The Cepheid Distance Scale: A Novel Method for Obtaining Mean Magnitudes from Single-epoch Observations

We present a novel technique for mapping single-phase observations of Cepheids in any given band into their time-averaged values, using strong priors on the known interrelations of the multiwavelength widths of Cepheid period–luminosity (PL) relations, combined with the physical ordering of individual Cepheids within and across the instability strip, as a function of temperature (or radius). The method is empirically calibrated and tested using high-precision published multiwavelength observations of Cepheids in the LMC. The example, given herein, takes a single-epoch B-band PL relation and transforms those random-phase observations to within ±0.05–0.06 mag of their time-averaged values. For high-precision single-phase data points, this method can transform single-phase magnitudes into mean magnitudes (without additional observations), bringing the statistical error budget for the PL relation at that wavelength down to the systematic floor. This technique is of particular importance for use with space-based facilities (e.g., Hubble Space Telescope or JWST) where limits on the availability of telescope time preclude dense phase coverage, often resulting in only single-epoch observations being available.

1. Cepheid Period-Luminosity (Leavitt Law) Relations: A Brief History Classical Cepheids and their period-luminosity relations have been known and used for at least a century.In that time we have learned a great deal about the stars themselves, in terms of their characteristic properties defined both by their limited range of periods, and by the shapes and amplitudes of their light (and color) curves, etc.We have also learned about the interrelations between their periods, time-averaged magnitudes, and colors, defining the slopes and intrinsic (but finite) widths of their period-luminosity (PL), period-color (PC), and period-luminosity-color (PLC) relations.We have used individual Cepheids to map the internal structure and topology of the Cepheid instability strip.As a result of this progress, we know the multiparameter, multiwavelength covariance matrix of the time-averaged magnitudes and colors of classical Cepheids.Some recent studies using this multidimensional matrix can be found in Freedman & Madore (2017) and Madore et al. (2017).
The knowledge required to build this matrix has taken considerable time to achieve, driven in large part by the detector technology available.Exploratory observations were first acquired in the blue and visual (using photographic plates, single-channel photometers, and eventually pixel array detectors), followed by near-infrared (NIR) data, both gathered from the ground, and most recently capped with near-and midinfrared observations made from space.

Dealing with Sparsely Sampled Light Curves
Dealing with sparse data for the study of Cepheid variables is not new.Telescope time is expensive.Although decades of photographic observations of Cepheids in Local Group galaxies had been obtained at Mount Wilson and Palomar by Edwin Hubble, Walter Baade, Henrietta Swope, and Allan Sandage through the 1970s, this type of telescope access was not generally available subsequently.Thus it has been necessary to develop new techniques to make optimal use of more sparsely sampled data.Using the Canada-France-Hawaii telescope (CFHT), Freedman (1986Freedman ( , 1988) ) began a program to observe known Cepheids in nearby Local Group galaxies using a newly available CCD camera equipped with broadband filters that included the conventional B and V wavelengths, but also redder R and I bands.With their greater sensitivity, wavelength coverage, linearity, and panoramic imaging, CCD observations opened up a new dimension of exploration into the use of Cepheids in their application to the extragalactic distance scale, particularly the ability to correct for the effects of interstellar dust.At the same time single-channel, NIR detectors were being made available on large telescopes, and the first observations of extragalactic Cepheids were being undertaken by groups at the University of Toronto (McGonegal et al. 1982;McAlary et al. 1983McAlary et al. , 1984) ) and later at SAAO in South Africa (Laney & Stobie 1986).
Although it was known (Wisńiewski & Johnson 1968), but not widely known,4 that individual Cepheids had progressively smaller amplitudes as one observed them further into the red, it still came as a surprise that the trend carried over to the PL relations themselves.Pioneering efforts by South African astronomers (Martin et al. 1979) (1988) also showed that the very sparsely sampled I-band PL relations were competitive, in their overall scatter, with preexisting (photographic) B and V time-averaged PL relations. 5This realization that templates were not only useful summaries of large numbers of observations (including those obtained in the past), but vital tools for obtaining time-averaged magnitudes, no matter how few data points were available.
Interestingly, the Fourier decomposition of a Cepheid light curve was first undertaken more than a century ago by Whittaker & Martin (1911) for RW Cas, and then decades later applied in a massive way by Cecilia Payne-Gaposchkin (1947).She calculated (in up to six different bandpasses) the first five Fourier harmonics for more than 50 galactic and extragalactic Cepheid and RR Lyrae variables (in the Milky Way, the Magellanic Clouds, and in ω Cen, respectively), in an initial attempt to better understand the systematics of the "Hertzsprung Progression."A quarter of a century later, Schaltenbrand & Tammann (1971) reintroduced Fourier component analysis as a computer-assisted tool to extract light-curve parameters for vast numbers of data points in a variety of photometric systems (323 Cepheids, 12,000 photoelectric observations all brought onto the standard UBV system, with 40 light-curve parameters uniformly extracted, with errors included).Thereafter, the production of templates became a  Freedman 1988).The plot illustrates how the difference between an instantaneous (random-phase) B-band magnitude and the mean magnitude for that Cepheid can be scaled to predict the correction needed to take the Rand I-band instantaneous magnitudes, to their respective individual mean magnitudes, so long as correctly scaled and appropriately shifted templates were available for all wavelengths being considered.
standard technique, with Simon & Lee (1981) reintroducing the methodology another decade later, followed by a new resurgence from Ngeow et al. (2003), Tanvir et al. (2005), Yoachim et al. (2009), Pejcha & Kochanek (2012), Inno et al. (2015) mostly in the optical, and recently by Chown et al. (2021) in the mid-infrared.For comparison with the Schaltenbrand and Tammann study, noted above, Pejcha and Kochanek reduced 177,000 photometric observations in 29 bands.Prior knowledge, distilled down into low-dimensionality fits have proven to be desirable and useful in many theoretical and observational applications of Cepheids to the understanding of the physics of these objects and their application to the galactic and extragalactic distance scale, allowing the interrelationship of various properties (across wavelengths) to be discovered, understood and applied (see Section 7: Interpretation in Payne-Gaposchkin 1947, and also Pejcha & Kochanek 2012 and references therein).

Relative Placement of Individual Cepheids across the Instability Strip
In general terms, we have learned from history and these broadband photometric surveys that individual Cepheids have the various shapes and amplitudes of their light curves that they do because of the underlying variations of their radii (i.e., surface area) and effective (surface) temperatures as a function of phase/time.Likewise we know that the PL relations have their range of periods, and their scatter/width in magnitude at fixed period, driven by the run of mean radius and mean surface brightness/temperature, both of which trend with period (see Madore & Freedman 1991 for additional details).
Simply stated, Cepheids generically lie within a twodimensional plane obeying a PLC relation, which is then externally constrained and bounded by additional physics (controlled primarily by temperature).As relatively high-mass stars evolve from the main sequence in color/temperature from the blue, they enter into the Cepheid instability strip and begin pulsating, and they then terminate their variability as they evolve further to the red, crossing out of the instability strip.To first order all of their properties, while they are identifiably inside of the Cepheid instability strip, are determined by the PLC, but bounded by independent physical constraints, defining the limits of the instability strip.
Consider a star entering the Cepheid instability strip for the first time.It will begin pulsating and be identifiable as a Cepheid sitting on the blue extreme of the instability strip.Marginalizing the three-dimensional PLC over color, the twodimensional PL relation will also show that same blue Cepheid to be the brightest Cepheid with that period; and, in the marginalized PC relation it will, of course, still be the bluest Cepheid at that same period.But the main point is that these statements will be true regardless of the bandpass used to observe that Cepheid.
In the course of its evolution, the star then traverses the instability strip at a fairly constant rate and at a roughly constant luminosity, and in doing so it will cross lines of increasing radii, changing its mean density, and thereby systematically changing (in this case increasing) its period.Evolution continues until the supergiant encounters the red edge of the instability strip, at which point it will be the reddest and now faintest example of a Cepheid having that period; a fact, again, that is independent of the wavelength at which that particular Cepheid is being observed.To generalize this, one can confidently say that the relative ordering of Cepheids across the PLC instability strip, (or within the corresponding marginalized PL and PC relations) is invariant and specifically, independent of wavelength.In mapping from the theoretical instability strip, as described by physical luminosities, temperatures, and radii, to their empirical counterparts of magnitudes, colors, and periods, the fact remains that their relative ordering inside these projections is preserved. 6hat does change, in going to observed quantities such as BVI or JHK magnitudes, is the sensitivity and scaling of those bandpasses due to color/temperature transformations, in particular.For example, it is now well known that the magnitude width of the optical B-band PL relation is significantly wider than the width of the NIR K-band PL relation, etc. (e.g., see Figure 3 in Madore & Freedman 1991, and more recently Figure 4 in Freedman & Madore 2024).This is primarily due to the lower sensitivity of the surface brightness (of any star, not just a Cepheid) to changes in temperature as a function of the wavelength at which the measurement is being made.And that systematically declining sensitivity with wavelength leads to the monotonically decreasing widths of the Cepheid PL relations with wavelength, as dramatically shown in Figure 2.
However, attention needs to be drawn to the fact that even a casual glance at Figure 2 also reveals something deeper than just the recognition of a decrease in width (or alternatively described as a decrease in scatter).As alluded to in the previous section, we can now see empirical evidence for the tight correspondence between the internal placement of each Cepheid within those relations, from band to band, from wavelength to wavelength.They are the same Cepheids, each with their unique periods, temperatures/colors, and radii/mean densities.It therefore should be expected that they would be distributed identically inside of each of the PL or PC relations.Seeing this tight correspondence, built with high-precision data, collected over a vast range of wavelengths (as exemplified by Figures 4, 5, 8, and 9) should not come as a surprise, but it is nevertheless, still impressive.Indeed, the tight correlation of residuals was first noted by Sandage & Tammann (1968) who plotted B and V magnitude residuals versus (B-V ) color residuals (their Figure 3) to demonstrate the physical (color/ temperature) basis of the finite widths of the respective PL relations.
Viewed from another perspective, there is a great deal of redundant information in these plots.Or, stated another way, there are now strong known priors, which are quantified and understood.Given what we know, it would be very disturbing to find Cepheids randomly switching their relative positions in their PL relations as a function of wavelength, so long as precision observations are used to construct these plots.

Relative Light-curve Variations as a Function of Wavelength
The self-similarity of the PL relations across wavelengths has a direct analog when we turn from the integrated properties (specifically mean magnitudes) to the underlying properties of individual Cepheid light curves themselves (shapes, amplitudes, phases, etc.) as a function of wavelength.Figure 3 shows two representative Cepheid multiwavelength light curves,7 inspired by the atlas of such multiwavelength light curves published over 55 yr ago by Wisńiewski & Johnson (1968).Quantitatively and qualitatively, the changes as a function of wavelength are both systematic and dramatic.Quantitatively, the U-band amplitudes can be in excess of 2 mag; while at the other end of the wavelength range covered here, the amplitudes, for the same star, in the K and L bands, for example, can be seen to have dropped by a factor of 10, to less than 0.2 mag, peak to peak.Qualitatively, the shape of a classical Cepheid light-curve transitions from an asymmetric waveform in the blue and ultraviolet: rapidly rising from minimum light to a sharply defined maximum, and then more slowly declining linearly back to a more broadly defined minimum.In the red, the light curve is symmetric, having a shape that, on close examination, is more cycloidal than simply sinusoidal.Moreover, the increasingly longer-wavelength light curves have their maxima asymptotically shifting to a later and later phase.
From a statistical perspective, we now know the multiparameter, multiwavelength covariance matrix of the light and color curves of individual classical Cepheids.With this vast store of information comes a great opportunity to optimize future observations, and retrospectively, to leverage sparsely sampled, existing data sets.
What follows is a logical consequence of the above arguments demanding, as well as demonstrating, multiwavelength self-consistency.The utility of this knowledge becomes clear in a case (as for Hubble Space Telescope (HST) or JWST observations, for example) where only limited numbers of observations are available due to time allocation constraints.It provides a means of significantly improving the precision of sparsely sampled data.

Quantitative Assessment
4.1.V-band Data Predicting the B-band Random-phase, Amplitude Corrections We first demonstrate this method by reducing the scatter in the single-epoch B-band PL relation, using the time-averaged V-band data to predict the necessary corrections.Figure 4 shows the tightly correlated time-averaged B and V positions of the Cepheids across the instability strip.Figure 5 shows the photometric scatter, amounting to ±0.043 mag around that correlation.
A quick estimate of the impact of constructing PL relations, based on single-epoch data, can be had by realizing that the amplitude of an average Cepheid at any given wavelength (see Figure 3) is comparable to (and for longer-period Cepheids, often slightly larger than) the magnitude width of the PL relation at that same wavelength (see Figure 2).That is, the  et al. 2011), to the near-infrared (JHK; Persson et al. 2004) and finally the optical (BVI from the "Three-hundred MilliMeter Telescope" (TMMT), which has been operating remotely in a queue-scheduled mode at the Las Campanas Observatory in Chile for the past 10 years, gathering BVI photometric data on nearby supernovae as well as Classical Cepheids in the Milky Way, LMC and SMC; A. Monson et al. 2024, in preparation).Individual PL ridge lines are shown as dashed lines, flanked on either side by the blue/bright and red/faint limits of the instability strip shown by ±2σ solid lines.The monotonic increase in the widths of the PL relations, top to bottom, long wavelength to short, can be readily seen.A closer look at the deviations of individual Cepheids with respect to the ridge line and with respect to one another in the B-band PL relation, for example, are repeated in detail and at high fidelity; however, simply in a progressively more compressed way in every one of the other PL relations, shrinking from blue to red, as should be the case.(Note that not all of the Cepheids plotted were observed in all of the bands shown.)temperature width of the instability strip (which, in large measure, drives the luminosity width of the PL relation) is closely matched by the temperature amplitudes of the Cepheids that populate it, which, in a similar way drive a large fraction of the luminosity amplitude of those Cepheids in the optical.If both of those quantities (amplitudes and full widths) are Figure 3.The change of amplitudes, light-curve shapes and the relative phases of maximum light for two typical Milky Way Cepheids, X Cyg, and S Nor, moving top to bottom, from the infrared to the ultraviolet.Notice that the light-curve asymmetry, prominent in the visual, blue, and ultraviolet (UBV), damps down into a more symmetric, cycloidal variation in the near-and mid-infrared (JHK and 3.6).As the amplitude drops, the peak magnitude moves progressively to later phases.Template matching must take all of these wavelength-dependent variations into account when using light curves at one wavelength to predict light curves at another.Inspired by Wisńiewski & Johnson (1968), the above plots were constructed from individual Cepheid photometry files compiled by one of us (B.F.M.) over the last 50 yr.Individual photometry files, including references to the original publications, can be obtained, upon request, from the first author.2).This example shows the detailed scaling of B-band residuals as a function of V-band residuals.The slope is well defined and the scatter is very low.Here the slope is DB/DV = 1.37 mag/mag, and the residual scatter around the fit is only ±0.043 mag.The slope defines the expansion/ compression factor in using either one of these PL relations to predict the width of the other and the relative placement of points therein.converted to equivalent sigmas (i.e., σ PL(B) and σ Ampl(B) ), they too will be approximately equal.Accordingly, the resulting scatter in the single-epoch PL relation (σ PL(B1) ) will, to first order, be the quadratic sum of the two sigmas, , or approximately 1.41 × σ PL(B) .Backing out the amplitude smearing will therefore decrease the statistical uncertainty on the apparent distance modulus from by about 30% (i.e., (1.4-1.0)/1.4= 0.29).This same correction will hold true for all wavelengths.
For the case at hand, the measured B-band scatter for the time-averaged data is σ B = ± 0.38 mag.However, the scatter in the single-epoch B-band PL relation is measurably larger, σ PL(B1) = ± 0.58 mag.And so in this case, the decrease in scatter, in taking out the random-phase noise, is 34% (i.e., (0.58-0.38)/0.58= 0.34).
A further consistency check can be made by testing if the individually predicted amplitude corrections are consistent with them having been randomly sampled from a Cepheid with a typical amplitude at that wavelength.The amplitude corrections applied in this example have a peak-to-peak range of ΔB = 1.88 mag (and an equivalent σ = ± 0.54 mag),8 which is slightly smaller than the value of ± 0.58 mag determined above.B-band amplitudes for long-period Cepheids in the Milky Way are shown in Figure 6; for the period range considered here the typical amplitude is about 1.6 mag (or an equivalent sigma of ±0.46 mag).It would appear that our small sample of long-period LMC Cepheids have B-band amplitudes that are about 20% larger than average.
The ultimate test of this method, however, is found in the comparison of the single-phase-corrected mean B magnitudes with the known time-averaged B magnitudes.These are shown in the lower two PL relations in Figure 7 (blue points), which should be contrasted with the more highly dispersed, single-epoch data in the next highest PL plot (red points).The bottom plot shows the random-phase-corrected data (blue points with black vertical tails terminating at the random-phase magnitudes) overplotted by the time-averaged data (open circles).Blue lines show PL fits to the amplitude-corrected data; red lines show the fits to the original single-phase data.The correspondence between the predictions and "ground truth" is reassuring.Quantitatively, the scatter found between the predictions and the known mean magnitudes is found, for this sample, to be ±0.057mag.
To put this decrease of scatter in perspective, for a Cepheid having a B-band amplitude of 1.88 mag (or an equivalent sigma of 0.54 mag, as given above), it would require N ∼100 additional observations to bring the sigma on the time-averaged B magnitude down a value of ±0.057 mag, as quoted above (i.e., 0.057 = 0.54/SQRT(N) for N = 98).

I-band Data Predicting the B-band Random-phase, Amplitude Corrections
We now apply the same methodology for reducing the scatter in the single-epoch B-band PL relation, but using the time-averaged I-band data to predict the necessary corrections.Figure 8 shows the more steeply sloped plot of the input correlation of the relative positions of the Cepheids across the instability strip, while Figure 9 shows the photometric scatter, amounting to ± 0.061 mag around that relation.
Figure 10 shows the resulting amplitude-corrected B-band PL relation at the bottom of the panel, and the time-averaged I-band PL relation used to generate the corrections, at the top of the panel.The only difference between this figure and Figure 7 is the decreased precision, as given in Figures 5 and 9.
However, it now needs to be stressed that the two examples given above are what might be called "worst case scenarios," in the following sense: using long-wavelength (red) data, where the and the widths of the PL relations are smallest, to predict amplitude corrections for shorter-wavelength (blue) PL relations will result in inflating the size of the photometric errors on the predicted amplitude corrections for the singleepoch data.Fortunately, the opposite effect is expected in the application of this method to available JWST data.In this case, using short-wavelength (blue) data to predict long-wavelength (red) corrections, works to our advantage by compressing the statistical/photometric errors in the short-wavelength templates, given that those amplitudes and attendant errors are both scaled down by the same factor before those corrections are applied.

From Random-phase to Time-averaged Magnitudes: A
Step-by-step Procedure Referencing Figures 7 and 10, we now lay out, in some detail, the simple steps used to map a single-phase Cepheid observation made in one band, into its time-averaged value, making use of fully time-averaged PL relations in one or more other bands.
Step 1.For each of the i = 1, m bands in which there are time-averaged PL relations, take the j = 1, n Cepheids, having observed periods, P( j), and time-averaged magnitudes, M(i, j), and subtract the best-fit mean PL relations from each of the data points, for each of the wavelengths, yielding (zero-mean) residuals R(i, j).
Step 2. Take the R(i, j) residuals in each of the distinct bands and divide them by the known total width9 of the intrinsic PL relation in that band, W(i).This will assign a normalized rank ordering (in each filter) to each of the Cepheids according to their normalized, relative-luminosity positions across the instability strip, i.e., RN(i, j) = R(i, j)/W(i).
Step 3. Average these normalized rankings, across all available bands with time-averaged data, to obtain a more Figure 8. Tight correlation of interband period-luminosity residuals (derived from Figure 2).This example shows the detailed scaling of B-band residuals as a function of I-band residuals.The slope is well defined and the scatter is very low.Here the slope is DB/DI = 1.89 mag/mag, and the residual scatter around the fit is only ± 0.061 mag.The slope defines the expansion/compression factor in using either one of these PL relations to predict the width of the other and the relative placement of points therein.robust wavelength-averaged estimate of the relative position of each Cepheid within and across the normalized instability strip.This, now, is the wavelength-neutral (and cross-band averaged) template ranking, ( ) ( ) = á ñ TR j RN i, j , which can be used to determine the corrections to the random-phase data, needed at any other wavelength, in order to bring each Cepheid back to its time-averaged magnitude.
Step 4. Now consider the single-epoch/random-phase data at our target wavelength observed in band m + 1. Fit the known slope of the PL relation at this wavelength to these M(m+1,j) data, and flatten these observations by subtracting the mean PL fit from each of the data points.This yields the trend-corrected residuals, R(m+1,j), in the target band m + 1.
Step 5. Take the normalized (unitless) template rankings TR( j) and multiply them by the known scaling factor, W(m+1) appropriate to the wavelength of the random-phase data.This product is our predicted (time-averaged) ranking (in magnitudes) PR(m+1, j) for each star across the instability strip in band m + 1.
Step 6.Take the predicted rankings (PR), and subtract them from the instantaneous magnitude residuals (M), to obtain the predicted random-phase-induced deviations, DN(m+1, j) = M (m+1, j)-PR(m+1, j).The DN represent the amplitude-induced deviations that individually scattered the data away from their time-averaged mean magnitudes.
Step 7. Reintroduce the slope and zero-point originally subtracted in Step 1, and solve for the new apparent zero-point, and the new scatter (both will change).This results in the phase-corrected PL relation in band m + 1.

Concluding Remarks
Noting the extremely tight correspondence between the relative widths of PL relations from one band to another, and the detailed relative placement of individual Cepheids within those PL relations (e.g., Figure 4), it is possible to use prior information obtained from high-precision data acquired at one or more wavelengths to predict the amplitude corrections needed to return the observed (single-phase) data point back to its time-averaged value.By analyzing very high signal-to-noise multiwavelength data for Cepheids in the LMC we have demonstrated the steps needed to be taken to recover time-averaged magnitudes at a given wavelength, where only single observations are available.The method as currently proposed returns amplitude corrections that, when applied, return the true, time-averaged magnitudes to within ± 0.05-0.06mag per star, and reproduces the global intrinsic dispersion within the PL relation, reducing the original, single-epoch-smeared dispersion down by 30%.This brings the Cepheid PL relations, based upon adequate numbers of Cepheids, down to a level of precision that is at, or below, the current zeropoint accuracy of Cepheid distance scale.With high-precision individual observations, we will have hit "the systematic floor," after which further progress will only come from addressing issues of accuracy (i.e., systematics), not sample size.
The methodology outlined in this paper is expected to be particularly useful when applied to single-epoch Cepheid data now being obtained with JWST.Virtually all of the Cepheids being used to calibrate the Hubble constant have multiepoch short-wavelength data, obtained for the discovery of variables and for their period determinations.The resulting PL relations are precisely what are needed here for predicting the random-phase amplitude corrections to the single-epoch data at the longer wavelengths being observed with NIRCam, for example.Applications and the necessary scaling relations between JWST and HST filters will be made available in forthcoming Chicago Carnegie Hubble Program (CCHP) papers.
photometers showed that the I-band PL relation was significantly narrower than both the B and V PL relations.And, somewhat later, Fernie (1975) briefly explored R and I photoelectric photometry, simply stating the advantages of moving to the red, in general, and tentatively commenting that the R-band PL relation was "remarkably tight."Two decades later, Laney & Stobie (1994) published VJHK PL relations for Cepheids in the LMC and SMC stating, without reference, that "The value of the dispersion systematically decreases from V to K as one might expect [emphasis added] . ..." As shown in Figure 1, obtaining PL relations based on single-epoch CCD exposures of Cepheids in IC 1613 Freedman (1988) also showed that the very sparsely sampled I-band PL relations were competitive, in their overall scatter, with preexisting (photographic) B and V time-averaged PL relations. 5This realization that templates were not only useful

Figure 1 .
Figure 1.The first example of the steps required to use a well-sampled light curve (upper plot) in the B band and predict the mean magnitudes of a Cepheid in the R and I bands, from a single-epoch observation in the latter two wavelengths (reproduced, with permission, fromFreedman 1988).The plot illustrates how the difference between an instantaneous (random-phase) B-band magnitude and the mean magnitude for that Cepheid can be scaled to predict the correction needed to take the Rand I-band instantaneous magnitudes, to their respective individual mean magnitudes, so long as correctly scaled and appropriately shifted templates were available for all wavelengths being considered.

Figure 2 .
Figure2.Eight period-luminosity relations for Cepheids in the Large Magellanic Cloud running (top to bottom) from the mid-infrared (4.5 and 3.6 μm;Scowcroft et al. 2011), to the near-infrared (JHK;Persson et al. 2004) and finally the optical (BVI from the "Three-hundred MilliMeter Telescope" (TMMT), which has been operating remotely in a queue-scheduled mode at the Las Campanas Observatory in Chile for the past 10 years, gathering BVI photometric data on nearby supernovae as well as Classical Cepheids in the Milky Way, LMC and SMC; A.Monson et al. 2024, in preparation).Individual PL ridge lines are shown as dashed lines, flanked on either side by the blue/bright and red/faint limits of the instability strip shown by ±2σ solid lines.The monotonic increase in the widths of the PL relations, top to bottom, long wavelength to short, can be readily seen.A closer look at the deviations of individual Cepheids with respect to the ridge line and with respect to one another in the B-band PL relation, for example, are repeated in detail and at high fidelity; however, simply in a progressively more compressed way in every one of the other PL relations, shrinking from blue to red, as should be the case.(Note that not all of the Cepheids plotted were observed in all of the bands shown.)

Figure 4 .
Figure4.Tight correlation of interband period-luminosity residuals (derived from Figure2).This example shows the detailed scaling of B-band residuals as a function of V-band residuals.The slope is well defined and the scatter is very low.Here the slope is DB/DV = 1.37 mag/mag, and the residual scatter around the fit is only ±0.043 mag.The slope defines the expansion/ compression factor in using either one of these PL relations to predict the width of the other and the relative placement of points therein.

Figure 5 .
Figure 5. Residual scatter in the predicted B-band PL relation after randomphase corrections to mean light have been applied.For our sample the residual scatter is σ = ±0.043mag.Dashed lines, on either side of zero, indicate 1σ deviations.

Figure 6 .
Figure 6.Period-amplitude distribution for Milky Way Cepheids.The range of Cepheid periods generally encountered in extragalactic studies (1.0 < log P < 1.8) is indicated by the two vertical red lines.The dashed red line marks a representative B-band amplitude of 1.6 mag for that restricted period range.(Diagram from data provided in Klagyivik & Szabados 2009).

Figure 7 .
Figure 7. Restoring single-epoch Cepheid photometry at one band, using a time-averaged PL relation, constructed at a different wavelength.In this case the V-band data at the top of the plot (for which there are many observations contributing to high-precision, time-averaged magnitudes) are used to predict the amount and the sign of amplitude-induced scatter contained in the single-epoch B-band PL relation (shown in the second plot down from the top).See Section 1.4 for a full description of the process and a detailed discussion of the remaining two PL relations at the bottom of the panel.Dashed lines are fits to the mean PL relations; solid lines are 2σ boundaries.

Figure 9 .
Figure 9. Residual scatter in the predicted B-band PL relation after randomphase corrections to mean light have been applied.For our sample the residual scatter is σ = ±0.061mag.Dashed lines, on either side of zero, indicate 1σ deviations.

Figure 10 .
Figure 10.Same as Figure 7, except that the I-band data, shown at the top of the panel, are being used to predict the corrections needed to bring the single-epoch Bband data back to their time-averaged positions.