Issue 
A&A
Volume 632, December 2019



Article Number  L1  
Number of page(s)  5  
Section  Letters to the Editor  
DOI  https://doi.org/10.1051/00046361/201936659  
Published online  29 November 2019 
Letter to the Editor
Vertical position of the Sun with γrays
Center for Astrophysics and Space Sciences, University of California, San Diego, 9500 Gilman Dr, La Jolla, CA 920930424, USA
email: tsiegert@ucsd.edu
Received:
9
September
2019
Accepted:
16
October
2019
We illustrate a method for estimating the vertical position of the Sun above the Galactic plane by γray observations. Photons of γray wavelengths are particularly well suited for geometrical and kinematic studies of the Milky Way because they are not subject to extinction by interstellar gas or dust. Here, we use the radioactive decay line of ^{26}Al at 1.809 MeV to perform maximum likelihood fits to data from the spectrometer SPI on board the INTEGRAL satellite as a proofofconcept study. Our simple analytic 3D emissivity models are lineofsight integrated, and varied as a function of the Sun’s vertical position, given a known distance to the Galactic centre. We find a vertical position of the Sun of z_{0} = 15 ± 17 pc above the Galactic plane, consistent with previous studies, finding z_{0} in a range between 5 and 29 pc. Even though the sensitivity of current MeV instruments is several orders of magnitude below that of telescopes for other wavelengths, this result reveals once more the disregarded capability of soft γray telescopes. We further investigate possible biases in estimating the vertical extent of γray emission if the Sun’s position is set incorrectly, and find that the larger the true extent, the less is it affected by the observer position. In the case of ^{26}Al with an exponential scale height of 150 pc (700 pc) in the inner (full) Galaxy, this may lead to misestimates of up to 25%.
Key words: Galaxy: general / Galaxy: structure / gamma rays: general / gamma rays: ISM / methods: statistical
© ESO 2019
1. Introduction
Measuring the vertical position of the Sun is interesting and important as it provides insights about Galactic kinematics, gravitational potentials, the history of the Solar System, and whether the Sun is located in a special position inside the Milky Way. Karim & Mamajek (2017) described the need for an absolute reference frame of the Galactic Coordinate System, historically, and in the context of current microarcsecondresolution astrometry. From their finding that the North Galactic Pole is 90.120 ° ±0.029° away from the dynamical centre of the Galaxy, that is, Sgr A*, the authors estimate that the Sun’s position above the Galactic midplane is currently 17 ± 5 pc to correct the absolute coordinate system. Thus, an accurate determination of the Sun’s absolute position will provide a more stable Galactic Coordinate System, which in return will allow for more precise measurements of Galactic kinematics and its overall structure.
With the subpercent measurement of the distance to the Galactic centre of R_{0} = 8178 ± 35 pc by Abuter et al. (2019; see also Do et al. 2019, finding R_{0} = 7946 ± 82 pc), it is now possible to move all the relative measurements, which typically use values between 7.5 and 9.0 kpc, on common grounds, and also to use different methods to coherently determine the vertical height of the Sun above the disc, z_{0}. In most studies intending to determine z_{0}, the counts of specific astrophysical objects, such as stars (e.g. Elias et al. 2006; Jurić et al. 2008; Majaess et al. 2009), globular or open clusters (e.g. Bonatto et al. 2006; Buckner & Froebrich 2014; Joshi et al. 2016) or magnetars (Olausen & Kaspi 2014) and their spatial and kinematic distributions are used. Alternatively, H I (Gum et al. 1960) or H II (e.g. Paladini et al. 2003) regions, interstellar dust (e.g. Mendez & van Altena 1998), or background star light (e.g. Freudenreich 1998) provide similar estimates. A median of 56 measurements during the last century provides an estimate for z_{0} of 17 pc, howver assuming different distances to the Galactic centre. Recent estimates, including measurements of the past ten years, obtain z_{0} values between 5 and 29 pc (Karim & Mamajek 2017), and show a large systematic spread among different as well as similar methods.
Measurements of the vertical position of the Sun above the Galactic plane mostly rely on the knowledge of an absolute reference frame: the IAUaccepted coordinate system from 1960 was defined with an accuracy of ≈0.1° (Blaauw et al. 1960; Gum et al. 1960). This value directly translates into an uncertainty for determining z_{0} in absolute terms. The statistics and precision of current measurements allow for a reduction of this uncertainty to ≲0.03°, that is, to a level where systematic differences become important again (e.g. Peretto & Fuller 2009; Simpson et al. 2012; Karim & Mamajek 2017). In addition, the assumption that the Galactic plane remains flat throughout may also introduce large systematic effects in various methods because it was shown that the Milky Way disc is warped as well (e.g. Skowron et al. 2019, estimating z_{0} = 14.5 ± 3.0 pc including a warp).
In this paper, we present a method for determining z_{0} based on the observed emission morphology of the ^{26}Al γray line at 1808.74 keV. Depending on the Sun’s vertical position, the fullsky appearance, and even more the emission peak, changes as a function of galactic latitude. Using analytic 3D emissivity models, in particular doubly exponential disc geometries, we derive lineofsight integrated morphologies, and fit for z_{0}. In Sect. 2 we explain the general geometry as well as the fitting procedure. Our results are presented in Sect. 3, followed by a study of possible biases in modelling the MeV sky latitudinally in Sect. 4. We conclude in Sect. 5.
2. Modelling the γray sky
2.1. ^{26}Al emission, lineofsight integration, and geometry
The radioactive nucleus ^{26}Al is ejected in winds of massive stars and their corecollapse supernovae (Diehl et al. 2006), and probably to a lesser extent in asymptotic giant branch stars and classical novae (Diehl et al. 2018a), into the interstellar medium. With a characteristic lifetime of τ ≈ 1.04 Myr, the nuclei experience β^{+} decay to an excited state of ^{26}Mg, and almost instantaneously deexcite by the emission of a 1808.74 keV γray photon. At the time of the decay, the ^{26}Al ejecta have been distributed quasihomogeneously around the massive stars in wind and supernovablown H I cavities (Krause et al. 2014). As massive stars are aligned with the spiral arms of the Galaxy, it has been suggested from the kinematics of the ^{26}Al line (Kretschmer et al. 2013), that the global emission is found in a bubblelike structure preceeding the arms (Krause et al. 2015).
Direct imaging with the most modern γray telescopes is not possible, and thus the physical parameters of a system cannot and should not be extracted by fits to an image. Instead, a (parametrised) model of the emission is to be convolved with the imaging response of the telescope, in order to perform maximum likelihood fits directly in the raw, photoncounting, and instrumentspecific data space.
While iterative methods, such as the Richardson–Lucy deconvolution (e.g. Knödlseder et al. 2005) or the maximum entropy method (e.g. Bouchet et al. 2015), have been used to produce images in the γray domain, their interpretation is on shaky grounds because the algorithms can suffer from a lack of objectivity (see, e.g. Allain & Roques 2006). To determine physical parameters, a full forwardmodelling approach, including the dominating instrumental background, the imaging response, and a geometrical or kinematic model should be preferred. With more data, details in the emission morphology are of course revealed. It nevertheless has been shown in many previous studies (e.g. Knödlseder et al. 1996; Oberlack et al. 1996; Diehl et al. 2006; Wang et al. 2009; Kretschmer 2011; Kretschmer et al. 2013; Siegert 2017) that to large extents, a simple threeparameter model can explain the fullsky ^{26}Al emission at 1.8 MeV, a doubly exponential disc:
In Eq. (1), ρ(x, y, z) is the instantaneous photon emissivity in 3D space, with (x, y, z) = (0, 0, 0) resembling the Galactic centre, in units of ph cm^{−3} s^{−1}, ρ_{0} is the normalisation, inherent to the total galactic 1.8 MeV luminosity and the received or measured flux at the point of the observer, R_{e} and z_{e} are the exponential scale radius and scale height, respectively, and is the Galactocentric radius.
In order to obtain an image from the distribution in Eq. (1), we performed lineofsight integrations, Eq. (4), starting from the position of the observer, here the Sun with (x_{0}, y_{0}, z_{0}), outwards. This requires a coordinate transformation,
with s being the sightline in galacticcoordinate direction (l, b), such that
and p_{0} = cosbcosl + cosbsinl. The lineofsight integration thus reads
where F_{los}(l, b) is the flux at longitude and latitude direction, (l, b), in units of ph cm^{−2} s^{−1} sr^{−1}, and which is not analytically solvable. For a fixed distance to the Galactic centre, R_{0}, and each combination of the scale dimension, (R_{e}, z_{e}), F_{los}(l, b) becomes a function of the observer position, z_{0}, only. Geometrically, changing z_{0} is equivalent to changing the coordinate frame latitudinally (see Fig. 1), resulting in an apparent “tilt” of the observed direction. As a result, the former (z_{0} = 0) symmetric emission profile, averaged over all longitudes, is skewed towards the opposite direction. In Fig. 2, the latitudinal emission profiles of the same exponential disc model with (R_{e}, z_{e}) = (3.5, 0.1) are shown for the symmetric case, and an observer position of z_{0} = +50 pc. Clearly, the emission peak is shifted towards negative latitudes if the observer is above the disc, here to about b = −3°. It is important to note that the flux in both examples is the same, but more spread out to higher latitudes in the case of an observer outside the disc. In addition, there is a certain level of “isotropic emission” in each of the cases (horizontal lines), which especially for codedmask telescopes in the soft γray regime is nearly impossible to detect^{1}. A slightly skewed morphology already reduces this isotropic part and generally allows for more precise measurements in the MeV range. The fullsky appearance of the two examples is shown in Fig. 3.
Fig. 1. Sketch illustrating the different lineofsight integration paths for the same apparent latitude angle, b/b′, as determined by the vertical position of the Sun, z_{0}, at fixed radial position, R_{0}. 

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Fig. 2. Longitudeaveraged lineofsight integration of the same exponential disc (R_{e}, h_{e}) = (3.5, 0.1) 3D emissivity model as a function of latitude for vertical positions of 0 (symmetric, orange) and +50 pc (violet). In the symmetric case, the emission peaks at b = 0°, as expected, and already for z_{0} = 50 pc, the peak is shifted 3° to negative latitudes. See text for further details and implications. 

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Fig. 3. Lineofsight integrated emissivities of the same exponential disc (R_{e}, h_{e}) = (3.5, 0.1) model as a function of longitude and latitude for z_{0} = 0 (left) and +50 pc (right). The images have been normalised to the symmetric case and thus contain the same absolute flux. Clearly, the contrast in the right image is higher, but the flux is more spread out into a larger number of pixels. 

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These detailed peculiarities can be used to determine z_{0} from γray emission alone. In the following Sect. 2.2, the basics about the spectrometer on board the International GammaRay Astrophysics Laboratory (INTEGRAL/SPI), the chosen data set, and model fitting in its raw photoncount data space are explained.
2.2. Instrument, data, and fit
The spectrometer SPI (Vedrenne et al. 2003) on board ESA’s INTEGRAL satellite (Winkler et al. 2003) measures photons in the energy range between 20 and 8000 keV with a 19element highpurity Ge detector camera. It uses a codedmask technique to distinguish between emission from the sky and typically high instrumental background radiation from the telescope and satellite themselves. SPI has a field of view of 16° and an angular resolution of 2.7°. The data space is thus a size19 vector of photon counts at different energies for individual pointed observations (“pointings”) of typically 30 min duration. Because we study one of the strongest γray lines, we summed all counts between 1805 and 1813 keV into one 8 keV broad energy bin, corresponding to three times the spectral resolution of SPI at 1.8 MeV. In this study, we do not aim for optimising or characterising the ^{26}Al emission morphology, nor are we interested in the detailed spectral shape of the emission line.
Our data set consists of 200 Ms of INTEGRAL/SPI observation time, distributed predominantly in the centre of the Galaxy as well as the Galactic plane, with a patchy exposure at higher latitudes^{2}. In total, more than 13 years of data are accumulated, resulting in N_{obs} = 92 867 pointings of consecutive size19 vectors. When the imaging response, R_{jp}, is applied for each pixel j ∈ (l, b)_{j} in each pointing p ∈ N_{obs}, to the emission model F_{los}(l, b)_{j}, this translates an image or model of the sky, , into the SPI data space for a particular set of observations:
In Eq. (5), the absolute flux normalisation, F_{norm}, is determined through a maximum likelihood fit, and z_{0} is a free parameter (see below). Understanding the instrumental background in γray astrophysics is key to any parameter inference. Details about robust and reliable background modelling with SPI can be found in Diehl et al. (2018b) and Siegert et al. (2019). Here, the background is modelled as a combination of contributions from the instrumental continuum, , as well as instrumental lines, , and their absolute levels are determined through a (timedependent) scaling parameter in a fit again. The total model thus writes
In this full forwardmodelling manner, the parameters F_{norm}, , and are straightforwardly determined by maximising the Poissonian likelihood function,
because they are linear parameters. However, the convolution of F_{los}(l, b; z_{0})_{j} with R_{jp} is computationally very expensive and would be required in each iteration of the fit to determine the unknown parameter z_{0} because it is changing the appearance of . We therefore fixed it for a particular fit, and evaluated the likelihood for a predefined set of z_{0} values. Here, we used a grid of 15 vertical extents, ranging between −20 and +130 pc in steps of 10 pc. These values are based on a canonical distance of 8.5 kpc to the Galactic centre. Because the problem is geometrically similar, a correction factor R_{0}/8.5 kpc = 0.9621 can be applied to each z_{0} value for a correct estimate given the more precise measurements of R_{0} by Abuter et al. (2019), or 0.9348 from the R_{0} estimate by Do et al. (2019).
3. Results
We used two examples to estimate the vertical position of the Sun from emission of the ^{26}Al decay line. The emission in both cases is characterised by the doubly exponential disc model, Eq. (1), but with two assumptions on the radial and vertical scale parameters: Wang et al. (2009) suggested an exponential scale height at 1.8 MeV of pc at a distance to the Galactic centre of 8.5 kpc. This estimate is based on a fixed scale radius of 3.5 kpc and only considers the sky region l< 60°, b< 30°. Siegert (2017) used the data set described in this work and performed exponential disc fits, including the full sky and varying both scale dimensions. This resulted in an estimate of R_{e} = 5.6 ± 0.6 kpc and z_{e} = 670 ± 190 pc for ^{26}Al. We scaled and reweighted the parameter sets of Wang et al. (2009) and Siegert (2017), resulting in the combinations (R_{e}, z_{e}) = (3.37, 0.15) and (4.81, 0.46), respectively. In a common reference system with R_{0} = 8.178 kpc, we finally performed the abovedescribed maximum likelihood fits. These two examples serve as a proofofconcept study that uses the two extreme values reported in the recent literature.
The probability density functions of z_{0}, given the two models, are shown in Fig. 4. The more concentrated emission model with smallscale dimensions results in a vertical position of the Sun above the Galactic midplane of 7 ± 9 pc. The largerscale dimensions, as suggested from the fullsky model fits, result in z_{0} = 15 ± 17 pc. These values are consistent with each other and also consistent with the median of 56 previous completely independent measurements using different techniques, giving ≈17 pc in a range between 5 and 29 pc (Karim & Mamajek 2017). While the model with a narrower emission morphology provides smaller uncertainties, it was shown that larger scale radii and heights are apparently needed to cover the entire^{3} Milky Way emission at 1.8 MeV (Siegert 2017). We therefore quote the larger uncertainty as a more conservative estimate. A more detailed model to describe the true ^{26}Al morphology in the Galaxy might even now result in a much more accurate estimate of z_{0}. The structural details of the ^{26}Al emission in the Milky Way are beyond the scope of this paper because the goal is to show the feasibility of such a measurement with currently available data in the soft γray regime
Fig. 4. Probability distribution functions for the vertical position of the Sun using INTEGRAL/SPI ^{26}Al data and two different exponential disc models. In this range of plausible models, the position is constrained between in the interval [−2, 32] pc, consistent with independent estimates. 

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4. Vertical emission extents and bias
This remarkable and unexpectedly good result leads to the question whether there might be biases in estimating the spatial extents of MeV γray emission. In particular, setting the vertical position of the observer incorrectly in a typical analysis, for example, intending to measure the exponential scale height, may result in over or underestimating the vertical extent of the emission.
We investigated this potential bias by fitting exponential disc models with z_{0} = 0 to simulations of models with z_{0} ≠ 0, and varying the scale height h_{e}. In this way, we determined an optimal scale height that differs from the input model if the correct position of z_{0} is not met. We chose three different sets of (R_{e}, z_{e}), (7.00, 0.30), (3.50, 0.10), and (5.00, 0.02), and performed a scan of h_{e} for each value of z_{0}. The resulting relative deviations for our three test cases are shown in Fig. 5. If the vertical position z_{0} = 0 is met, the relative deviation from the correct scale height is by definition equal to zero. Because this problem is again symmetric, we only show positive values of z_{0}. The larger the true scale height, the smaller the relative deviation from using an incorrect observer position. In the case of (R_{e}, z_{e}) = (7.00, 0.30), for example, the relative deviation inside the z_{0}position interval determined from γrays is up to 20%. This value is comparable to the uncertainty of the ^{26}Al scale height determined by Siegert (2017) (< 30%) and should therefore be carefully considered when new or updated measurements in the MeV regime are discussed. In the case of smaller true scale heights, the effect of choosing an incorrect observer position is even more severe and ranges up to 50% (125%) when scale heights as small as 100 pc (20 pc) are considered.
Fig. 5. Relative deviations, True − Fitted/True × 100%, of doubly exponential and change throughout for consistency disc models as a function of z_{0} and z_{e} (z_{e} = 300 pc orange, 100 pc violet, and 20 pc pink), fitted by a z_{0} = 0 model, assuming the correct R_{e}. The yellow region determines the 1σ uncertainty from the z_{0} fit from 1.8 MeV data alone, see Sect. 3. By definition, the z_{0} = 0 case yields a relative deviation of zero. 

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5. Conclusion
Using 13 years of ^{26}Al data at 1.8 MeV from INTEGRAL/SPI, we determined the vertical position of the Sun above the Galactic midplane to z_{0} = 15 ± 17 pc in a proofofconcept study. This is a remarkable result, both in terms of accuracy and precision: current soft γray telescopes suffer from the “MeV gap” in sensivity (e.g. De Angelis et al. 2018; Timmes et al. 2019; Tomsick et al. 2019), which most of the time allows them to only study the sources inside, or the Milky Way itself. The statistical uncertainties on z_{0} we estimate from only using INTEGRAL/SPI data and a simple firstorder analytic geometric model are only a factor of a few away from the most precise measurements using star counts or H II regions (17 ± 2 pc), which show a systematic spread of ≈12 pc (Karim & Mamajek 2017), however. A more detailed geometric model of ^{26}Al would already provide similar uncertainties in z_{0} with current instrumentation. In consequence, the accuracy of this measurement can imply biases in determining other emission parameters, such as its vertical extent (latitudinal or exponential scale height). We find that specifically for ^{26}Al, the relative uncertainty of scale heights as a function of z_{0} can be up to 25% (or higher for smaller scale heights than ^{26}Al). This value is comparable to the uncertainties from scaleheight measurements themselves, and may lead to misestimates (in both directions) of the true scale height. Consequently, z_{0} should be carefully considered in future modelling of the MeV γray sky.
We note that Compton telescopes are barely affected by this circumstance because they locate the emission based on the more unique Compton scattering response. See e.g. the Compton Spectrometer and Imager, COSI (Kierans et al. 2016; Tomsick et al. 2019).
See http://www2011.mpe.mpg.de/gamma/instruments/integral/spi/www/public_data/index.html for a current exposure map.
Combinations of small and large scale heights provide an estimate of how much emission comes from the solar vicinity (Siegert 2017), but this is very degenerate in the SPI data space and was thus not considered here as a third case.
Acknowledgments
Thomas Siegert is supported by the German Research Society (DFGForschungsstipendium SI 2502/11).
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All Figures
Fig. 1. Sketch illustrating the different lineofsight integration paths for the same apparent latitude angle, b/b′, as determined by the vertical position of the Sun, z_{0}, at fixed radial position, R_{0}. 

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In the text 
Fig. 2. Longitudeaveraged lineofsight integration of the same exponential disc (R_{e}, h_{e}) = (3.5, 0.1) 3D emissivity model as a function of latitude for vertical positions of 0 (symmetric, orange) and +50 pc (violet). In the symmetric case, the emission peaks at b = 0°, as expected, and already for z_{0} = 50 pc, the peak is shifted 3° to negative latitudes. See text for further details and implications. 

Open with DEXTER  
In the text 
Fig. 3. Lineofsight integrated emissivities of the same exponential disc (R_{e}, h_{e}) = (3.5, 0.1) model as a function of longitude and latitude for z_{0} = 0 (left) and +50 pc (right). The images have been normalised to the symmetric case and thus contain the same absolute flux. Clearly, the contrast in the right image is higher, but the flux is more spread out into a larger number of pixels. 

Open with DEXTER  
In the text 
Fig. 4. Probability distribution functions for the vertical position of the Sun using INTEGRAL/SPI ^{26}Al data and two different exponential disc models. In this range of plausible models, the position is constrained between in the interval [−2, 32] pc, consistent with independent estimates. 

Open with DEXTER  
In the text 
Fig. 5. Relative deviations, True − Fitted/True × 100%, of doubly exponential and change throughout for consistency disc models as a function of z_{0} and z_{e} (z_{e} = 300 pc orange, 100 pc violet, and 20 pc pink), fitted by a z_{0} = 0 model, assuming the correct R_{e}. The yellow region determines the 1σ uncertainty from the z_{0} fit from 1.8 MeV data alone, see Sect. 3. By definition, the z_{0} = 0 case yields a relative deviation of zero. 

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In the text 
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