ApJL, 160, L147Babcock, H. W. 1947, ApJ, 105, 105—. 1949a, ApJ, 110, 126—. 1949b, The Observatory, 69, 191Bagnulo, S., Fossati, L., Landstreet, J. D., & Izzo, C. 2015, A&A, 583,A115Bagnulo, S., Landi Degl’Innocenti, E., Landolfi, M., & Leroy, J. L. 1995,A&A, 295, 459Bagnulo, S., Landi Degl’Innocenti, M., Landolfi, M., & Mathys, G. 2002a,A&A, 394, 1023Bagnulo, S., Landolfi, M., Mathys, G., & Landi Degl’Innocenti, M. 2000,A&A, 358, 929Bagnulo, S., Szeifert, T., Wade, G. A., Landstreet, J. D., & Mathys, G.2002b, A&A, 389, 191Bagnulo, S., Wade, G. A., Donati, J.-F., et al. 2001, A&A, 369, 889Bruntt, H., Kervella, P., M´erand, A., et al. 2010, A&A, 512, A55Calamai, G., Landi Degl’Innocenti, E., & Landi Degl’Innocenti, M. 1975,A&A, 45, 297Catalano, F. A., Renson, P., & Leone, F. 1993, A&AS, 98, 269Deutsch, A. J. 1970, ApJ, 159, 985Donati, J.-F., Semel, M., Carter, B. D., Rees, D. E., & Collier Cameron, A.1997, MNRAS, 291, 658Kurtz, D. W., Elkin, V. G., & Mathys, G. 2007, MNRAS, 380, 741Landi Degl’Innocenti, E., & Landolfi, M. 2004, Astrophysics and SpaceScience Library, Vol. 307, Polarization in Spectral Lines,doi:10.1007 / / aas:1997103Mathys, G., Stehl´e, C., Brillant, S., & Lanz, T. 2000, A&A, 358, 1151Mestel, L. 1999, Stellar magnetism (Oxford : Clarendon)Preston, G. W. 1967, ApJ, 150, 871—. 1971, PASP, 83, 571Ryabchikova, T., Nesvacil, N., Weiss, W. W., Kochukhov, O., & St¨utz, C.2004, A&A, 423, 705Schwarzschild, M. 1950, ApJ, 112, 222Semel, M., & Li, J. 1996, SoPh, 164, 417Semel, M., Rees, D. E., Ram´ırez V´elez, J. C., Stift, M. J., & Leone, F.2006, in Astronomical Society of the Pacific Conference Series, Vol.358, Astronomical Society of the Pacific Conference Series, ed.R. Casini & B. W. Lites (San Francisco, CA: ASP), 355Stibbs, D. W. N. 1950, MNRAS, 110, 395Stift, M. J., Leone, F., & Cowley, C. R. 2012, MNRAS, 419, 2912Tinbergen, J., & Rutten, R. 1992, A User’s Guide to WHTSpectropolarimetryTrigilio, C., Leto, P., Umana, G., Buemi, C. S., & Leone, F. 2011, ApJL,739, L10Trigilio, C., Leto, P., Umana, G., Leone, F., & Buemi, C. S. 2004, A&A,418, 593Wade, G. A., Donati, J.-F., Landstreet, J. D., & Shorlin, S. L. S. 2000,MNRAS, 313, 823Wol ff , S. C., & Wol ffff