The Sirius System and Its Astrophysical Puzzles: Hubble Space Telescope and Ground-based Astrometry^∗

Figure 1 illustrates a typical WFPC2 frame, containing a severely overexposed image of the primary star and a well-exposed image of Sirius B lying to the lower left. Our WFC3 frames have a similar appearance.

Zoom In Zoom Out Reset image size

For the astrometric measurements of these frames, we followed an essentially identical procedure to that described in B15 for our analysis of overexposed WFPC2 F1042M images of Procyon; thus, we do not repeat all of the details here but only give a brief outline. In particular, in order to build oversampled representations of the F1042M PSF for their study of Procyon, B15 included and discussed an analysis of all the available WFPC2 data on Sirius. As Figure 1 shows, the diffraction spikes exhibit quasi-periodic variations in intensity as a function of distance from the center. B15 found that this structure varies from epoch to epoch (probably because of a dependence on the exact location of A within the field of view), making it difficult to construct a usable oversampled PSF for the unsaturated outer regions of the images. Instead, we used a procedure of fitting straight lines to the unsaturated portions of the diffraction spikes and defining their intersection point to be the centroid of Sirius A. For the unsaturated images of Sirius B, centroids were determined from a more conventional technique of PSF fitting, again as discussed in detail in B15. In order to calibrate the systematic offset in the centers using the two different techniques, we obtained unsaturated F1042M images of the star 109 Virginis, an A0 V star with a color very similar to that of Sirius, but sufficiently faint that both unsaturated and saturated images could be compared directly. The resulting offsets were applied to the spike intersection points to place them in the same system as the direct centroids of Sirius B.

A similar approach was used for the WFC3 observations of Sirius in F953N, which is described in detail in B15 for the analysis of images of Procyon obtained in the same filter. As for the WFPC2 data, we also analyzed and discussed all of the Sirius data then available with WFC3 as direct support of the Procyon results in B15. Again, we fitted straight lines to the diffraction spikes to determine the centroid of A, and used PSF fitting to find the location of B in each image. The offset between the two methods was determined from the unsaturated and saturated calibration frames we obtained for the A3 V star HD 23886. However, one difference from the WFPC2 frames is that the diffraction spikes in F953N do appear to have a consistent appearance at all epochs. Thus, for the WFC3 frames, we could use an alternative method of developing a deep mean PSF for the outer regions, based on a large number of frames of Sirius, Procyon, and HD 23886. This allowed us to determine the positions of Sirius A using PSF fitting instead of the diffraction-spike intersection point. The positions of Sirius B were again found from a conventional unsaturated PSF-fitting method. The systematic offset correction between the two PSF-based centroids was determined from the unsaturated and saturated calibration observations of HD 23886.

The process of converting the astrometry from the image plane to the absolute J2000 frame is, again, described in B15. It is based on an adopted plate scale and the known orientation on the sky of the image y-axis. Our final HST astrometric results are given in Table 2. For WFC3, we present the results from both methods—the diffraction-spike intersection and the PSF fit—as well as the weighted means of the two. Note that the PAs are referred to the equator of J2000, not to the equator of the observation epoch that is the usual practice for ground-based measurements.

A long-term program of photographic astrometry of the Sirius system was started in 1965 by I.W.L.,¹⁷ using the 26 inch refractor of the USNO in Washington, DC. These observations used a hexagonal aperture mask, which causes intensity to drop off rapidly in certain directions around a bright star, while producing six bright “spikes” at other PAs (e.g., Aitken 1935, p. 60; van Albada 1962). Proper mask orientation allows close companions to be detected that would not otherwise be resolved easily. Along with the hexagonal mask, an objective grating with extremely fine, evenly spaced wires was added. This produced good first- and second-order images of the primary star, which could be measured and used to locate its centroid. Figure 2 shows a digitized version of a typical photographic plate from this series of observations.

Zoom In Zoom Out Reset image size

Trailed exposures were also taken on the same plates to define the east–west direction. Details of the observations and astrometric measurements of the separation and PA on 56 nights between 1965 and 1969 were published by Lindenblad (1970). A later paper (Lindenblad 1973) presented additional measurements between 1969 and 1972. Subsequently, this observing program was continued until 1984, but unfortunately these plates had never been measured or the results published. During this interval, about 160 usable photographic observations were obtained, on 66 different nights.

Astrometry of these plates was carried out by M.S.-M. by digitizing them on the StarScan (Zacharias et al. 2008) machine at USNO. A centroiding algorithm was used for the images of Sirius A and B, with the mean position of the two first-order images of A defining its location. A plate scale of $20\buildrel{\prime\prime}\over{.} 8476\,{\mathrm{mm}}^{-1}$ was adopted.

These measures were corrected for the “Ross effect” (see Lindenblad 1970), whereby the blackened portions of the photographic emulsions dry faster than the non-blackened, causing nearby portions of the emulsion to contract differentially. The correction was determined by using measures of the second-order images. These images are far enough away from the bright primary that they are essentially undisturbed by the Ross effect. In the absence of contraction, the distance between the second-order images would be twice the distance between the first-order images, and the departure from this relation allows calculation of the correction (van Albada 1962, 1971).

Table 3 presents the results of these measurements. The PAs are for the equator of the observation epoch.

We have collected and critically examined all published measurements of the Sirius visual binary of which we are aware (to which we add the new HST and USNO data presented in this paper). These data are discussed in the Appendix, along with excerpts from the associated full electronic versions of the tables.

The first step in our determination of orbital elements was to adjust all of the measurements, both HST and ground based, to a J2000 standard equator and epoch. We used the formulations given by van den Bos (1964) in order to correct for (1) precession (except for the HST measures, which are already in the J2000 frame), (2) the change in direction to north due to proper motion, (3) the changing viewing angle of the three-dimensional orbit due to proper motion, and (4) the steadily decreasing distance of the system due to radial velocity (RV).

Except for precession, these corrections are small relative to the observational uncertainties for the ground-based data, and are also small for the HST data because their epochs are all so close to 2000.0. We verified our coding by showing that it reproduces the values presented for Sirius by van den Bos (1960), if we used his input parameters. For the actual corrections, we used the parallax from Section 6.1 below, the Hipparcos proper motion (van Leeuwen 2007), and for the space-motion correction, we used an initial solution for the orbital elements of the visual binary. The RV of the center of mass of the Sirius system can be determined from the observed RVs of Sirius A and the relative orbit. There have been several such determinations, beginning with Campbell (1905), who obtained $-7.4\,\,\mathrm{km}\,{{\rm{s}}}^{-1}$. Here we use our relative orbit and all available RV data from 1903 to 1995 to determine a system velocity of $-7.70\,\,\mathrm{km}\,{{\rm{s}}}^{-1}$. (Details of the RV determination will be presented in a separate forthcoming spectroscopic paper on Sirius B.) To obtain the true radial component of the space motion, we must correct this for the gravitational redshift (GR) of Sirius A, yielding $-8.47\,\,\mathrm{km}\,{{\rm{s}}}^{-1}$.

In the Appendix, we tabulate both the input observed PAs and separations, and those corrected to J2000.

We determined elements for the visual orbit via a seven-parameter fit to the combined set of J2000-corrected HST and ground-based measurements. This fit employed a ${\chi }^{2}$ minimization procedure, as described in detail for Procyon in B15. For the HST data, we used the WFPC2 and WFC3 measures¹⁸ in Table 2, with the (small) corrections to J2000 applied.

We divided the measurements into three groups, based on the methods of observation: visual micrometer; photographic, CCD, and mid-infrared (MIR); and HST. Then we assigned different uncertainties to each of the three data sets, determined through an iterative procedure based on a comparison with the simultaneous orbit fits, as follows. (1) Few of the micrometer observations contained explicit uncertainty estimates, so we set uniform uncertainties by forcing the reduced ${\chi }_{\nu }^{2}$ to equal unity for this set of measurements. On the assumption that observations taken with a larger telescope are typically more precise and that observing techniques improved over time, we then scaled the uncertainties proportionally to telescope aperture size by computing a two-variable linear fit, and again forced ${\chi }_{\nu }^{2}$ to unity. (2) For the photographic and ground-based CCD+MIR measurements, we applied uniform uncertainties that are isotropic in two dimensions to force the reduced ${\chi }_{\nu }^{2}$ to 1 for this set of measurements. (3) We did not scale the HST uncertainties, but left them at the values given in Table 2.

Following these adjustments to the measurement uncertainties of the first two groups of data, we re-computed the fit to the entire set of ground-based and HST measurements. We used a sigma-clipping algorithm to reject measurements for which the separation and/or PA was discrepant by more than four times the standard deviation of the residuals. We then rescaled the uncertainties according to the methods described above, and repeated the sigma-clipping algorithm until no further measurements were rejected. The final parameters for the visual orbit are listed in Table 4. Uncertainties were computed from the diagonal elements of the covariance matrix. The electronic table and associated text in the Appendix indicate which observations ended up being rejected.

For the HST data, we found a ${\chi }^{2}$ per degree of freedom of ${\chi }_{\nu }^{2}=1.87$. We believe this excess over unity is plausibly due to minor systematic effects not included in our error determinations, such as changes in plate scale due to the occasional adjustments in telescope focus, telescope “breathing” due to thermal effects in orbit, or small errors in the telescope roll angle, or conceivably they could have an astrophysical origin (discussed below in Section 7).

We also experimented with a fit in which we fixed the orbital period based on the entire set of data covering over 150 years, but then determined the remaining parameters using only the high-precision HST data. However, this resulted in the parameter uncertainties actually increasing by factors of about 1.6–4, and a reduction in a by only 0.8 mas, which is small compared to its uncertainty. Therefore, we consider the values in Table 4 to be the best estimates.

In Figure 3 (top panel), we plot all of the measurements, color-coded according to their source (visual micrometer, photography, CCD and MIR, and HST). The solid-line ellipse shows our orbital fit. For clarity, in the bottom panel of Figure 3, we omit the numerous visual measures. The top panel shows that the visual measurements frequently had very large errors, often of an arcsecond or even more. There is a tendency, especially at the smaller separations, for the separation to be systematically overestimated by the visual observers. The bottom panel shows that even the CCD and MIR measures had a bias toward overestimation of the separation. (Note that the semimajor axis had also been systematically overestimated for Procyon in the historical data, as shown by Girard et al. 2000 and B15.)

Zoom In Zoom Out Reset image size

The observational errors for the HST data are too small to be visible in Figure 3, so in Figure 4 we zoom in on a plot of the HST measures only (red dots), along with the calculated positions (open blue circles) based on our orbital solution. But once again the deviations from the orbital fit to the exquisite HST astrometry are too small to be seen.

Zoom In Zoom Out Reset image size

In Figure 5 we plot the residuals of our HST measurements as a function of time relative to the orbital fit, in the directions of right ascension (top panel) and declination (bottom panel). In right ascension, the fit agrees with all of the measures within 1σ error bars, except for a ∼2.5σ residual for a single observation. There appears to be no jump resulting from the change of cameras in 2009. There is a slightly larger scatter in the declination residuals, with two noticeable departures at the beginning and end of 2006 of about 2σ and 3.7σ. Because the PAs on those dates were around 100°, a residual in declination of these sizes could in principle be due to an incorrect telescope roll angle, at a level of about $0\buildrel{\circ}\over{.} 08$. Normally, the HST roll angle is known to about ±$0\buildrel{\circ}\over{.} 003$, but in very exceptional cases¹⁹ the uncertainty can approach $0\buildrel{\circ}\over{.} 1$.

Zoom In Zoom Out Reset image size

In addition to the period, P, and the semimajor axis, a, of the relative orbit listed in Table 4, we need two further quantities in order to determine dynamical masses for both stars: the absolute parallax of the system, π, and the semimajor axis, a_A, of the absolute orbital motion of Sirius A on the sky.

For the parallax, we used two independent determinations: (1) GG78 calculated the parallax from measurements of over 300 photographic plates obtained at the Yerkes and Allegheny Observatories between 1917 and 1977, combined this result with four earlier published determinations, and corrected the result from relative to absolute to yield a parallax of $0\buildrel{\prime\prime}\over{.} 3777\pm 0\buildrel{\prime\prime}\over{.} 0031;$ (2) the absolute parallax was measured by the Hipparcos mission to be $0\buildrel{\prime\prime}\over{.} 37921\pm 0\buildrel{\prime\prime}\over{.} 00158$ (van Leeuwen 2007). These results are in good agreement, and we adopt their weighted mean, as given in the top part of Table 5.

GG78 also determined the semimajor axis of Sirius A’s orbital motion from the Yerkes and Allegheny material, and their result is presented in the bottom section of our Table 5. However, the calculation needs to be repeated using our new determination of the elements of the relative orbit. Fortunately, GG78 tabulated the individual measurements of the photocenter location (their Table 6), allowing us to repeat the determination of a_A; we found that one observation, at 1966.824, was very discrepant, possibly due to typographical error, and it was deleted. Our new adopted value of a_A is given in the final entry in Table 5.

The final column in Table 6 lists the dynamical masses that result from our adopted parameters. For comparison, the second column gives the masses derived by van den Bos (1960) and the third column lists the values from GG78. We used the usual formulae for the total system mass, $M={M}_{A}+{M}_{B}={a}^{3}/({\pi }^{3}\,{P}^{2})$, and for the individual masses, ${M}_{A}=M(1-{a}_{A}/a)$ and ${M}_{B}=M\,{a}_{A}/a;$ in these equations the masses are in ${M}_{\odot }$, a and π in arcseconds, and P in years. Our total system mass, $3.081\,{M}_{\odot }$, is about 3.6% lower than was derived by GG78, and our individual masses, $2.063\,{M}_{\odot }$ and $1.018\,{M}_{\odot }$, are smaller by similar factors.²⁰ These differences are due almost entirely to a 2.5% reduction in our adopted a³, and a 1.0% reduction in ${\pi }^{-3}$. Compared to the earlier study by van den Bos, our total system mass is lower by 3.4%, even though his value of $a=7\buildrel{\prime\prime}\over{.} 50$ is very close to our measurement. In this case, the difference is due mostly to our 3.1% reduction in ${\pi }^{-3}$ relative to the value used by van den Bos.

Table 6.  Dynamical Masses for Sirius System

Quantity	van den Bos (1960)	Gatewood & Gatewood (1978)	This Paper
Total mass, ${M}_{A}+{M}_{B}$	3.20 M⊙	3.196 ± 0.083 M⊙	3.081 ± 0.034 M⊙
Mass of Sirius A, MA	2.15 M⊙	2.143 ± 0.056 M⊙	2.063 ± 0.023 M⊙
Mass of Sirius B, MB	1.05 M⊙	1.053 ± 0.028 M⊙	1.018 ± 0.011 M⊙

Download table as:  ASCIITypeset image

In Table 7 we present the error budgets for our derived masses of Sirius A and B, based on the random uncertainties of each of the parameters.²¹ As the table shows, the mass errors are entirely dominated by the uncertainty in the parallax. Unfortunately, this error is unlikely to be reduced in the near future, because Sirius is far too bright for its parallax to be measured by the current Gaia mission (D. Pourbaix and S. Jordan 2016, private communications).

Table 7.  Error Budgets for Sirius System Dynamical Masses

Quantity	Value	Uncertainty	$\sigma ({M}_{A})$ $({M}_{\odot })$	$\sigma ({M}_{B})$ $({M}_{\odot })$
Absolute parallax, π	0.3789	±0.0014 arcsec	0.023	0.011
Semimajor axis, a	7.4957	±0.0025 arcsec	0.0024	0.0007
Semimajor axis for A, aA	2.4761	±0.0045 arcsec	0.0018	0.0018
Period, P	50.1284	±0.0043 years	0.0004	0.0002
Combined mass uncertainty	⋯	⋯	0.023	0.011

Download table as:  ASCIITypeset image

In Sections 8–9, we discussed the two components of Sirius as if they have evolved independently. But—even though we find consistent ages for Sirius A and B under this assumption—independent single-star evolution may not have been the case. In the present-day orbit, the separation of Sirius A and B ranges from 31.5 au at maximum to 8.1 au at periastron. If the progenitor of B had a mass of ∼5.0–$5.6\,{M}_{\odot }$, as discussed in Section 8, then the total mass of the system was reduced from an original ∼7.1–$7.7\,{M}_{\odot }$ to its present $3.08\,{M}_{\odot }$, due to mass lost from the progenitor of B. Under the assumption that this mass loss was isotropic and on a timescale long compared to the orbital period (cf. Burleigh et al. 2002, Section 2) and ignoring any interactions between the stars, this implies that the periastron separation was only ∼1.5–1.6 au in the progenitor binary. This is smaller than the radius attained in the AGB phase of a ∼5.0–$5.6\,{M}_{\odot }$ star, and thus the two must almost certainly have interacted in the past. However, the binary did not enter into a common-envelope event, and the orbit did not even tidally circularize as might have been expected—it still has an eccentricity of 0.59. Moreover, as discussed in Sections 8–9, Sirius B appears to be a normal WD and Sirius A shows no apparent departure from single-star evolution nor any obvious signs of a past interaction.

Several authors, including BM08, have pointed out that Sirius is by no means unique: there are many detached binary systems in which one component is a WD that clearly interacted with the primary star in the past, yet they still have eccentric orbits. For instance, barium stars, in which processed material from an AGB star (now faded to a WD) is present on the surface of a companion, are often in eccentric orbits (Izzard et al. 2010 and references therein). Another example of a wide evolved binary that avoided circularization is the Procyon system, consisting of an F5 subgiant and a DQZ WD companion in a 40.8 year orbit, with an eccentricity of 0.40 (see B15). BM08 (see also references therein for earlier theoretical considerations) have modeled binary systems in which the orbit is initially significantly eccentric before the more massive component reaches the AGB. Mass loss from the AGB star is enhanced at each periastron passage, producing a growth rate in eccentricity larger than the rate of tidal circularization. In Section 4.1 of BM08, they explore parameter space to see whether the Sirius system itself can be reproduced through this eccentricity-pumping mechanism, and they are able to do so. The amount of material accreted by Sirius A from the progenitor of B in the successful scenarios is about 0.05–$0.1\,{M}_{\odot }$. In a later paper, L11 raised the possibility that as much as ∼$0.5\,{M}_{\odot }$ was accreted by Sirius A. However, a possible objection to significant mass transfer from the companion is that the rotational velocity of Sirius A is small compared to most A-type stars (Section 1), indicating that it avoided being spun up by accretion to a short rotation period.

Since the photospheric composition of A appears to have been modified by diffusive processes, it is difficult to apply strong constraints on accretion scenarios based on chemical abundances at its surface. However, L11, and earlier Richer et al. (2000), suggested that the abundance patterns, in particular deficiencies of C and O, and near-solar N, are at least qualitatively consistent with accretion of CNO-processed material having occurred at the time Sirius B was an AGB star. The enrichments of the heavy elements Sr, Y, Zr, and Ba found by L11 (3–30 times solar), and of Cu and heavier elements found by Cowley et al. (2016; generally 10–100 times solar), may also be indicative of accretion of ${\text{}}s$-process elements. In addition, any mass accretion might have resulted in helium enrichment of the atmosphere, which would increase element abundance ratios relative to hydrogen.

We made an exploratory theoretical study, in which an additional Tycho model was run to assess whether a past accretion event could have produced a detectable effect on the evolution of Sirius A. A star initially of mass $1.96\,{M}_{\odot }$, with a metallicity of $0.9\,{Z}_{\odot }$, was evolved for 100 Myr, the approximate time for a $5.6\,{M}_{\odot }$ companion star to evolve to the thermally pulsing AGB phase. At that point, steady accretion at a rate of ${10}^{-7}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ was implemented for a duration of 1 Myr, yielding a final mass for the star of $2.06\,{M}_{\odot }$. After the accretion was terminated, the model readjusted on a thermal timescale and then converged onto an evolutionary track with a normal shape and rate of evolution. This track had a slightly lower ${T}_{\mathrm{eff}}$ and larger radius at a given luminosity than a non-accreting model starting at a mass of $2.06\,{M}_{\odot }$. This particular scenario results in a poorer fit to the observed stellar parameters of Sirius A than the single-star models discussed in Section 9; however, the unexceptional shape and evolutionary timescale of the post-interaction track makes it difficult to rule out a similar accretion history for Sirius A.

PK12 considered an alternative scenario, in which eccentric binaries with compact components like Sirius are descended from systems that were initially triple. The onset of mass loss as the most massive component evolves could trigger an orbital instability, leading to ejections, interactions, or even physical collisions. In Section 7 of PK12, they consider the specific case of Sirius. PK12 ran theoretical simulations of triple systems, with a third star orbiting an inner binary having initially a small orbital eccentricity. They were able to reproduce the current Sirius system in cases where the third star was eventually ejected, and the inner binary pumped to high eccentricity. This scenario is, in principle, testable, since it predicts existence of an ejected third star as a very wide common-proper-motion companion.

Over a century ago, Hertzsprung (1909) pointed out that Sirius appears to share the space motions of the bright A-type “Dipper” stars in Ursa Major. A substantial literature has developed since then in which many authors have assigned Sirius to membership in this Ursa Major Group (UMaG). A detailed investigation of the UMaG, and a summary of earlier work, is given in a classical paper by Roman (1949). Eggen (1960) stated that Sirius is known to be a member of the UMaG, indeed calling it the “Sirius Group,” or later (Eggen 1992) the “Sirius Supercluster.” Soderblom & Mayor (1993) reviewed the literature on moving groups up to 1993, and considered Sirius a “probable” member of the UMaG.

However, King et al. (2003, hereafter K03) made an updated examination of stellar memberships in the UMaG, based on a large collection of new data, including precise new parallaxes from the Hipparcos mission. This led to a “clean” sample of definite UMaG members and demotion of Sirius to an “uncertain” membership category. In Figure 9 we plot the observational H–R diagram (M_V versus B − V) for the “certain” UMaG members, using the data in K03 (their Table 5) for stars with a membership class of “Y” (i.e., “yes,” indicating definite membership). Also plotted are the data for Sirius A from the same table. K03 derived an age of the UMaG of 500 ± 100 Myr based on Y² isochrones. We essentially verified this result by using the Y² isochrones database, together with the accompanying interpolation tool²⁷ (Demarque et al. 2004), to find a solar-composition isochrone that provides a reasonable fit to the UMaG data; this is the 550 Myr isochrone plotted in Figure 9. The figure shows that Sirius A lies well to the left and below the main-sequence turnoff of the UMaG, in accordance with the younger age implied by our discussion in Sections 8–9. We calculated another Y² isochrone, for an age of 220 Myr and metallicity $[\mathrm{Fe}/{\rm{H}}]=-0.07$, consistent with our findings in Section 9. This isochrone passes directly through the Sirius point, as shown in Figure 9. We conclude that either Sirius is too young to be a member of the UMaG, or that if it is a member, it is conceivably a blue straggler or other exotic object.

Zoom In Zoom Out Reset image size

K03 quote a metallicity of $[\mathrm{Fe}/{\rm{H}}]=-0.09$ for the UMaG, and in a more recent paper, Ammler-von Eiff & Guenther (2009) state that UMaG members have [Fe/H] lying in the range −0.14 to +0.06. The slightly subsolar bulk metallicity of Sirius A that we deduced in Section 9 is reasonably consistent with these values, so it is difficult to exclude Sirius from the moving group solely on the basis of its composition.

In Section 8, we adopted two versions of the IFMR from a recent paper by C16, in order to obtain the initial mass of the progenitor of Sirius B, from which we determined its evolutionary timescale. Here we consider the reverse approach, in which we constrain the total age of Sirius B to be equal to that found in Section 9 for the age of Sirius A. Then we take the difference between that age and the cooling age of the WD to be equal to the pre-WD evolutionary timescale for the progenitor. Using our evolutionary codes, we finally infer the progenitor’s mass that yields this timescale.

The ages of Sirius A from YREC and the Tycho code are 237 and 247 Myr, respectively (Section 9). Taking the YREC age and subtracting the 126 Myr cooling age of Sirius B (Section 8), we find a pre-WD evolutionary timescale for the progenitor of about 111 Myr. YREC gives this timescale for a progenitor mass of $\sim 4.8\,{M}_{\odot }$.

We also ran Sirius B progenitor models using the Tycho code for masses of 4.85, 5, 5.1, 5.6, and $6.0\,{M}_{\odot }$, at a metallicity of $0.9\,{Z}_{\odot }$. These simulations were terminated during the thermally pulsing AGB (TP-AGB), at which point computational times become long. A conservative estimate is that continuation of the evolution all the way through removal of the H envelope would increase the measured ages by only ∼2%. Including a 2% adjustment for the end of the TP-AGB, the lifetimes of these models, starting from the ZAMS, are 145, 132, 112, 100, and 81 Myr, respectively. The age of Sirius A, based on the Tycho code, is 247 Myr. Subtracting the 126 Myr WD cooling age yields a progenitor pre-WD timescale of 121 Myr, corresponding to a progenitor mass of $5.06\,{M}_{\odot }$.

There is a significant discrepancy in stellar lifetimes in this mass range between Tycho and YREC. It arises from the treatment of convection. The additional turbulent entrainment beyond the thermodynamic boundary of the convective core that is a feature of the 321D algorithm in Tycho is predicted to increase with stellar mass, resulting in longer main-sequence lifetimes and larger He cores. The final mass of the WD in Tycho is also larger than in YREC for a given progenitor mass, due to the larger He core and the larger extent of convection during He burning. Thus the relatively small differences in the $2.06\,{M}_{\odot }$ Sirius A models become much more marked in a $5\,{M}_{\odot }$ model.

We note that the exact final WD mass predicted by the Tycho code depends on details of the TP-AGB evolution, but the models allow us to constrain the progenitor mass needed to yield a $1.018\,{M}_{\odot }$ WD at this metallicity to about $5.0\pm 0.1\,{M}_{\odot }$, in agreement with the lifetime argument above.

Our result (${M}_{\mathrm{initial}}=4.8$–$5.06\,{M}_{\odot }$ and ${M}_{\mathrm{final}}=1.018\,{M}_{\odot }$) lies well within the observational scatter for WDs in open clusters shown in the IFMR of Figure 8 in C16. Our results are also consistent with the earlier study of L05, who inferred a Sirius A age of 225–250 Myr and an initial mass for Sirius B of ${5.06}_{-0.28}^{+0.37}\,{M}_{\odot }$. Thus, again, we see no direct evidence for a departure from normal single-star evolution.

We have assembled what we believe to be a complete compilation of all published historical measurements of the PA and angular separation of Sirius B relative to the primary star. Our tabulation is based on a critical review of measures contained in the Washington Double Star Catalog (WDS)²⁸ maintained at the USNO and from our additional literature searches.

The complete tabulation is presented in the electronic version of this paper and will also be available at the VizieR website.²⁹ Notes at the end of the tabulation give extensive commentary on the historical observations. Table 8 in the present paper shows an excerpt from the full table, with some of the columns omitted for clarity. The table presents the date of observation (Besselian year), measured PA and separation, PA and separation corrected to J2000 (as described in Section 5.1) and their adopted uncertainties, an observer code, a code for method of observation, telescope aperture, and a code for notes and remarks. A second tabulation contains the bibliographic codes (BibCodes) and full literature references corresponding to the observer codes; an excerpt from this tabulation is given in Table 9.

The visual micrometer observations did not always include a contemporaneous measurement of both PA and separation. In Table 8, these omissions are listed with a value of −99.0 and the associated errors are set to zero. The adopted uncertainties were determined as described in Section 5.2, but are listed with a value of 0.0 for measurements that were rejected from the orbit fit.

Many early publications provided measures averaged over multiple nights or even an entire observing season for the purpose of reducing computational labor in subsequent analyses. With modern computers, there is no need for such averaging, so we opted to present the individual measures whenever available. However, if an observer reported more than one measurement on a given night, we did compute the mean position for that night. If the original publication only reported a mean across several nights, we tabulate that mean as reported.

Some early results were reported in more than one publication. We identified these by listing additional reference codes in the electronic version of the table. There were several cases where the results were slightly different from one publication to another; we identify these instances in the notes column and provide an explanation for selecting the values listed in the table. Here we discuss one specific case in more detail: Struve (1893) attempted to correct his measurements for systematic errors by measuring artificial double stars, and he published both the original and corrected measures for the companion of Sirius. Aitken (1935, p. 61) cautions that the variances of the PAs and separations compared with those of other observers are often larger for Struve’s corrected values than for the originals. However, in the case of Struve’s measurements of Sirius, we in fact found that the mean residuals in the PAs did improve when we used Struve’s corrected values, while the separation residuals did not change significantly; we therefore chose to use Struve’s corrected values.

Another set of measurements that we discuss in more detail are the USNO photographic observations by Lindenblad (1970, 1973). He published data for the raw individual measures, but only corrected the seasonal means for emulsion contraction (see Section 4.1). The average time span covered by his mean measures is ∼70 nights. We chose to apply Lindenblad’s tabulated scale factors in order to compute corrected separations and PAs for the individual measurements. We then averaged the measures for plates taken on the same night, reducing the number of individual measurements from 157 down to 77 observations on unique dates.

The electronic table contains a total of 2354 measurements. Of these, two observations by G. M. Searle at 1866.93 and 1867.03 (1882, Harvard Annals, 13, p. 36) were rejected because his notes indicate that they do not refer to Sirius B. Additionally, the two HST measurements by Schroeder et al. (2000) were replaced by our reanalysis of the same images, as described in Section 3. The table also includes eight attempted observations wherein the binary was unresolved, or where no formal measurement of the binary position was reported, including the initial discovery by the Clarks.

A total of 2350 measurements remain for inclusion in our initial orbital fit, contributed by 135 distinct observers. There are 1915 visual micrometer observations, 407 photographic observations and only 28 using “modern” techniques. Remarkably, aside from the HST observations reported here, astrometry of Sirius B has been almost entirely neglected by professional astronomers for more than the past three decades, subsequent to a final 1986 photographic observation reported by Jasinta & Hidayat (1999). The only exception of which we are aware is a single measurement in 2005 derived from MIR observations with the Gemini South telescope (Skemer & Close 2011). Ten measurements by two amateur astronomers based on CCD frames obtained with small telescopes between 2008 and 2016 have been reported, their most recent publications being Anton (2014) and Daley (2016).

In our orbital solution described in Section 5.2, we used a sigma-clipping algorithm to reject badly discrepant measurements. We rejected a total of 67 observations: 59 micrometer, five photographic, and three amateur CCD measurements. The rejected observations are identified in the Notes column of Table 8. As mentioned in the notes to the electronic version, many of these observations were of dubious quality to begin with.