X-ray Spectro-polarimetry with Photoelectric Polarimeters
aa r X i v : . [ a s t r o - ph . I M ] M a r X-ray Spectro-polarimetry with Photoelectric Polarimeters
T. E. Strohmayer
X-ray Astrophysics Lab, Astrophysics Science Division, NASA’s Goddard Space FlightCenter, Greenbelt, MD 20771
ABSTRACT
We derive a generalization of forward fitting for X-ray spectroscopy to in-clude linear polarization of X-ray sources, appropriate for the anticipated nextgeneration of space-based photoelectric polarimeters. We show that the inclu-sion of polarization sensitivity requires joint fitting to three observed spectra, onefor each of the Stoke’s parameters, I ( E ), U ( E ), and Q ( E ). The equations forStoke’s I ( E ) (the total intensity spectrum) are identical to the familiar case withno polarization sensitivity, and for which the model-predicted spectrum is ob-tained by a convolution of the source spectrum, F ( E ′ ), with the familiar energyresponse function, ǫ ( E ′ ) R ( E ′ , E ), where ǫ ( E ′ ) and R ( E ′ , E ) are the effective areaand energy redistribution matrix, respectively. In addition to the energy spec-trum, the two new relations for U ( E ) and Q ( E ) include the source polarizationfraction and position angle versus energy, a ( E ′ ), and ψ ′ ( E ′ ), respectively, and themodel-predicted spectra for these relations are obtained by a convolution withthe “modulated” energy response function, µ ( E ′ ) ǫ ( E ′ ) R ( E, E ′ ), where µ ( E ′ ) isthe energy-dependent modulation fraction that quantifies a polarimeter’s angu-lar response to 100% polarized radiation. We present results of simulations withresponse parameters appropriate for the proposed PRAXyS
Small Explorer ob-servatory to illustrate the procedures and methods, and we discuss some aspectsof photoelectric polarimeters with relevance to understanding their calibrationand operation.
Subject headings: polarization — methods: data analysis — techniques: polari-metric — X-rays: general — instrumentation: polarimeters
1. Introduction
In comparison with imaging, timing and spectroscopic measurements, polarization re-mains the “missing piece of the puzzle” of observational X-ray astrophysics. To date, the 2 –only measurement in the 2 - 10 keV band is still the ≈
20% polarization fraction inferredfor the Crab nebula (Weisskopf 1978). The dearth of additional detections has largely beenbecause of a lack of instruments sensitive enough to make such observations. However, withthe advent of micro-pattern gas detectors which can directly leverage the photoelectric effectto infer linear polarization (Costa et al. 2001; Black et al. 2004, 2010), it is likely that thissituation will change in the not-too-distant future. Indeed, a number of X-ray polarime-try mission concepts have been proposed in the last few years. Among these are severalphotoelectric effect polarimeters sensitive in the 2 - 10 keV band, including the Polarimeterfor Relativistic Astrophysical X-ray Sources (
PRAXyS ) Small Explorer (Hill et al. 2014;Jahoda et al. 2016), and the Imaging X-ray Polarimetry Explorer (
IXPE , Weisskopf et al.2016). Also, the proposed
PolSTAR experiment (Krawczynski et al. 2015) would pair ahard X-ray mirror similar to that flown on
NuSTAR with a passive scattering element toprovide broad band X-ray polarimetry from ≈ −
50 keV. Above 10 keV Compton scatter-ing becomes competitive with the photoelectric effect, and several instruments have recentlybeen developed to exploit this to enable polarimetry in the hard X-ray band. Among theseare the balloon-borne payloads
X-Calibur (Beilicke et al. 2014) and
PoGOLite (Chauvin etal. 2016). Additionally, ESA has recently selected the X-ray Imaging Polarimetry Explorer(
XIPE , Soffitta et al. 2016) for study and possible implementation as ESA’s M4 mediumclass mission. Finally, we note that shortly after submission of this paper, NASA selected
IXPE for implementation in the 2020 timeframe as a Small Explorer mission (SMEX).In anticipation of the further opening of the polarization window in the X-ray band itis timely to explore the question how one can properly generalize X-ray spectroscopic ob-servations to include linear polarization of X-ray sources. That is, what is the additionalcomputational “machinery” required to do X-ray spectro-polarimetry from space observato-ries. In this paper we outline in some detail how to infer the physical properties of sourceswhich include linear polarization properties with space-based photoelectric polarimeters.This includes a generalization of the standard methods of X-ray spectral “forward fitting” toinclude polarization properties, as well as a discussion of the detector calibration informationthat is needed.We also discuss some aspects of photo-electric polarimeters relevant to understandingtheir calibration and operation, and we present the results of simulations that illustrate theprocedures and methods using spectro-polarimetric capabilities appropriate for the proposed
PRAXyS
Small Explorer observatory. We note that the methods described here shouldalso prove applicable to data expected from NASA’s recently selected
IXPE small explorerobservatory, as well as other future X-ray polarimetry missions. 3 –
2. Background
For the simpler, well-known case where X-ray detectors have no polarization sensitivity,an observation of an astrophysical source, at least in the context of spectroscopy, can becharacterized by a physical input (intrinsic) source spectrum, and the detector’s energyresponse function, which quantifies the rate at which photons of intrinsic energy E ′ areobserved in detector energy channel E . The energy response function is often broken upinto two components, the energy redistribution matrix, R ( E ′ , E ), which is defined as theprobability that a photon of energy E ′ is detected in channel E , and the effective area, ǫ ( E ′ ), which quantifies the detector’s collecting area as a function of energy. The count ratespectrum predicted to be observed in a detector is then the convolution, O ( E ) = Z E ′ F ( E ′ ) ǫ ( E ′ ) R ( E ′ , E ) dE ′ , (1)where F ( E ′ ) is the intrinsic source spectrum, that is, the number of source photons withenergies between E ′ and E ′ + dE ′ . The standard, “forward fitting” procedure for doingX-ray spectroscopy is then to introduce some model parameterization for F ( E ′ ), use thatmodel and the response functions to generate predicted detector channel spectra, and thenconstrain the model’s parameters by comparing these to the actual observed spectra using astatistical procedure such as χ fitting (see, Lampton, Margon & Bowyer 1976). For example,the XSPEC software package is a commonly used tool in the X-ray astrophysics communityto implement this procedure (Arnaud 1996).
3. Polarimetry
The generalization to spectro-polarimetry adds an additional observable, the sky angle, ψ , of the polarization vector of each detected photon. A convenient way to define the skyangle is to construct a coordinate system on the sky with x and y axes defined as localNorth and East, respectively. The polarization angle is then taken as the azimuthal angleincreasing in the counter-clockwise direction when looking at the sky. The z -axis pointsfrom the source to the observer, forming a right-handed coordinate system. It is standardto define local North as having an angle of zero, and then an angle pointing East is π/ π . We note that this description is consistent with the IAU convention forpolarimetry (IAU 1974; see also Hamaker & Bregman 1996).Whereas with no polarization sensitivity the detector response matrix is a two dimen-sional function, for polarimetry it becomes a four dimensional function, X ( E ′ , E, ψ ′ , ψ ), that 4 –describes the rate at which photons with intrinsic energies E ′ and intrinsic polarization an-gles ψ ′ are observed by a detector with energy E and a 100% polarization along the angle ψ .A good polarimeter will be one where the observed angular distribution in ψ of the counts inany detector energy channel (or range of channels) is well-peaked around the intrinsic angle ψ ′ , the phrase “more strongly modulated” around ψ ′ also comes to mind.We consider a general X-ray source as an incoherent ensemble of photons with wave-vectors ~k = ˆ z . Such an ensemble can always be represented as a superposition of 100%polarized photons. In this more general case the observed “spectrum” is a function of bothenergy and sky angle. For a given energy channel (or range of channels) the distribution ofobserved sky angles is commonly referred to as a modulation curve, which can be written ina general way as, O ( E, ψ ) = Z E ′ Z ψ ′ H ( E ′ , ψ ′ ) X ( E ′ , E, ψ ′ , ψ ) dE ′ dψ ′ , (2)where H ( E ′ , ψ ′ ), is now a more general intrinsic source spectrum that describes the numberof source photons with energy E ′ between E ′ and E ′ + dE ′ , and with polarization anglesbetween ψ ′ and ψ ′ + dψ ′ . We will introduce several general source descriptions specific tolinear polarization, including a Stokes parameter description, shortly.In practice, a detector system will produce a polarization angle measurement in a coor-dinate system defined by the detector’s geometry, mode of operation, and orientation. Wecall this the detector frame, and we refer to intrinsic and measured polarization angles inthis frame as φ ′ and φ , respectively. The orientation of the detector with respect to the skycoordinate system defines a mapping which relates the sky angle and detector frame angles.We define this mapping as ρ ( t ), such that ψ ′ = φ ′ + ρ ( t ), and ψ = φ + ρ ( t ). If the detectororientation is fixed in time, then the mapping is simply a constant offset, ρ . If however thedetector is rotated at a constant rate about the line of sight (say, by the planned rolling ofa spacecraft around the line of sight), then the mapping is a simple linear function of time, ρ ( t ) = Ω t , where Ω is the angular roll rate of the spacecraft.Now, let’s begin by exploring an ansatz where the detector response function, X ( E ′ , E, φ ′ , φ ),can be factored into two terms, the standard energy response function, and a new term, V ( E ′ , φ ′ , φ ), that describes the angular response of the polarimeter. This term effectivelydescribes how well a photon’s intrinsic polarization angle φ ′ can be measured. With thisdescription the full response function can be written, X ( E ′ , E, φ ′ , φ ) = ǫ ( E ′ ) R ( E ′ , E ) V ( E ′ , φ ′ , φ ) . (3)Specifically, the angular term, V ( E ′ , φ ′ , φ ), describes the probability for a photon of intrinsicenergy E ′ and intrinsic polarization angle φ ′ to be observed with a polarization angle φ . 5 –We note that there could be additional ways of considering how to construct X ( E ′ , E, φ ′ , φ ),however, we argue that the above is reasonable for several reasons. First, It is known thatgrazing incidence X-ray optics essentially do not alter the polarization properties of inci-dent photons at levels relevant for this discussion (see Almeida & Pillet 1993; Chipman etal. 1992, 1993; Hill et al. 2016). Second, the dominant photoelectric cross-section in therelevant energy band (ionization of K shell electrons in an s state) is not dependent on thepolarization angle of the photon, and while some materials can display linear dichroism,a variation in the absorption strength with polarization direction, this is generally a smalleffect and only confined to narrow bands around some absorption edges (see for example,Collins 1997; Bannister et al. 2006). Thus, to good approximation the relevant physicsassociated with ǫ ( E ′ ) and R ( E ′ , E ) is largely independent of the photon polarization direc-tion. Finally, one will ultimately determine the response function in an empirical fashion, bycalibrating a detailed physical model of the detector system against actual measurements.For example, one can carry out laboratory measurements whereby 100% polarized beamsof photons of known energy and polarization angle illuminate the detector. The observedmodulation curves for different input photon energies can then be used as a direct estimatorof the angular distribution V ( E ′ , φ ′ , φ ), and one can explore empirically the extent to whichequation (3) is realized in practice.For photoelectric polarimeters, which infer the angular direction of an ejected photoelec-tron, the angular response function is generally of the form, V ( E ′ , φ ′ , φ ) = v ( E ′ , ( φ − φ ′ )) = C ( A ( E ′ ) + B ( E ′ ) cos ( φ − φ ′ )), where C is an overall normalization factor (see, for example,Costa et al. 2001). From this expression one can define the energy-dependent modulationfraction, µ ( E ′ ) ≡ ( V max − V min ) / ( V max + V min ) = B ( E ′ ) / (2 A ( E ′ ) + B ( E ′ )) (Costa et al.2001; Strohmayer & Kallman 2014). Alternatively, one can also write this response usingan equivalent Stokes formalism as, V ( E ′ , φ ′ , φ ) = C ( i ( E ′ ) + u ( E ′ ) sin(2 φ ) + q ( E ′ ) cos(2 φ )),where now the energy-dependent modulation fraction is µ ( E ′ ) = ( u ( E ′ ) + q ( E ′ )) / /i ( E ′ ),and φ ′ = 1 / − ( u ( E ′ ) /q ( E ′ )). A proper angular response function must be normalizedsuch that the integrated probability gives unity. It is straightforward to show that for thefunction V ( E ′ , φ ′ , φ ) = C ( A ( E ′ ) + B ( E ′ ) cos ( φ − φ ′ )), and with φ ranging from 0 to π , C = ( π ( A ( E ′ ) + B ( E ′ ) / − . First, we emphasize that photoelectric polarimeters are only sensitive to linear polar-ization, and not, circularly polarized radiation. That is, a purely circularly polarized X-raybeam would appear “unpolarized,” to such a detector, in the sense that the observed an- 6 –gular distribution of polarization position angles would be identical to that produced by acompletely unpolarized beam. Thus, when we refer to an unpolarized spectrum or flux, thistechnically also includes any circularly polarized component of the source flux. For the caseof linear polarization the spectro-polarimetric properties of X-ray sources can be describedin several equivalent ways. A convenient description uses the so-called Stokes parameterdecomposition. Here one can define the source spectrum as, H ( E ′ , ψ ′ ) = F ( E ′ ) + W ( E ′ ) + Z ( E ′ ) , (4)where W ( E ′ ) and Z ( E ′ ) describe the linear polarization properties of the source, and F ( E ′ )is the so-called total intensity energy spectrum. The fractional polarization amplitude isthen given by, a ( E ′ ) = ( W ( E ′ ) + Z ( E ′ )) / /F ( E ′ ), and the source polarization positionangle is ψ ′ ( E ′ ) = 1 / − ( W ( E ′ ) /Z ( E ′ )). An equivalent description can be given usingthe un-polarized, h ( E ′ ), and polarized, g ( E ′ ), spectra, H ( E ′ , ψ ′ ) = h ( E ′ ) + g ( E ′ ) , (5)and one must also define the energy-dependent polarization position angle, ψ ′ ( E ′ ). In thiscase the polarization amplitude is a ( E ′ ) = g ( E ′ ) / ( h ( E ′ )+ g ( E ′ )). A convenient feature of thisprescription is that each spectral component has a simple angular dependence. By definition,the un-polarized spectrum, h ( E ′ ), is uniformly distributed with intrinsic polarization angles ψ ′ ranging from 0 to π , and the polarized spectrum, g ( E ′ ), only has intrinsic angles ψ ′ ( E ′ ) = ψ ′ ( E ′ ). These two descriptions being equivalent, it is straightforward to show that thepolarized spectrum is g ( E ′ ) = ( W ( E ′ ) + Z ( E ′ )) / , and the un-polarized spectrum is h ( E ′ ) = F ( E ′ ) − ( W ( E ′ ) + Z ( E ′ )) / .When generating physical source models researchers may find it more convenient tocompute the total spectrum, F ( E ′ ), as well as the energy-dependent polarization amplitude, a ( E ′ ), and position angle, ψ ′ ( E ′ ). In such a case it is then straightforward to determine theStokes spectra, W ( E ′ ) and Z ( E ′ ) from the above equations. Doing this one finds, W ( E ′ ) = F ( E ′ ) a ( E ′ ) tan(2 ψ ′ ( E ′ ))(1 + tan (2 ψ ′ ( E ′ ))) / = F ( E ′ ) a ( E ′ ) sin(2 ψ ′ ( E ′ )) = g ( E ′ ) sin(2 ψ ′ ( E ′ )) (6)and Z ( E ′ ) = F ( E ′ ) a ( E ′ )(1 + tan (2 ψ ′ ( E ′ ))) / = F ( E ′ ) a ( E ′ ) cos(2 ψ ′ ( E ′ )) = g ( E ′ ) cos(2 ψ ′ ( E ′ )) . (7) We can now use equation (2) above to determine the observed spectrum for a particularsource description and detector response functions. For the source description we will use 7 – H ( E ′ , ψ ′ ) = h ( E ′ ) + g ( E ′ ) δ ( ψ ′ − ψ ′ ( E ′ )) since each component has a well-defined angulardistribution, which will simplify the integrations over ψ ′ . Here, the Dirac delta functionrestricts the intrinsic polarization angles, ψ ′ , for the polarized spectrum to the position anglefor the source, ψ ′ ( E ′ ). We will also use V ( E ′ , φ ′ , φ ) = C ( A ( E ′ ) + B ( E ′ ) cos ( φ − φ ′ )). Notethat with this form for V ( E ′ , φ ′ , φ ), we have that cos ( φ − φ ′ ) = cos (( ψ − ρ ( t )) − ( ψ ′ − ρ ( t ))) =cos ( ψ − ψ ′ ), and it suffices to write the response function directly in terms of ψ and ψ ′ ,although we emphasize that more complex detector angular response functions are possible.Substituting these expressions into equation (2) gives, O ( E, ψ ) = Z E ′ Z ψ ′ ( h ( E ′ ) + g ( E ′ ) δ ( ψ ′ − ψ ′ ( E ′ ))) ǫ ( E ′ ) R ( E ′ , E ) C ( A ( E ′ )+ B ( E ′ ) cos ( ψ − ψ ′ ( E ′ ))) dE ′ dψ ′ . (8)For the term proportional to the unpolarized component, h ( E ′ ), we can rewrite the integralas, Z E ′ h ( E ′ ) ǫ ( E ′ ) R ( E ′ , E ) π ( A ( E ′ ) + B ( E ′ ) / Z ψ ′ (cid:2) A ( E ′ ) + B ( E ′ ) cos ( ψ − ψ ′ ( E ′ )) (cid:3) dψ ′ dE ′ , (9)where we have explicitly included C given above. The integration over ψ ′ is now straightfor-ward, and is simply equal to π ( A ( E ′ ) + B ( E ′ ) / ψ dependence inte-grates out. This is as expected, since for the assumed form of V ( E ′ , ψ ′ , ψ ) = v ( E ′ , ( ψ − ψ ′ )),where µ ( E ′ ) is a function of E ′ only (and not ψ ′ ), an unpolarized source produces a uniformdistribution in observed angle ψ . We emphasize that if this is not the case, and the modu-lation fraction is also a function of the intrinsic polarization angle, ψ ′ , then one should notexpect an unpolarized source to produce a uniform (flat) angular distribution in observedangle ψ (we discuss this further in § Z E ′ h ( E ′ ) ǫ ( E ′ ) R ( E ′ , E ) dE ′ . (10)Thus, as intuition would suggest, the unpolarized term looks exactly like the analogous casewith no polarization sensitivity.The remaining term involving the polarized spectrum, g ( E ′ ), is, Z E ′ Z ψ ′ g ( E ′ ) δ ( ψ ′ − ψ ′ ( E ′ )) ǫ ( E ′ ) R ( E ′ , E ) π ( A ( E ′ ) + B ( E ′ ) / (cid:2) A ( E ′ ) + B ( E ′ ) cos ( ψ − ψ ′ ( E ′ )) (cid:3) dψ ′ dE ′ . (11)In this case the angular integration is simplified by the delta function, which restricts thesource polarized photons to have the intrinsic angle ψ ′ ( E ′ ). This reduces the integral to, Z E ′ g ( E ′ ) ǫ ( E ′ ) R ( E ′ , E )( A ( E ′ ) + B ( E ′ ) / (cid:2) A ( E ′ ) + B ( E ′ ) cos ( ψ − ψ ′ ( E ′ )) (cid:3) dE ′ , (12) 8 –where we have picked up a factor of π from the delta function integration. Now, using thedefinition of µ ( E ′ ) above it can be shown that A ( E ′ )( A ( E ′ ) + B ( E ′ ) /
2) = 1 − µ ( E ′ ) , (13)and B ( E ′ )( A ( E ′ ) + B ( E ′ ) /
2) = 2 µ ( E ′ ) . (14)With these substitutions we can re-write equation (12) in the form, Z E ′ g ( E ′ ) ǫ ( E ′ ) R ( E ′ , E ) (cid:2) (1 − µ ( E ′ )) + 2 µ ( E ′ ) cos ( ψ − ψ ′ ( E ′ )) (cid:3) dE ′ . (15)Combining all the terms from equation (8) we have the result, O ( E, ψ ) = Z E ′ (( h ( E ′ ) + g ( E ′ )(1 − µ ( E ′ ))) ǫ ( E ′ ) R ( E ′ , E ) (16)+2 g ( E ′ ) µ ( E ′ ) ǫ ( E ′ ) R ( E ′ , E ) cos ( ψ − ψ ′ ( E ′ )))) dE ′ . (17)This integrand is of the form α ( E ′ , E ) + β ( E ′ , E ) cos ( ψ − ψ ′ ( E ′ )) if we make the identifica-tions, α ( E ′ , E ) = ( h ( E ′ )+ g ( E ′ )(1 − µ ( E ′ ))) ǫ ( E ′ ) R ( E ′ , E ), and β ( E ′ , E ) = 2 g ( E ′ ) µ ( E ′ ) ǫ ( E ′ ) R ( E ′ , E ).Now, with the help of some trigonometric identities it is straightforward to show that, α ( E ′ , E ) + β ( E ′ , E ) cos ( ψ − ψ ′ ( E ′ )) = (cid:18) ( α ( E ′ , E ) + β ( E ′ , E )2 (cid:19) (18)+ (cid:18) β ( E ′ , E )2 sin(2 ψ ′ ( E ′ )) (cid:19) sin(2 ψ ) + (cid:18) β ( E ′ , E )2 cos(2 ψ ′ ( E ′ )) (cid:19) cos(2 ψ ) (19)(see, for example, Strohmayer & Kallman 2014). If we further define I ( E ′ , E ) = ( α ( E ′ , E ) + β ( E ′ , E ) / U ( E ′ , E ) = ( β ( E ′ , E ) /
2) sin(2 ψ ′ ( E ′ )), and Q ( E ′ , E ) = ( β ( E ′ , E ) /
2) cos(2 ψ ′ ( E ′ )),then equation (16) can be expressed in the familiar Stokes form O ( E, ψ ) = I ( E ′ , E ) + U ( E ′ , E ) sin(2 ψ ) + Q ( E ′ , E ) cos(2 ψ ) , (20)with, I ( E ) = Z E ′ ( h ( E ′ ) + g ( E ′ )(1 − µ ( E ′ )) + g ( E ′ ) µ ( E ′ )) ǫ ( E ′ ) R ( E ′ , E ) dE ′ (21)= Z E ′ ( h ( E ′ ) + g ( E ′ )) ǫ ( E ′ ) R ( E ′ , E ) dE ′ (22) U ( E ) = Z E ′ g ( E ′ ) µ ( E ′ ) ǫ ( E ′ ) R ( E ′ , E ) sin(2 ψ ′ ( E ′ )) dE ′ , (23) 9 –and Q ( E ) = Z E ′ g ( E ′ ) µ ( E ′ ) ǫ ( E ′ ) R ( E ′ , E ) cos(2 ψ ′ ( E ′ )) dE ′ , (24)where now we have explicitly included the integration over E ′ . These equations relate theintrinsic source properties (spectral and linear polarization properties defined by h ( E ′ ), g ( E ′ )and ψ ′ ( E ′ )) to the modulation curve observed by a polarization sensitive detector charac-terized by three response functions, the traditional energy response functions ǫ ( E ′ ) (effectivearea) and R ( E ′ , E ) (energy redistribution matrix), and the energy dependent modulationfraction µ ( E ′ ), which encompasses the detector’s polarization sensitivity. With the help ofequations (6) and (7) these can also be written as, I ( E ) = Z E ′ F ( E ′ ) ǫ ( E ′ ) R ( E ′ , E ) dE ′ (25) U ( E ) = Z E ′ W ( E ′ ) µ ( E ′ ) ǫ ( E ′ ) R ( E ′ , E ) dE ′ (26)and Q ( E ) = Z E ′ Z ( E ′ ) µ ( E ′ ) ǫ ( E ′ ) R ( E ′ , E ) dE ′ . (27)In thinking further about modeling X-ray sources including their linear polarizationproperties, the above discussion outlines a path. An observation in a particular energy chan-nel, E (or range of channels), is a background-subtracted counts or count rate modulationcurve of the form, O ( E, ψ ) = I ( E ) + U ( E ) sin(2 ψ ) + Q ( E ) cos(2 ψ ) . (28)One can perform a χ fit to the observed modulation curve for each energy channel, pro-ducing the three observed Stoke’s spectra, with their associated uncertainties. One can thendefine source models, using, for example, parameterizations for F ( E ′ ), a ( E ′ ), and ψ ′ ( E ′ )(recall that the polarized spectrum g ( E ′ ) = F ( E ′ ) a ( E ′ )), generate predicted spectra us-ing equations (25), (26) and (27), and then carry out χ minimization to find the sourceparameters which best fit the observed spectra in a statistical sense. This is entirely anal-ogous to the simpler case with no polarization sensitivity, except that the generalization tospectro-polarimetry requires the joint fitting of three observed spectra, one for each of theStokes parameters. One model spectrum, F ( E ′ ), is folded through the full detector responsefunction, ǫ ( E ′ ) R ( E ′ , E ), and the two new spectra, W ( E ′ ) = F ( E ′ ) a ( E ′ ) sin(2 ψ ′ ( E ′ )) and Z ( E ′ ) = F ( E ′ ) a ( E ′ ) cos(2 ψ ′ ( E ′ )) are folded through the “modulated response” function, µ ( E ′ ) ǫ ( E ′ ) R ( E ′ , E ).The forward fitting procedure implemented by XSPEC can in principle accommodatethis process with a few simple additions. For example, F ( E ′ ) is a physical energy spectrum, 10 –and the XSPEC package includes many such options. In order to compute the model-predicted spectra U ( E ) and Q ( E ) one would require model parameterizations for a ( E ′ ) and ψ ′ ( E ′ ). Such models do not yet exist in the current XSPEC implementation, but they couldbe easily added. In XSPEC parlance they would be relatively simple multiplicative modelcomponents. Further, the additional detector modulation function, µ ( E ′ ), is very much likean effective area function, which can be included in the XSPEC implementation as a so-called “ancillary response function” (an “arf” file), so this could easily be incorporated inthe same way.
4. Detector Considerations
In the discussion above we assumed that the detector’s angular response function sat-isfied the condition that V ( E ′ , φ ′ , φ ) = v ( E ′ , ( φ − φ ′ )), that is, we assumed that for a givenphoton energy E ′ all intrinsic polarization angles, φ ′ , produce the same modulation fraction.This need not necessarily be the case. A simpler way to say this is that a detector systemcould in principle measure some intrinsic polarization angles better than others.Photoelectric polarimeters work by imaging the charge track of a photoelectron producedwhen an X-ray photon is absorbed in the detection gas (see Costa et al. 2001; Black et al.2004). The charge track must be drifted some distance and then detected in a pixellateddetector/readout system. Detectors of this type have been developed in two basic geometries.One, which we call an “imaging” polarimeter, drifts the charge track in the same directionas the incident photon beam (Costa et al. 2001). The other, known as a “time projectionchamber” (TPC) polarimeter, drifts the charge in a direction orthogonal to the photonbeam (Black et al. 2004). When a charge track drifts it also diffuses, smearing out thetrack. If it drifts too far before being detected diffusion will completely erase the directional(polarization) information within the track. In order to keep diffusion to a reasonable levelan imaging polarimeter must have a relatively shallow layer of detection gas (unless diffusioncan be reduced in some other fashion). The pixellated readout system makes up the bottomof this layer. This provides a limit to the efficiency of detectors constructed in this geometry.By contrast, a TPC polarimeter can contain a much greater depth of absorbing gas (andtherefore have a greater efficiency, other things being equal), since the tracks are drifted tothe side of the detection volume. However, the position of the track within the field of view isbetter-sampled in the imaging geometry, so by centroiding the track or reconstructing it, onecan estimate where it ocurred on the sky (hence the imaging appellation). Since the trackdrifts to the side in a TPC the location information regarding where it interacted within thevolume is at least partially lost. 11 –Several effects in the photoelectron track imaging and reconstruction process can, inprinciple, result in some intrinsic angles producing higher modulations. For example, thedetector readout formats have some intrinsic, pixellated geometry. Some employ eitherhexagonal or Cartesian geometries. Since the track image is generally of modest resolution,angles corresponding to symmetries of the underlying readout geometry could in principlebe better resolved than others. This could result in particular intrinsic angles producing ahigher modulation than others.Another possible cause of variation in the modulation with intrinsic angle results frompotential drift asymmetries. Since tracks that drift for longer have a greater time to diffuse,the longer a track drifts the poorer, on average, is the accuracy with which the track anglecan be measured. For one thing this means that the modulation fraction in such a detectoris a function of the drift distance. Since X-rays are absorbed over a range of drift distances,the detector’s angular response will be an average over the modulation as a function of driftdistance, weighted by the relative number of photons absorbed at each distance. Since theaverage depth at which an X-ray is absorbed is also a function of X-ray energy, the totaldistribution of drift distances will depend to some level on the intrinsic photon energy. Asnoted above, these effects are included in any experimentally determined angular responsefunction.Considering a single interaction point for simplicity, there is a potential drift asymmetryintroduced by the geometry of a TPC compared to an imaging polarimeter. In an imagingpolarimeter the photoelectrons are preferentially ejected in a plane that is parallel to thedetector plane. Thus, for the imaging polarimeter, most of the tracks are drifted for the samedistance. This symmetry is broken in the TPC geometry, since the photoelectrons are nowpreferentially ejected in a plane which is perpendicular to the detection plane. In this casehalf the tracks are ejected toward the detection plane and half away from it. Those directed atthe detection plane will travel for a somewhat shorter distance (time), and will suffer slightlyless diffusion than those directed away from the detection plane. If we measure track anglesfrom 0 to 2 π with 0 (and 2 π ) representing tracks directed straight at the detection planeand π those directed straight away from the detection plane, then those with intrinsic angleof zero diffuse slightly less on average than those with angle π . Thus, the charge trackswith angle 0 can be expected to be slightly “sharper” (better resolved) and therefore theirangles somewhat more accurately determined. While it is beyond the scope of this paper toexplore such effects in detail, the discussions above establish that it should not be totallyunexpected for a photoelectric polarimeter to be described by an angular response functionthat does not exactly satisfy V ( E ′ , φ ′ , φ ) = v ( E ′ , ( φ − φ ′ )), although in many circumstancesthe assumption of uniformity is likely to be a very good one. 12 –Nevertheless, it is instructive to explore the non-uniorm response case a bit further,and below we will show that in such a case the response to an unpolarized source does notproduce a flat modulation curve. We can demonstrate this by introducing a slightly moregeneral response function. First, the properly normalized “uniform” response function usedabove can be expressed as, V ( E ′ , φ ′ , φ ) = (1 /π ) (cid:0) (1 − µ ( E ′ )) + 2 µ ( E ′ ) cos ( φ − φ ′ ) (cid:1) . (29)We can introduce a simple “non-uniform” response by defining µ ( E ′ , φ ′ ) = µ ( E ′ ) η ( φ ′ ), where η is a slowly varying function of φ ′ . We can mimic an illustrative non-uniform effect asdescribed above by defining η = 1 − (2 d/π ) | φ ′ | . Here, d is just a small constant factor thatdefines the size of a linear change in the modulation as φ ′ ranges from 0 to ± π/
2. In this casethe modulation would be a maximum for φ ′ = 0 and drop to 1 − d at φ ′ = ± π/
2. Figure 1compares the response functions in these uniform and non-uniform cases. The black curvesshow uniform response functions (with µ ( E ′ ) = 0 .
5) for 5 equally spaced values of φ ′ . Onecan see that the peaks shift for different φ ′ values, but the maximum values of the responseare equal (as expected for a uniform response). The dashed black line is equal to the integralof the uniform response over φ ′ (minus 0.68 to plot it within the same y range, the integral isunity by definition), and is flat as expected. The red curves in Figure 1 show a non-uniformresponse function for the same φ ′ values, and with µ = 0 . d = 0 .
3. It is fairly easy tosee how the modulation amplitude drops as φ ′ moves away from 0. The dashed red curveshows the integral over all φ ′ for the non-uniform response function (again minus 0.68 forplotting purposes), and it is evidently not flat, but peaks where the response shows thelargest modulation amplitude. It should be emphasized that the value of d chosen in thiscase is purely for illustrative purposes only, and is not meant to represent any particulardetector system.While some might consider such an effect a detector “systematic,” this is really notan accurate description, as it is simply a part of “how the detector works,” and one canstill generate predicted modulation curves, and do spectro-polarimetry based on equation(2), though, depending on the complexity of an actual non-uniform response, the resultingobserved modulation curves may not have a simple analytical representation, and numericalevaluation of the angular integrals in (2) could be necessary. In any case, the problem isstill well defined mathematically. It should be noted that in the above discussion we wereconsidering the observed response in a detector’s frame. If the detector is fixed (not rotating),then a similar response to unpolarized flux would be evident in the sky frame as well, thoughperhaps with some constant offset in angle specified by the mapping from detector to skyframes.Since polarization properties have not been measured for most astrophysical sources, 13 –a commonly raised concern is that a “non-uniform” response to unpolarized flux might befalsely claimed as a significant polarization measurement. But this again assumes that allpolarimeters will produce flat modulation curves when illuminated with unpolarized flux, andthis is only true if the response is uniform as described above. The key issue, as with anyobservational claims, is that they be based on accurate and reliable instrument calibrations.Thus, polarimeters with non-uniform responses can also be effective instruments, however,the potential risk that a non-uniformity could lead to a false polarization claim argues formore careful attention to detector modeling, calibration and monitoring in such a case.While we emphasize that the best way to mitigate against a false polarization claim isa proper understanding of ones detector response function, one can also “enforce” flatnessof the response to unpolarized flux in the sky frame by rotation of the detector about theline of sight. As outlined earlier, rotation of the detector provides a mapping from sky angleto detector frame angle of the form ψ ′ = φ ′ + Ω t , where Ω is the angular rotation rate of thespacecraft. Spacecraft rotation tends to enforce the condition that all intrinsic sky anglesare measured at all intrinsic detector frame angles. This has the effect of “smoothing out”a non-uniform response to unpolarized flux of the kind described in §
5. Simulated Observations and Data Modeling
We now walk through an example of spectro-polarimetric fitting using simulated ob-servations with parameters and response functions appropriate for observations with theproposed
PRAXyS
Small Explorer mission (Iwakiri et al. 2016). As is appropriate in thiscase, we consider a uniform angular response function, such that the modulation function, µ , is only a function of intrinsic photon energy, E ′ (see eqn 29).We first define a source model using the total spectrum, F ( E ′ ), the polarization ampli-tude, a ( E ′ ), and the polarization position angle, ψ ′ ( E ′ ). As an illustrative example wechoose model parameters consistent with the known spectrum and polarization proper-ties of the Crab nebula (see Weisskopf et al. 1978). We use a power-law photon spec-trum, F ( E ′ ) = C n E ′− α , with index α = 2 .
1. For the polarization properties we assumesimple linear dependencies with energy. We take a ( E ′ ) = a + ∆ a ( E ′ − . ψ ′ ( E ′ ) = ψ ′ − ∆ ψ ′ ( E ′ − . a = 0 .
19 and ψ ′ = 156 deg, which are ap- 14 –proximately consistent with the measured values at 2 . E ′ to explore the sensitivity of a PRAXyS observation to such changes. Forthis purpose we take ∆ a = 0 .
01 keV − and ∆ ψ ′ = 1 deg keV − .For the response functions we use an effective area curve and quantum efficiency ap-propriate for the proposed PRAXyS
Small Explorer (SMEX) observatory, and we use amodulation function, µ ( E ′ ) based on measurements obtained with and detector simulationsof the PRAXyS polarimeter (Iwakiri et al. 2016). For this demonstration example we em-ploy a simplified diagonal redistribution matrix, and since the Crab is a bright source weignore backgrounds. Figure 2 shows the resulting full (black) and modulated (red) effectiveareas (in units of cm ) used in our simulations, as well as the energy dependent modulationfunction, expressed as a percentage (green).To simulate an observation we carry out the following procedures. We assume a con-stant source count rate equal to that expected from the Crab nebula, and first assign arandom event time. We take the total spectrum and fold it through the full response matrix,obtaining a predicted count rate spectrum in energy channel space. We then convert thatto a cumulative distribution function and use the so-called transformation method to makerandom energy channel draws. Once we have the energy of the photon, we draw a randomdeviate between 0 and 1 and use a ( E ′ ) defined above to assign it to the “polarized” or “unpo-larized” angular distributions the correct fraction of times. For example, if it is “polarized,”then we assign it the correct sky position angle, ψ ′ ( E ′ ), for it’s energy. If it’s “unpolarized,”we assign it a sky position angle that is a random draw from a uniform distribution of skyangles. Finally, we “detect” the event’s observed sky angle using the appropriate angularresponse function for its energy (see equation 29). In this way we can build up a simulatedobservation of N events.We also allow for the possibility of uniform angular rotation of the spacecraft. Withrotation included a few additional steps are required for the simulations. First, after assigningan event’s intrinsic sky angle we find the corresponding detector angle for this sky anglebased on the event time and the mapping from sky to detector angle. Then, we assign anobserved detector angle using the appropriate angular response (modulation) function, andthen, finally, we place that detector frame angle back on the sky (detected sky angle) usingthe same mapping. 15 – The proposed baseline science plan for
PRAXyS would result in millions of countsdetected from the Crab nebula and pulsar, so we illustrate the capabilities and methodswith a simulation for 6 million detected counts from the nebula. For the χ fitting we firstgroup the energy bins such that each new channel has at least 1 . × counts. For eachenergy channel we then determine the three Stokes parameters that describe its observedmodulation curve. This can be done in two effectively equivalent ways. One is to define M angular bins and then bin up the events into a modulation curve and do χ fitting (see, forexample, Strohmayer & Kallman 2014). However, in this case one can also use a bin-freeestimator for the Stokes parameters, such that, I ( E ) = N E , U ( E ) = 2 P N E i =1 sin(2 ψ i ), and Q ( E ) = 2 P N E i =1 cos(2 ψ i ) (see, for example, Montgomery & Swank 2015; Kislat et al. 2015b).Here, N E is just the total number of events in energy channel E . For I ( E ) the uncertaintyis simply σ I = ( N E ) / , and for Q ( E ) and U ( E ) we use σ Q = σ U = (2 N E ) / . We used bothestimates and found they give consistent results. Here we present results using the bin-freemethod.With the three observed Stokes energy spectra we can now carry out χ fitting to con-strain the parameters of the spectro-polarimetric model described above. We use equations(22), (23) and (24) to determine the model-predicted spectra, and we jointly fit the simulateddata to all three spectra. We use a spectral model written in IDL along with a least-squaresfitting routine also developed within IDL that is based on MINPACK-1 (Markwardt 2009).The model has six free parameters, two each for the power-law energy spectrum, F ( E ′ ), thepolarization amplitude, a ( E ′ ), and the polarization position angle, ψ ′ ( E ′ ). Figures 3 through8 summarize results of the simulation with the parameters described above. Figures 3 and4 show the resulting Stokes spectra, I(E) (Figure 3), Q ( E ) and U ( E ) (Figure 4), along withthe best fitting model spectra (solid curves running through the data points). In addition tothis we also plot the difference between the data and best fitting models. For this examplewe have 84 total spectral bins (28 for each Stokes spectrum) and 6 free parameters for atotal of 78 degrees of freedom. We find an acceptable minimum χ value of 77.6.Figure 5 shows a “residuals” plot of χ = (Data − Model) /σ data . The inferred fractionalpolarization amplitude (top) and position angle (bottom) versus energy are shown in Figure6. Here, the top panel shows the observed modulation amplitude in each energy channel, a ( E ) = ( Q ( E ) + U ( E ) ) / /I ( E ) (red points), and the data points (black square symbols)show the inferred polarization amplitude, a p ( E ) = a ( E ) /µ avg ( E ) with 1 σ error bars. Here, µ avg ( E ) is the mean modulation in each grouped energy channel. The inferred polarizationposition angle versus energy, ψ ′ ( E ) = 1 / − ( Q ( E ) /U ( E )), is shown in the bottom panel.The solid (black) line in each panel is the best fitting polarization model (either amplitude or 16 –position angle), and the dashed lines show the input models used to generate the simulation(the “true” model). The red symbols in both panels show the existing polarization measure-ments of the Crab nebula as an indication of the current state of knowledge (Weisskopf etal. 1978). Finally, in Figures 7 and 8 we show the derived confidence regions for both thepolarization amplitude parameters (Figure 7) and the position angle (Figure 8). We showcontours drawn at ∆ χ = 2 . .
61, which correspond to confidence levels of 68 . PRAXyS would be extremely sensitive to such variations. For example, Figures 7 and 8indicate that with 6 million detected photons, variations in fractional amplitude at the levelof 1% keV − , and position angles at the level of 1 deg keV − , can clearly be detected. We have presented a generalization of the standard “forward fitting” procedure for X-ray spectroscopy to include linear polarization of X-ray sources. When the angular responseof the polarimeter is “uniform,” in the sense that for a given photon energy all intrinsicphoton polarization angles produce the same fractional modulation, then the polarizationsensitivity introduces two additional observed spectra, related to the Stokes U ( E ) and Q ( E )parameters. Thus, joint fitting of three observed spectra can yield constraints on spectro-polarimetric source models. The computation of the predicted spectra as a convolution ofthe source spectral model with the detector energy response function maintains the samefamiliar form, however, for the new U ( E ) and Q ( E ) energy spectra the appropriate detec-tor response function is the “modulated” response, µ ( E ′ ) ǫ ( E ′ ) R ( E ′ , E ), which is just thetraditional energy response function multiplied by the detector’s energy-dependent modula-tion function, µ ( E ′ ). The additional functionality required for spectro-polarimetry is thusrelatively straightforward, and could be incorporated within exiting X-ray spectral softwaretools, as for exaxmple, XSPEC, with relatively simple modifications.Several previous studies have also explored aspects of X-ray spectral analysis in thecontext of polarimetry. For example, Kislat et al. (2015a) described an iterative “un-folding” method based on Bayes’ Theorem to obtain model-independent estimates of theenergy spectrum and polarization properties, and they presented simulated results with thismethod appropriate for the X-Calibur hard X-ray Compton scattering polarimeter. In ad- 17 –dition, Krawczynski (2011) explored a maximum likelihood analysis method for Comptonpolarimeters based on measuring both the azimuthal and polar angles of the scattered pho-tons. There are similarities between these methods and the forward-folding procedure wedescribe here. For example, they both account for energy-dependent effects with a multidi-mensional response function that models how input and output observables (such as energyand position angle) are related through the detection process. The iterative procedure ap-pears to be more computationally intensive, but in principle, returns model-independentestimates of source spectra and polarization properties. On the other hand, forward foldingcan likely be more easily incorporated into existing software tools (such as XSPEC), andwhile not strictly model-independent, it still enables important insights regarding sourceproperties.While it is beyond the scope of this paper to derive results appropriate for all currentlyproposed X-ray polarimeters, the basic methods discussed here should also be applicableto instruments working in the hard X-rays and gamma-rays. However, it is possible thatadditional observables may need to be included in the response functions. For example, inaddition to measuring the energy and azimuthal scattering angles, the hard X-ray scatter-ing experiments, such as
X-Calibur , benefit from also measuring the polar scattering angle(Krawczynski 2011). Moreover, issues associated with uniformity of the response functions,as discussed in §
4, would have to be explored for specific detector systems.The author acknowledges helpful discussions with Keith Jahoda, Craig Markwardt, TimKallman, and Jean Swank. We also thank the anonymous referee for a helpful review of thispaper. 18 –
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3. The dashed curves show the integral overall φ ′ for both the uniform (black) and non-uniform (red) response functions (minus 0.68 tofit on the same vertical scale). The non-uniform response is evidently not flat, but peakswhere the response shows the largest modulation amplitude. See the discussion in § µ ( E ′ ) (green) expressed as a percentage. 22 –Fig. 3.— Stokes spectrum I ( E ) for our 6 million count Crab nebula simulation described in §
5. The black symbols and error bars show results of the simulated data, and the solid redcurve is the best fitting model for Stokes I ( E ). The horizontal red curve running throughzero is the data minus the best-fitting model. See the discussion in § U ( E ) (lower) and Q ( E ) (upper) for our 6 million count Crab nebulasimulation described in §
5. The black symbols with error bars show the simulated data, andthe solid curves show the best-fitting models for U ( E ) (red) and Q ( E ) (green). The greenand red histograms running through zero are the data minus the model for each spectrum.See the discussion in § χ = (Data − Model) /σ data for the best-fitting model for our 6million count Crab nebula simulation described in §
5. The fit to all three spectra are shownin terms of “bin number,” with 28 bins for each spectrum in the order I ( E ), U ( E ) and Q ( E ).For example, bins 0 - 27 correspond to the residuals for Stokes I ( E ). See the discussion in § §
5. Here, the observed modulation amplitude (green) is a ( E ) =( Q ( E ) + U ( E ) ) / /I ( E ), and the inferred polarization amplitude is a p ( E ) = a ( E ) /µ ( E ).The solid (black) line is the best fitting polarization amplitude model, and the dashed lineshows the input amplitude model used to generate the simulation (the “true” model). Bot-tom: Observed position angle versus energy for our 6 million count Crab nebula simulation.The solid (black) line is the best fitting position angle model, and the dashed line showsthe input model used to generate the simulation (the “true” model). In both panels the redsymbols show the existing polarization measurements of the Crab nebula as an indication ofthe current state of knowledge (Weisskopf et al. 1978). See § A , and ∆ A ,derived from our 6 million count Crab nebula simulation described in §
5. We show contoursdrawn at ∆ χ = 2 .
3, and 4.61 which correspond to confidence levels of 68 .
3, and 90%,respectively. The green square symbol marks the input (“true”) values. See the discussionin §§
3, and 90%,respectively. The green square symbol marks the input (“true”) values. See the discussionin §§ ψ , and∆ ψ for our 6 million count Crab nebula simulation described in §
5. We show contoursdrawn at ∆ χ = 2 .
3, and 4.61 which correspond to confidence levels of 68 .
3, and 90%,respectively. The green square symbol marks the input (“true”) values. See the discussionin §§