# Light Curves For Ten R Coronae Borealis Stars For Longer Than a Century:
Secular Evolution, Dip Statistics, and a General Model for the Shape of
Isolated Light Curve Dips

Bradley E. Schaefer1,
1Department of Physics and Astronomy, Louisiana State University, Baton Rouge,
Louisiana, 70803, USA
E-mail<EMAIL_ADDRESS>
###### Abstract

R Coronae Borealis stars (RCBs) are cool supergiants that display non-periodic
deep dips in brightness. Recently, a group of ‘Hot RCB stars‘ has been
discovered to be fast evolving across the HR diagram, as these stars leave the
RCB region, with brightness changes at the rate of $\sim$1 mag/century.
Perhaps cool RCB stars can also be seen evolving, either increasing in
temperature as they evolve to become Hot RCB stars, or perhaps increasing in
luminosity as the stars arrive at the RCB region. To seek these changes, the
only possible method is to extract archival data going back more than a
century, looking for the brightness changes associated with the evolution. I
have measured and extracted 323,464 magnitudes (mostly from the Harvard plates
and from the AAVSO) for ten cool RCB stars, all with over a century for the
light curves, all consistently calibrated to a modern magnitude system. For
times away from any dips, these light curves are flat to within the typical
uncertainty of $\pm$0.10 mag/century. That is, I see no significant evolution.
I also have collected a large database of light curve dips and their
properties. From this, the light curves for all the well-observed isolated
dips have the same shape, featuring a flat slope for the few days immediately
after the minima. Further, I derive a general model for the shape of the light
curve for all isolated RCB dips, with a simple equation accurately describing
the observed recovery to maximum light.

###### keywords:

stars: evolution – stars: variables – stars: supergiants – stars: individual:
UX Ant, R CrB, V742 Lyr, W Men, VZ Sgr, V3795 Sgr, Z UMi, HV 12842, SP77 65-2,
IRAS 05374-6355

††pubyear: 2023††pagerange: Light Curves For Ten R Coronae Borealis Stars For
Longer Than a Century: Secular Evolution, Dip Statistics, and a General Model
for the Shape of Isolated Light Curve Dips–References

##  1 INTRODUCTION

The R Coronae Borealis stars (RCBs) are cool, hydrogen-deficient, carbon-rich
supergiants that are normally near-constant in brightness, yet suffer
occasional randomly-timed deep minima of durations months-to-decades (Clayton
1996; 2012). The deep minima are caused when the star ejects a thick carbon
dust cloud, which temporarily obscures the photosphere of the star until the
cloud slowly dissipates. RCBs are rare, with 147 RCBs currently known in the
Galaxy and the Magellanic Clouds (Tisserand et al. 2020).

RCBs have two proposed evolutionary paths (Iben et al. 1996; Saio & Jeffery
2002).. The first path is with the coalescence of a helium and a carbon-oxygen
white dwarf in a binary system, to form an RCB star. The second path is when a
post-AGB star undergoes a final helium shell flash and puffs up to form an RCB
star. The observational evidence heavily favours the WD-merger scenario
(Clayton et al. 2007, 2011; Clayton 2012).

Most RCB stars have Teff= 5000–8000 K, but there are five RCB stars that are
significantly hotter (15,000–25,000 K) (De Marco et al. 2002). Bateson (1978)
first noticed that one of these stars, DY Cen, seemed to be undergoing a
longterm secular decline in brightness. De Marco et al. (2002) found that
three of these stars, DY Cen, MV Sgr, and V348 Sgr, have been fading over the
last century, implying an evolutionary movement from right-to-left across the
top of the HR diagram. I have confirmed and extended their discovery (Schaefer
2016) by collecting very long light curves for four Hot RCBs from archival
plates at Harvard College Observatory plus modern CCD photometry reported
through the American Association of Variable Star Observers (AAVSO). For
example, DY Cen increased in temperature from 5800 K in 1906, to 7500 K in
1932 (at which time the dips in brightness ceased), to 19,400 K in 1987, and
then to 24,800 K in 2010. So we are seeing rapid steady evolutionary changes
in real-time, as one of the very few such cases observed in all astronomy.

The Hot RCB stars are evolving quickly away from the cool RCB region in the HR
diagram (supergiants with $\approx$ 5000–8000 K). This raises the question of
whether we can watch a cool RCB star evolving into a Hot RCB star? Or
similarly, perhaps we can catch the evolution of the progenitor as it
approaches the RCB state (Lauer et al. 2019)? These various arrivals and
departures from the RCB state have timescales of centuries and millennia, so
they are potentially observable with a century-long light curve. The only way
to get information on timescales approaching a century is to use the various
collections of archival photographic plates, for which the oldest and largest
collection is at Harvard College Observatory, stretching back to 1889. The RCB
brightness will change due to the evolution of the luminosity, as well to the
evolution of the surface temperature. The speed of evolution might perhaps be
fastest for the most massive and/or the hottest of the cool RCB stars (Lauer
et al. 2019).

So a reasonable plan to seek the recent evolutionary changes of the cool RCB
stars is to look for secular changes in their century-long light curves. The
Harvard plates provide good coverage over the entire sky to 16th mag and
deeper from roughly 1889–1954 and 1969–1989. To get the light curve extended
to 2023, I have used the CCD data recorded by the many observers reporting to
the AAVSO. Care must be taken to cast all of the magnitudes onto a consistent
scale, and to evaluate the dip frequency correctly in spite of sparse data.
This paper reports my results on secular changes in the brightness and dip-
frequency for ten cool RCB stars with light curves longer than a century using
Harvard and AAVSO data. I also make various statistical studies of the dips
themselves. Further, I derive a general model of the shape of isolated dip
light curves, and test this with my long-term light curves.

##  2 Observations

Ten cool RCB stars, see Table 1, have been selected based on several criteria.
For example, R CrB itself was selected for the availability of a high quality
and well-observed light curve going back to 1843. W Men, HV 12842, and SP77
65-2 were chosen for being amongst the brightest RCB stars in the LMC, so that
they must be amongst the most luminous RCB stars. And I chose V3795 Sgr, VZ
Sgr, and IRAS 05374-6355 as currently being the hottest of the cool RCBs, so
perhaps most likely to be starting to evolve more quickly. I selected UX Ant,
V742 Lyr, and Z UMi simply for being relatively bright and inside the sky
region for which data are publicly available. I selected against Galactic RCB
stars that were relatively faint or in crowded fields. My target stars are
listed in Table 1 in the order as from the General Catalog of Variable Stars,
and then alphabetically by designation for those with no GCVS name.

Table 1: Target RCB stars and Journal of Observations

RCB star | Galaxy | $\mu$ | $B_{\rm 1950}$ | $M_{\rm B}$ | $T_{\rm eff}$ | $\log[\epsilon(H)]$ | Years | B mags | Visual mags | Max mags
---|---|---|---|---|---|---|---|---|---|---
UX Ant | Milky Way |  $\sim$17 | 12.52 |  $\sim$ -4.8 | 7000 | 6.6 | 1890 – 2023 | 1342 | 1979a | 1064
R CrB | Milky Way | 10.7 | 6.75 | -4.1 | 6900 | 6.6 | 1784 – 2023 | … | 298868 | 24696b
V742 Lyr | Milky Way | 15.8 | 13.17 | -2.9 |  $\lesssim$5000 | … | 1893 – 2022 | 935 | … | 562
W Men | LMC | 18.5 | 14.25 | -5.4 | 6700 | 6.0 | 1890 – 2023 | 1256 | 2377a | 959
VZ Sgr | Milky Way | 14.8 | 11.08 | -4.8 | 7000 | 6.2 | 1889 – 2023 | 1320 | 7308a | 820
V3795 Sgr | Milky Way |  $\sim$15 | 12.07 |  $\sim$ -5.9 | 7000 |  $<$4.1 | 1889 – 2023 | 699 | 4195a | 508
Z UMi | Milky Way | 13.3 | 12.01 | -1.7 | 5250 |  $<$7 | 1892 – 2023 | 2738 | … | 1444
HV 12842 | LMC | 18.5 | 14.12 | -4.6 | … | 6.3 | 1900 – 2011 | 131 | … | 59
IRAS 05374-6355 | LMC | 18.5 | … | … | … | … | 1900 – 2011 | 113 | … | 0
SP77 65-2 | LMC | 18.5 | 15.44 | -3.4 | … | … | 1894 – 2011 | 297 | … | 223

aThese visual magnitudes were used only to define the times of dips. bThe
number is for the number of nights that are certainly at maximum light, that
is, this light curve has been nightly averaged.

Various fundamental properties of the target stars are collected into Table 1.
The second column of Table 1 marks the host galaxy, while the third column
gives the distance modulus ($\mu$) in magnitudes. The distances are from Gaia
DR3 (Gaia Collaboration et al. 2023), Montiel et al. (2018), and Kijbunchoo et
al. (2011), while the LMC RCBs have the distance modulus from Freedman et al.
(2001). The next column give the average $B$ magnitude for the year 1950, as
derived later in this paper. The fifth column gives the calculated absolute
magnitude in $B$, with the addition of extinction values taken from Schlafly &
Finkbeiner (2011). The next column lists the effective surface temperature, in
degrees K, as taken from Jurcsik (1996) and Kijbunchoo et al. (2011). The
seventh column lists the hydrogen abundance in logarithmic units,
$\log[\epsilon(H)]$, as calculated by Pandey, Hema, & Reddy (2021). Measured
values were collected from Pandey et al. (2021), Kipper & Klochkova (2006),
and Jurcsik (1996).

Nearly all of my collected observations were taken from either of two sources;
the Harvard archival photographic plates and the AAVSO observers. The first
major source is magnitudes recently measured from the archival photographic
plates at the Harvard College Observatory. This archive contains the majority
of the world’s deep plates with wide coverage. Indeed, the coverage is the
entire sky, north and south, recording all stars to 14 mag with thousands of
plates, and recording all stars to 16 mag with hundreds of plates, and
recording many stars to 18 mag with tens of plates. Harvard has intense
coverage from 1889 to 1954 and from around 1969 to 1989. Almost all of these
plates return magnitudes in the modern B magnitude system, while the few ‘red’
and ‘yellow’ plates are ignored here. The Harvard plates are an awesome and
unique source, providing thousands of magnitudes for all my RCB stars with a
photometric accuracy of typically 0.10 mag. Importantly for my project, for
all the stars other than R CrB itself, Harvard plates provide the only means
in the world for getting a light curve from 1889 into the 1930s.

The magnitudes from the Harvard plates have been derived with two methods. The
first method is the traditional direct by-eye visual comparison of stellar
image sizes as made by looking at the plate on a light table through a
magnifying loupe. The observer makes comparisons of the RCB star image radius
versus the radii of nearby comparison stars of known magnitude. This method
returns a photometric accuracy of typically 0.10 mag under ordinary good
conditions. Full details of the measure of the Harvard plates are given in
Schaefer (2016). The second method is produced by the huge program called
Digital Access to a Sky Century at Harvard (DASCH) run by the Harvard College
Observatory (Grindlay et al. 2012), J. Grindlay Principal Investigator. The
DASCH program has been making high-quality digital scans of all Harvard
plates, with roughly half completed to date. These scans are available
publicly on-line. Further, the DASCH program has analyzed all point sources on
all plates with a very sophisticated photometry analysis (Tang et al. 2013) so
as to create a light curve for every star. These light curves are publicly
available on-line111http://dasch.rc.fas.harvard.edu/lightcurve.php. The
typical photometric accuracy for the DASCH magnitudes is 0.10 mag for good
conditions. Importantly, both by-eye and DASCH magnitudes have been measured
many times to have a close-to-zero offset from each other (and from
independent photometry) and with effectively identical one-sigma measurement
errors. Importantly, both by-eye and DASCH magnitudes are made with APASS
comparison stars in the Johnson B-magnitude system, and the plate material has
spectral sensitivity effectively identical to the Johnson B-magnitude system,
so the resultant magnitudes are all confidently in the modern B-magnitude
system. These two points mean that both by-eye and DASCH photometry can be
seamlessly combined with modern CCD magnitudes to form one consistent very-
long-term light curve.

For completeness, the DASCH light curves include magnitudes that should not be
used when top-quality photometry is required. For example, a small fraction of
the plates have obvious plate defects of many types at the position of the RCB
star, and these constitute most of the outliers from a smooth light curve.
Further, I have rejected multiple-exposure plates and plates that are not the
usual blue sensistive. These non-useable magnitudes are readily recognized
from the quality flags and parameters in the DASCH database. Further, I have
systematically rejected all magnitudes that are within 0.30 mag of the plate
limit, and all magnitudes with a quoted one-sigma error bar of $>$0.30 mag. I
have examined by-eye all plates (looking at either the glass plate or the
DASCH scan) with outlier magnitudes, and about half of the remainder. So all
my DASCH magnitudes have been put through a high quality control. This
procedure is ‘painful’ for R CrB itself, because all of the plates at maximum
light (with $B$$\sim$6.75) are brighter than the DASCH limit of near 8.0 mag.
That is, for stars brighter than this limit, DASCH has problems with
calibration stars that are usually far away where various field effects create
systematic errors. The result is that the DASCH light curve for R CrB has the
only reliable magnitudes for when it is in dip with $B$$\gtrsim$9 mag.
Fortunately, for the very long light curve of R CrB, we can use the extensive
visual light curve, which is available nearly continuously from 1843 to today,
plus 1784–1796.

My second major source of magnitudes is from the AAVSO, as collected from
observers and telescopes worldwide since 1911. I have taken AAVSO magnitudes
from four of their primary sets of data. The first AAVSO data set is their
archived CCD magnitudes in the modern B band, mostly from the 1990s to 2023.
These are the very large number of magnitudes taken through a B-band filter,
usually with relatively small telescopes, by hundreds of observers worldwide,
all with the the usual standard procedures. This data set provides the bulk of
the RCB coverage from the 1990s to 2023. The AAVSO data sets are freely
available on-line222https://www.aavso.org/data-download. The second AAVSO data
set is their AAVSO Photometric All-Sky Survey (APASS). This survey covers all
stars in the sky, north and south, down to roughly 17 mag with good
photometric accuracy in the B and V bands. With this, APASS provides a few
recent magnitudes for my RCB stars. APASS also provides the magnitude
calibration and comparison stars used for all the other observers (the DASCH
project, my by-eye magnitudes from the Harvard plates, and the AAVSO CCD
observers). The third AAVSO data set that I have used is the collection of
visual magnitudes for R CrB itself as taken widely from the published
literature. This third data set is what extends the R CrB light curve from the
founding of the AAVSO in 1911 back all the way to Sir John Herschel in 1843.
These visual magnitudes have identical color terms as later visual data (the
human eye’s spectral sensitivity does not vary) and the comparison stars are
close to the modern values, so the pre-AAVSO light curve is essentially
identical in nature as the normal AAVSO magnitudes reported after 1911. These
archival visual magnitudes are recorded in the same AAVSO database for R CrB.
Importantly, the typical one-sigma uncertainty for a single visual magnitude
estimate of R CrB is $\pm$0.15 mag, yet when combined with many other visual
measures within the same time interval, the uncertainty on the average becomes
$<$0.05 mag for all of the seasonal averages. Importantly, the intrinsic
variations of the target stars is always $>$0.1 mag (due to pulsations and
residual dips), so the accuracy of the time-binned visual light curve is
already more than good enough for measuring any evolutionary effects. The
fourth AAVSO data set that I have used is the vast collection of visual
observations for four other RCB stars. These observations have mainly been
used to specify the time intervals that the RCB star is certainly not inside a
dip, for purposes of dip statistics.

I also used a set of published visual magnitudes from 1784–1796 from before
the original discovery of R CrB by Pigott & Engelfield (1797). These
observations are reported as explicit comparisons with multiple comparison
stars. To convert to modern visual magnitudes (i.e., directly comparable to
the AAVSO visual magnitudes), I first converted the Johnson V magnitudes for
each comparison star to visual magnitudes by the measured relation of Stanton
(1999). Then Pigott’s comparisons were applied to get a visual magnitude for R
CrB. The light curve shows two intervals (1784–1785 and 1796) over which the
star was apparently at maximum light. These magnitudes are valuable because
they extend my at-maximum light curve by 59 years, which makes for a
substantially more sensitive measure of any long-term secular trend.

An additional small source of magnitudes was the ASAS-3 survey333All Sky
Automated Survey, http://www.astrouw.edu.pl/asas/?page=aasc for IRAS
05374-6355 from 2002-2011. The reason for needing these magnitudes is that I
do not have any Harvard or AAVSO magnitudes after 1989. Apparently, the star
was deep in minima for most of the time.

With these sources of magnitudes, I have collected century-long light curves
for the ten RCB stars. The magnitudes and coverage are tabulated in Table 1.
The last three columns give the total number of magnitudes in the B-band (from
Harvard plates and AAVSO CCD measures), in the visual band, and the total
number of magnitudes for the RCB star that are certainly at maximum.

##  3 RCB Light Curves

I have collected magnitudes for each of my ten cool RCB stars, each with
longer than a century of coverage. All individual magnitudes are given in the
Table 2, for which most of the data lines appear only in the Supplementary
Material. (However, the full AAVSO light curve for R CrB itself, plus the
later visual magnitudes for four RCB targets used for extending the dip
statistics, are publicly available in the AAVSO on-line database, and so are
not included in the Table.) The first column identifies the RCB star, and then
the next two columns give the time of each observation in Julian date (JD) and
fractional year. The fourth column identifies the source of the magnitude. For
the Harvard data, this is given as the letters for the plate series followed
by the plate number. The fifth column gives the B-band magnitude and its
1-sigma error bar. The last column gives my evaluation of whether the RCB star
is at maximum light (‘Max’), where I have taken a very conservative criterion.
If star cannot be confidently stated to be at maximum, then I designate the
measure by ‘Dip’, even though the star might be close to maximum light.

Table 2: RCB magnitudes (full table with 8895 magnitudes is in the Supplementary Material) RCB star | JD | Year | Source | Magnitude | Max?
---|---|---|---|---|---
UX Ant | 2411507.54 | 1890.38 | B5226 | 13.37 $\pm$ 0.07 | Dip
UX Ant | 2411519.49 | 1890.41 | B5351 | 12.80 $\pm$ 0.26 | Dip
UX Ant | 2411522.51 | 1890.42 | B5353 | 13.50 $\pm$ 0.20 | Dip
UX Ant | 2411883.58 | 1891.41 | B6126 | 12.30 $\pm$ 0.07 | Max
UX Ant | 2412210.68 | 1892.31 | B7442 | 12.62 $\pm$ 0.25 | Max
$\ldots$ |  |  |  |  |
SP77 65-2 | 2444663.97 | 1981.16 | DSB689 | 15.40 $\pm$ 0.27 | Max
SP77 65-2 | 2444699.89 | 1981.26 | DSB726 | 15.20 $\pm$ 0.38 | Max
SP77 65-2 | 2444997.99 | 1982.08 | DSB856 | 15.20 $\pm$ 0.20 | Max
SP77 65-2 | 2447186.00 | 1988.07 | DSB2461 | 15.00 $\pm$ 0.20 | Max
SP77 65-2 | 2455563.00 | 2011.00 | APASS | 15.54 $\pm$ 0.05 | Max

A variety of criteria can be proposed to determine when an RCB star is in a
dip, or when it is at maximum light. A primary problem is how to define
maximum light, even for a perfect light curve, because the stars come out of
each dip with what appears to be an asymptotic approach to some constant level
that we can label as ‘maximum’. (This is just due to the geometric dilution of
the expanding dust cloud never going to zero extinction.) So where should we
draw the line? This is further confused by the intrinsic pulsational
variations of the parent star. So any constant-magnitude threshold would
reject either times when the star is at maximum light yet still with some
substantial dust extinction (thus introducing a bias to the bright side) or
times when the star is still recovering from a dip (thus providing a bias to
the faint side). The observational situation is often made greatly worse when
we deal only with sparsely sampled light curves.

For identifying dips and maxima, I started off by flagging magnitudes more
than 1.5 mag fainter than an approximate maximum. This gave a list of
recognized dips. These dips occasionally consisted of multiple deep dips that
had run into overlapping times. I then expanded each time interval so as to
include any measured magnitudes that were within 0.5 mag of the local maximum.
I then further expanded the time interval, perhaps by months, so that we can
be highly certain that the star has recovered to maximum light. The result is
the identification of which magnitudes are certainly from maximum light (see
the last column in Table 2).

The plots showing the full light curves just display the usual structure for
RCB stars. Rather, given my original purpose of looking for secular changes, I
have extracted only those magnitudes that are certainly at maximum light, and
these are binned them together into contiguous intervals. For the binning
intervals, I have taken only complete calendar years that have no magnitudes
inside a dip. The inside-dip identification was made extremely conservatively,
with the dip intervals expanded somewhat to be sure (see last column of Table
2), and then the further selection of only pristine years provides further
insolation from dips.

I constructed binned light curves of the RCB stars certainly at maximum light,
and these are the primary data product. Each binned magnitude is a straight
average of the magnitudes in each time interval. The 1-sigma uncertainty is
taken as the RMS of the included magnitudes divided by the square root of
$N_{\rm obs}$, with $N_{\rm obs}$ being the number of magnitudes in the
average. In cases of large $N_{\rm obs}$, the quoted error bar is limited to
0.01 mag, in recognition that the real systematic uncertainties are something
close to this. My binned and averaged light curves at maximum are displayed in
Figure 1 for R CrB and in Figure 2 for the other targets, and tabulated in
Table 3.

Figure 1: Binned at-maximum light curves for R CrB. The extraction of the magnitudes at maximum produces a binned light curve that has coverage for 239 years with 298868 magnitudes, so this cannot be improved for any star even far into the future. Before 1924, the scatter is fairly large, likely resulting from ordinary photometric problems. After 1924, the large number of magnitudes and their good calibration makes for a scatter with an RMS of near 0.02 mag. For my primary science question, we get a very accurate measure of any secular change due to evolution. For the full interval of 1784–2023, the best linear fit gives a slope -0.029$\pm$0.045 mag per century, which is to say that we see no significant evolution in this best case. This line is the thick black line, while the flat gray line provides a guide for the eye. If we just look at the 1924–2023 century, the best-fitting line has a close-to-zero slope, indicating no significant evolution. Figure 2: Binned at-maximum light curves for 8 RCB stars. This is a plot of the light curves in Table 3 and their best line fits in Table 4. Here are plots for eight RCB stars (each displayed with an integer offset from the $B_{\rm 1950}$ values, with the thin gray lines providing a reference for a perfectly flat line, and with the thick black lines showing the best linear fits. Such a plot for R CrB is in Figure 1, while IRAS 05374-6355 was never seen at maximum light. This is my primary observational result. The idea is to use these plots and fits to seek secular brightness changes from evolution as the RCB stars either enter or leave the RCB region of the HR diagram. The primary result is that none of the RCB stars shows any significant secular trend. That is, these stars are all evolving at a rate such that their B magnitudes change by less than roughly 0.10 mag per century. The individual points show variations around any best-fitting curve, far larger than any measurement or systematic errors, and I argue that these must be due to intrinsic variability of the star, for example from small unrecognized dust dips or from ordinary RCB pulsations. Table 3: RCB binned light curves at maximum light RCB star | Years | $\langle Year\rangle$ | $N_{\rm obs}$ | $B_{\rm max}$
---|---|---|---|---
UX Ant | 1902–1903 | 1903.1 | 23 | 12.55 $\pm$ 0.03
UX Ant | 1907–1910 | 1908.9 | 27 | 12.48 $\pm$ 0.03
UX Ant | 1920–1929 | 1925.1 | 52 | 12.51 $\pm$ 0.03
UX Ant | 1933–1935 | 1934.7 | 63 | 12.48 $\pm$ 0.02
UX Ant | 1939–1942 | 1941.3 | 195 | 12.50 $\pm$ 0.01
UX Ant | 1945–1952 | 1947.3 | 214 | 12.42 $\pm$ 0.01
UX Ant | 1970–1972 | 1971.2 | 17 | 12.55 $\pm$ 0.05
UX Ant | 1978–1989 | 1985.1 | 114 | 12.49 $\pm$ 0.02
UX Ant | 2006 | 2006.3 | 36 | 12.61 $\pm$ 0.01
UX Ant | 2015 | 2015.3 | 11 | 12.66 $\pm$ 0.01
UX Ant | 2019–2023 | 2022.9 | 82 | 12.61 $\pm$ 0.01
R CrB | 1784–1785 | 1784.8 | 3 | 6.24 $\pm$ 0.02 a
R CrB | 1795 | 1795.6 | 8 | 5.88 $\pm$ 0.04 a
R CrB | 1843–1844 | 1844.0 | 30 | 6.06 $\pm$ 0.02 a
R CrB | 1847–1849 | 1848.1 | 24 | 6.00 $\pm$ 0.02 a
R CrB | 1853 | 1853.5 | 9 | 6.06 $\pm$ 0.07 a
R CrB | 1856 | 1856.6 | 20 | 6.25 $\pm$ 0.04 a
R CrB | 1875 | 1875.3 | 33 | 6.24 $\pm$ 0.03 a
R CrB | 1882–1885 | 1884.2 | 26 | 6.33 $\pm$ 0.04 a
R CrB | 1898–1899 | 1899.4 | 33 | 6.07 $\pm$ 0.03 a
R CrB | 1901–1902 | 1901.8 | 210 | 5.83 $\pm$ 0.02 a
R CrB | 1906–1908 | 1907.9 | 80 | 5.92 $\pm$ 0.02 a
R CrB | 1913–1914 | 1914.0 | 472 | 5.97 $\pm$ 0.01 a
R CrB | 1916 | 1916.5 | 219 | 6.03 $\pm$ 0.01 a
R CrB | 1919–1920 | 1920.1 | 399 | 6.25 $\pm$ 0.01 a
R CrB | 1922 | 1922.5 | 312 | 6.28 $\pm$ 0.01 a
R CrB | 1925–1933 | 1929.6 | 2976 | 6.05 $\pm$ 0.01 a
R CrB | 1936–1937 | 1937.0 | 634 | 6.08 $\pm$ 0.01 a
R CrB | 1941 | 1941.5 | 271 | 6.05 $\pm$ 0.01 a
R CrB | 1944 | 1944.5 | 269 | 6.04 $\pm$ 0.01 a
R CrB | 1951 | 1951.5 | 291 | 6.04 $\pm$ 0.01 a
R CrB | 1954 | 1954.5 | 313 | 6.06 $\pm$ 0.01 a
R CrB | 1957–1959 | 1958.5 | 924 | 6.04 $\pm$ 0.01 a
R CrB | 1970–1971 | 1971.0 | 705 | 6.04 $\pm$ 0.01 a
R CrB | 1979–1982 | 1981.0 | 1434 | 6.07 $\pm$ 0.01 a
R CrB | 1987 | 1987.5 | 358 | 6.00 $\pm$ 0.01 a
R CrB | 1990 | 1990.5 | 362 | 6.05 $\pm$ 0.01 a
R CrB | 1992 | 1992.5 | 362 | 6.06 $\pm$ 0.01 a
R CrB | 1997 | 1997.5 | 362 | 6.06 $\pm$ 0.01 a
R CrB | 2002 | 2002.5 | 362 | 6.07 $\pm$ 0.01 a
R CrB | 2004–2006 | 2005.5 | 1082 | 6.03 $\pm$ 0.01 a
R CrB | 2021–2023 | 2022.0 | 695 | 6.07 $\pm$ 0.01 a
V742 Lyr | 1893–1894 | 1893.8 | 3 | 13.23 $\pm$ 0.01
V742 Lyr | 1897–1899 | 1898.8 | 5 | 13.10 $\pm$ 0.09
V742 Lyr | 1901–1903 | 1902.3 | 10 | 12.97 $\pm$ 0.09
V742 Lyr | 1914–1918 | 1916.7 | 11 | 12.95 $\pm$ 0.09
V742 Lyr | 1935 | 1935.5 | 68 | 13.44 $\pm$ 0.03
V742 Lyr | 1937–1939 | 1938.6 | 125 | 13.09 $\pm$ 0.02
V742 Lyr | 1944–1946 | 1945.8 | 47 | 13.11 $\pm$ 0.06
V742 Lyr | 1948–1950 | 1949.5 | 68 | 13.06 $\pm$ 0.03
V742 Lyr | 1967–1968 | 1967.9 | 4 | 12.95 $\pm$ 0.03
V742 Lyr | 1974–1976 | 1975.8 | 42 | 13.48 $\pm$ 0.04
V742 Lyr | 1978 | 1978.5 | 13 | 13.19 $\pm$ 0.05
V742 Lyr | 1983 | 1983.5 | 23 | 13.59 $\pm$ 0.04
V742 Lyr | 1985–1989 | 1987.3 | 89 | 13.22 $\pm$ 0.03
V742 Lyr | 2009–2014 | 2012.4 | 27 | 12.91 $\pm$ 0.02
V742 Lyr | 2016 | 2016.8 | 8 | 13.28 $\pm$ 0.03
V742 Lyr | 2019 | 2019.7 | 29 | 13.18 $\pm$ 0.02
W Men | 1890–1895 | 1893.8 | 6 | 14.28 $\pm$ 0.10
W Men | 1899–1901 | 1900.6 | 8 | 14.35 $\pm$ 0.04
W Men | 1909–1917 | 1915.3 | 8 | 14.13 $\pm$ 0.05
W Men | 1924–1927 | 1926.0 | 48 | 14.24 $\pm$ 0.02
W Men | 1929 | 1929.2 | 42 | 14.28 $\pm$ 0.02
Table 4: continued

RCB binned magnitudes  RCB star Years $\langle Year\rangle$ $N_{\rm obs}$
$B_{\rm max}$ W Men 1930–1933 1932.1 129 14.23 $\pm$ 0.02 W Men 1938–1939
1939.2 108 14.24 $\pm$ 0.01 W Men 1945–1946 1945.9 169 14.28 $\pm$ 0.01 W Men
1948–1954 1950.0 140 14.28 $\pm$ 0.02 W Men 1970–1971 1971.4 6 14.04 $\pm$
0.09 W Men 1978–1989 1983.6 48 14.17 $\pm$ 0.04 W Men 2013 2013.7 15 14.23
$\pm$ 0.01 W Men 2016–2023 2021.4 42 14.26 $\pm$ 0.02 VZ Sgr 1889–1893 1891.9
12 11.17 $\pm$ 0.10 VZ Sgr 1898–1899 1899.2 17 11.37 $\pm$ 0.11 VZ Sgr
1903–1904 1904.0 68 11.03 $\pm$ 0.03 VZ Sgr 1907–1908 1908.1 59 10.91 $\pm$
0.02 VZ Sgr 1920–1923 1922.3 71 11.18 $\pm$ 0.03 VZ Sgr 1938–1941 1940.2 124
10.99 $\pm$ 0.03 VZ Sgr 1945–1948 1946.8 68 11.07 $\pm$ 0.03 VZ Sgr 1971–1982
1978.0 27 11.09 $\pm$ 0.04 VZ Sgr 2006 2006.6 9 11.20 $\pm$ 0.02 VZ Sgr
2010–2011 2011.0 11 11.01 $\pm$ 0.02 VZ Sgr 2015–2016 2015.6 11 11.02 $\pm$
0.05 V3795 Sgr 1889–1899 1894.6 29 11.95 $\pm$ 0.04 V3795 Sgr 1900–1908 1904.3
22 12.00 $\pm$ 0.05 V3795 Sgr 1910–1913 1911.9 32 12.10 $\pm$ 0.03 V3795 Sgr
1917–1926 1922.7 83 12.32 $\pm$ 0.03 V3795 Sgr 1928–1933 1930.8 150 12.19
$\pm$ 0.02 V3795 Sgr 1940–1942 1941.9 103 12.14 $\pm$ 0.02 V3795 Sgr 1953–1954
1954.1 7 11.89 $\pm$ 0.09 V3795 Sgr 1970 1970.5 9 11.94 $\pm$ 0.09 V3795 Sgr
1981–1989 1985.7 26 12.02 $\pm$ 0.04 V3795 Sgr 1995–2001 1998.2 1627 12.15
$\pm$ 0.01 V3795 Sgr 2010–2013 2012.3 87 12.00 $\pm$ 0.03 V3795 Sgr 2022–2023
2023.3 21 11.96 $\pm$ 0.01 Z UMi 1892 1892.3 1 11.90 $\pm$ 0.11 Z UMi
1897–1898 1897.6 15 11.91 $\pm$ 0.04 Z UMi 1904 1904.5 15 11.86 $\pm$ 0.04 Z
UMi 1907–1909 1908.7 33 11.91 $\pm$ 0.04 Z UMi 1913–1914 1913.9 50 11.93 $\pm$
0.03 Z UMi 1925–1926 1925.8 44 12.03 $\pm$ 0.05 Z UMi 1931–1934 1933.5 143
12.15 $\pm$ 0.02 Z UMi 1940 1940.3 64 12.00 $\pm$ 0.02 Z UMi 1953 1953.2 5
11.93 $\pm$ 0.09 Z UMi 1970–1973 1972.3 14 12.27 $\pm$ 0.05 Z UMi 1977–1978
1977.9 36 12.05 $\pm$ 0.02 Z UMi 1985 1985.5 48 12.01 $\pm$ 0.02 Z UMi
1987–1989 1988.1 95 12.02 $\pm$ 0.02 Z UMi 2010 2010.4 198 12.01 $\pm$ 0.01 Z
UMi 2018 2018.3 32 12.32 $\pm$ 0.04 Z UMi 2021–2023 2021.6 60 12.04 $\pm$ 0.02
HV12842 1923–1924 1923.8 3 14.12 $\pm$ 0.02 HV12843 1944–1949 1946.4 17 14.02
$\pm$ 0.03 HV12844 1971 1971.7 4 14.15 $\pm$ 0.05 HV12845 1978–1985 1983.2 12
14.30 $\pm$ 0.07 HV12846 1986–1989 1987.6 18 14.17 $\pm$ 0.07 SP77 65-2
1900–1905 1903.3 5 15.53 $\pm$ 0.12 SP77 65-3 1924–1929 1928.3 58 15.37 $\pm$
0.03 SP77 65-4 1930–1933 1931.9 19 15.44 $\pm$ 0.07 SP77 65-5 1934–1936 1935.1
73 15.46 $\pm$ 0.03 SP77 65-6 1942–1943 1943.2 26 15.55 $\pm$ 0.05 SP77 65-7
1949–1950 1949.9 12 15.34 $\pm$ 0.07 SP77 65-8 1971–1988 1979.6 9 15.41 $\pm$
0.11 SP77 65-9 2011 2011 1 15.45 $\pm$ 0.05
aAll R CrB magnitudes are consistently in the visual magnitude system.

The light curves are modern B-band magnitudes from the Harvard archival plates
plus the AAVSO CCD B-band magnitudes. However, there is one exception, where
visual magnitudes are used. The first exception is that all of the R CrB light
curve is consistently constructed in the visual band. I could have constructed
a light curve from 1889 to 2023 in the B-band, but then there would always be
worries about connecting this to the pre-1889 visual light curve, and since
the primary task is looking for secular trends, I sought to eliminate such
questions by using only visual data for the entire 1784-2023 interval.

##  4 RCB Evolution

My primary goal is to test for secular changes of RCB stars caused by the
evolution as a star either approaches or leaves the RCB region of the HR
diagram. For this, I am looking for systematic changes across the last century
from the binned at-maximum light curves in Table 3 and Figs 1 and 2. Even a
brief examination of my figures shows that there are no large or obvious
secular changes on any long time-scale. Still, I should quantify this result.

I expect that evolutionary changes will be on a sufficiently long time-scale
such that the changes will be smooth across one century. That is, I should be
looking for linear variations. This is best done as a chi-square fit for the
individual light curves in Table 3. The model is taken as $B_{\rm
linear}=B_{\rm 1950}+S(Y-1950)/100$, where $B_{\rm 1950}$ is the magnitude in
the year 1950, $S$ is the slope in units of magnitudes per century, and $Y$ is
the year of the observation. This model has been centred in 1950 so as to make
for uncorrelated errors, and the units of magnitudes per century makes for
convenient and understandable values. My use of chi-square fitting allows for
the usual calculation of uncertainties on $B_{\rm 1950}$ and $S$ as being the
largest changes over which the $\chi^{2}$ rises from its best-fitting minimum
value by 1.00 for the 1-sigma error bar. The cause of the scatter about the
best linear fits could be any or all of (1) unrecognized systematic errors,
(2) small or short unrecognized dust dips included in the at-maximum light
curve, and (3) ordinary pulsational variations being sampled imperfectly.

With this, I have made linear fits to my binned at-maximum light curves, with
the results tabulated in Table 4. The first column is the RCB star. I have no
fit for IRAS 05374-6355, because I have never found it at maximum light at
anytime since 1900. The next two columns give the best-fitting $B_{\rm 1950}$
and $S$, along with their 1-sigma error bars calculated with the added
intrinsic variance. The fourth column is the observed RMS scatter (in
magnitudes) of the averaged magnitudes about the best-fitting line. My best-
fitting lines are plotted in Figs 1 and 2.

Table 5: Linear fits to century-long at-maximum light curves RCB star |  $B_{\rm 1950}$ (mag) |  $S$ (mag/cen) | RMS (mag)
---|---|---|---
UX Ant | 12.52 $\pm$ 0.02 | 0.12 $\pm$ 0.05 | 0.05
R CrB | 6.06 $\pm$ 0.02 | -0.03 $\pm$ 0.05 | 0.11
V742 Lyr | 13.17 $\pm$ 0.06 | 0.10 $\pm$ 0.14 | 0.20
W Men | 14.25 $\pm$ 0.02 | -0.03 $\pm$ 0.05 | 0.08
VZ Sgr | 11.08 $\pm$ 0.04 | -0.02 $\pm$ 0.08 | 0.14
V3795 Sgr | 12.07 $\pm$ 0.04 | -0.04 $\pm$ 0.10 | 0.13
Z UMi | 12.01 $\pm$ 0.03 | 0.13 $\pm$ 0.07 | 0.09
HV 12842 | 14.12 $\pm$ 0.04 | 0.21 $\pm$ 0.15 | 0.08
SP77 65-2 | 15.44 $\pm$ 0.03 | 0.00 $\pm$ 0.10 | 0.07

Now we can look for evolution as a slope $S$ being significantly different
from zero. A positive $S$ value indicates a fading of the star, while a
negative value indicates a brightening of the star. A star approaching the RCB
region on the HR diagram will come from the bottom, so it should be
brightening. A star leaving the RCB region will be shifting to higher
temperature at nearly-constant luminosity, and this will increase the B-band
brightness as the bolometric correction changes. So the expectation is that
any evolutionary change should result in a negative $S$ value.

From Table 4, we only see $S$ values within 2.3-sigma of zero. The largest
deviations from zero slope are all with a positive slope. In no case can we
point to any significant, or even suggestive, secular change. So this is my
result, we have nice century-long light curves at maximum for nine RCB stars,
and we see no evidence of evolution, with a typical uncertainty of 0.10
magnitudes per century.

##  5 Dip Statistics

Long-term RCB light curves can also allow the measure of properties and
statistics for the dips themselves. Little work has been published on such a
plan. The only work that I can find is a study in 1996 by J. Jurcsik (Jurcsik
1996). He was able to collect light curves for 27 RCB stars, with a median
coverage of 64 years in length, for a total of 358 dips. From this, he only
derived one dip statistic, the average time between dips ($\langle\Delta
T\rangle$). Jurcsik found that this $\langle\Delta T\rangle$ is strongly
correlated with relative abundance of hydrogen in the RCB stellar atmosphere.
The relation is that a low hydrogen abundance points to a short average time
between dips. Correspondingly, a high hydrogen abundance goes along with a
long time between dips. Jurcsik’s excellent work provides a characteristic
clue as to the dip-formation mechanism.

###  5.1 Dip Dates, Durations, and Depths

My century-long light curves can also be used to provide a thorough data set
on RCB dips and their statistics. A dip was taken to be an interval over which
at least one measured magnitude was fainter than 1.5 mag below the average
maximum light, and each dip is distinct from adjacent dips by the star
recovering to around maximum for some time between. Each dip consists either
of single isolated dips with just one brightness minimum, or multiple dips
caused by multiple mass ejections leading to complex dips with multiple local
minima in brightness. Table 5 collects some properties on all the dips
(isolated and complex) that I have recognized, 174 in total. The first column
is the RCB star, and subsequent columns give the Julian date (JD) of the dip,
the duration of the dip in days, and the faintest observed B magnitude
($B_{\rm min}$). For this, I have not listed any one dip for IRAS 05374-6355,
as I only see a continuous dip from 1900 to 2011. In general, I took the JD of
the dip to be the date with the deepest observed measured magnitude, which I
took to be $B_{\rm min}$. I took the dip duration to be the interval from the
first to last time for which the star was more than 1.5 mag fainter than its
average maximum magnitude.

Table 6: RCB dips RCB star | JD | Duration | $B_{\rm min}$
---|---|---|---
UX Ant | 2413246 | <438 | >14.02
UX Ant | 2414037 | <413 | >14.41
UX Ant | 2415073 | <335 | 14.20
UX Ant | 2421349 | <690 | >14.45
UX Ant | 2426455 | <105 | >15.24
UX Ant | 2428719 | 329 | 16.50
UX Ant | 2431109 | 62 | 15.00
UX Ant | 2451881 | 431 | >15.00
UX Ant | 2452440 | 335 | >14.60
UX Ant | 2454298 | 1155 | >18.80
UX Ant | 2456085 | 32 | 15.28
UX Ant | 2457800 | >108 | 16.10
R CrB | 2377016 | 246 | >7.8
R CrB | 2395200 | <150 | 8.00
R CrB | 2397543 | >80 | 10.00
R CrB | 2398240 | >4 | 8.00
R CrB | 2398791 | 270 | 10.70
R CrB | 2399793 | 37 | 10.70
R CrB | 2400196 | 162 | 11.27
R CrB | 2400987 | 40 | 12.70
R CrB | 2402137 | 2318 | 13.30
R CrB | 2405077 | 1335 | 13.90
R CrB | 2407205 | >80 | 9.70
R CrB | 2407760 | >40 | 13.50
R CrB | 2410497 | 29 | 7.60
R CrB | 2411865 | >300 | 8.80
R CrB | 2413041 | >510 | 10.30
R CrB | 2413409 | <40 | 9.50
R CrB | 2416202 | <140 | 7.98
R CrB | 2416289 | 18 | 8.53
R CrB | 2416991 | <430 | 10.00
R CrB | 2418503 | 350 | 13.80
R CrB | 2418762 | 116 | 12.73
R CrB | 2418974 | 140 | 11.80
R CrB | 2419249 | 253 | 12.10
R CrB | 2419532 | 16 | 7.85
R CrB | 2420729 | 33 | 8.70
R CrB | 2421397 | 399 | 13.70
R CrB | 2422768 | 15 | 9.40
R CrB | 2422973 | 40 | 9.79
R CrB | 2423557 | 59 | 12.43
R CrB | 2424070 | 17 | 8.93
R CrB | 2427829 | 212 | 12.01
R CrB | 2429235 | 422 | 13.57
R CrB | 2430673 | 122 | 12.50
R CrB | 2431542 | 15 | 7.85
R CrB | 2431807 | 37 | 9.50
R CrB | 2431877 | 24 | 9.70
R CrB | 2432173 | 105 | 11.60
R CrB | 2432999 | 352 | 14.10
R CrB | 2434360 | 107 | 11.00
R CrB | 2437053 | 153 | 12.57
R CrB | 2437207 | 48 | 11.80
R CrB | 2437944 | 166 | 13.20
R CrB | 2438593 | 1454 | 14.85
R CrB | 2440422 | 13 | 8.03
R CrB | 2441424 | 93 | 12.40
R CrB | 2442098 | 133 | 11.45
R CrB | 2442735 | 115 | 11.13
R CrB | 2443281 | 303 | 13.44
R CrB | 2445624 | 240 | 13.83
R CrB | 2446348 | 44 | 10.48
R CrB | 2447429 | 199 | 11.45
Table 7: continued

RCB dips  RCB star JD Duration $B_{\rm min}$ R CrB 2447774 28 8.48 R CrB
2449297 28 8.27 R CrB 2450042 324 13.60 R CrB 2451121 105 8.58 R CrB 2451203
54 9.67 R CrB 2451422 132 13.70 R CrB 2451901 48 12.39 R CrB 2452700 51 12.75
R CrB 2454948 3592 15.10 R CrB 2458777 32 8.51 R CrB 2460113 53 10.15 V742 Lyr
2415151 <732 >15.54 V742 Lyr 2419308 >1300 >15.72 V742 Lyr 2424801 >4070
>17.27 V742 Lyr 2428343 <504 >15.52 V742 Lyr 2430617 1105 >17.16 V742 Lyr
2432347 <533 >15.92 V742 Lyr 2433778 <621 16.10 V742 Lyr 2437851 >31 >14.96
V742 Lyr 2440390 >1570 >15.45 V742 Lyr 2443396 30 >14.71 V742 Lyr 2444100 1466
>15.08 V742 Lyr 2445873 <350 >15.08 V742 Lyr 2457253 <664 15.64 V742 Lyr
2458010 334 15.97 V742 Lyr 2458982 397 18.65 W Men 2413878 109 >15.78 W Men
2416378 479 16.00 W Men 2417966 <1500 >16.90 W Men 2422610 <2100 >16.27 W Men
2423466 >30 15.83 W Men 2425598 39 >16.96 W Men 2427807 >140 16.13 W Men
2430726 760 16.26 W Men 2443931 50 16.00 W Men 2445398 30 16.00 W Men 2447202
334 16.80 W Men 2450453 40 15.70 W Men 2451880 30 15.50 W Men 2454213 >410
>15.90 W Men 2456900 180 16.80 W Men 2457931 >240 >16.70 VZ Sgr 2415533 >220
13.72 VZ Sgr 2417361 >16 12.70 VZ Sgr 2418917 2595 14.99 VZ Sgr 2424655 144
14.01 VZ Sgr 2426064 635 15.28 VZ Sgr 2427305 >6 13.19 VZ Sgr 2428444 <18
12.74 VZ Sgr 2430823 122 13.73 VZ Sgr 2431338 50 15.38 VZ Sgr 2433071 <370
14.75 VZ Sgr 2446225 770 14.80 VZ Sgr 2446320 36 14.02 VZ Sgr 2446863 4 12.80
VZ Sgr 2447362 450 16.90 VZ Sgr 2448528 13 13.30 VZ Sgr 2449212 74 15.40 VZ
Sgr 2449301 26 13.30 VZ Sgr 2450609 195 15.20 VZ Sgr 2452160 116 14.30 VZ Sgr
2454712 440 14.20 VZ Sgr 2456157 89 13.90

Table 8: continued

RCB dips  RCB star JD Duration $B_{\rm min}$ VZ Sgr 2456384 283 14.91 V3795
Sgr 2418457 <640 13.54 V3795 Sgr 2428395 94 13.83 V3795 Sgr 2431619 385 13.82
V3795 Sgr 2441153 >410 14.40 V3795 Sgr 2444023 <140 13.60 V3795 Sgr 2448766
445 16.30 V3795 Sgr 2452781 49 >15.30 V3795 Sgr 2452933 146 15.00 V3795 Sgr
2453183 99 14.70 V3795 Sgr 2453479 173 >16.00 V3795 Sgr 2457064 441 18.10 Z
UMi 2413364 10 13.63 Z UMi 2414826 >5 13.53 Z UMi 2415925 >5 >15.55 Z UMi
2419092 <200 13.52 Z UMi 2422376 755 >14.76 Z UMi 2424945 <130 13.58 Z UMi
2425681 <400 >14.15 Z UMi 2428016 180 >15.19 Z UMi 2428520 240 >15.15 Z UMi
2428931 120 >15.72 Z UMi 2429457 184 >14.15 Z UMi 2430076 872 16.25 Z UMi
2431529 305 >14.82 Z UMi 2432021 230 >14.82 Z UMi 2432646 328 15.60 Z UMi
2433397 266 >15.28 Z UMi 2434120 <250 14.53 Z UMi 2440269 596 >15.44 Z UMi
2442490 598 >15.58 Z UMi 2443956 495 >15.01 Z UMi 2445208 150 >14.65 Z UMi
2446552 <50 >14.44 Z UMi 2454555 380 18.53 Z UMi 2455641 128 15.20 Z UMi
2456002 567 14.19 Z UMi 2457249 480 16.53 Z UMi 2458821 274 18.78 HV12842
2424409 1482 >16.40 HV12842 2427061 1845 >17.50 HV12842 2429620 <750 15.70
HV12842 2430611 <400 16.10 HV12842 2433979 420 >16.70 SP77 65-2 2413878 <2228
>17.00 SP77 65-2 2429703 1244 >17.50 SP77 65-2 2431850 1481 17.20 SP77 65-2
2433979 >400 >17.00

The dip properties have a variety of problems, even if we have a perfect light
curve with hourly sampling. There is no unique or standard means of
identifying a dip, with this being critical when two or more dips overlap. For
overlapping dips, I have no way to say whether it is one dip with a complex
structure or many dips for each minimum. And I have no good definition of the
duration, with many varied definitions possible. Further, a deep problem is
that dips return to maximum only asymptotically, and there is no good
criterion for knowing when the dip ends.

The dip properties have large problems because the light curves are not
perfect, being often sparsely sampled and the deepest magnitudes often being
limits. Many of the RCB star dips (but not for those of R CrB itself) have the
time of true minimum missed, and hence I only have a lower limit on $B_{\rm
min}$ and my date of minimum might be days or months different from the real
JD of faintest light. And the duration often can only have a poor upper or
lower limit. (For example, when I have two plates taken a day apart showing a
definite dip, but the closest-in-time plates are only in the preceding and
following observing season, should we call the duration as 1 day, $>$1day, or
$<$320 days? All these possible statements quote a numerical value that poorly
represent the real situation, and many of the durations have similar very
large uncertainties.) Further, the sparse light curves mean that I inevitably
miss many entire dips, hence making for a substantial distortion in the
statistics of the dip-to-dip interval distribution. My light curves have gaps
every year due to the stars’ conjunction with the Sun. Further, my light
curves suffer from the notorious Menzel Gap from roughly 1954–1969, caused by
the Harvard Director stopping the plate program so he could fund his own solar
research. And my fainter RCB stars have a further long gap from the end of the
Harvard plates in 1989 up to the start of the AAVSO light curve.

For the collected dips, I have calculated the median dip duration in days,
$\langle D\rangle$, and the median dip depth in magnitudes,
$\langle\delta\rangle$. The use of the median statistic is a reasonable way to
handle the many limits on the durations and depths. These are tabulated in
Table 6.

Table 9: Statistics on RCB dips RCB star | $\langle D\rangle$ | $\langle\delta\rangle$ | $N_{\rm lun}$ | $F_{\rm lun}$ | $N_{\rm season}$ | $F_{\rm season}$ | $\langle\Delta T\rangle$ | $P_{\rm lun}$ | $\langle T_{\rm HWHM}\rangle$
---|---|---|---|---|---|---|---|---|---
UX Ant | 329 | 3.2 | 391 | 0.05 | 90 | 0.08 | 2200 | 0.012 | 34
R CrB | 111 | 5.2 | 1877 | 0.29 | 176 | 0.52 | 800 | 0.030 | 19
V742 Lyr | 698 | 2.9 | 393 | 0.33 | 84 | 0.46 | 2500 | 0.020 | …
W Men | 140 | 1.9 | 302 | 0.06 | 66 | 0.15 | 2000 | 0.016 | 120
VZ Sgr | 122 | 3.0 | 392 | 0.12 | 84 | 0.26 | 1300 | 0.028 | 10
V3795 Sgr | 160 | 2.6 | 243 | 0.04 | 77 | 0.07 | 3500 | 0.010 | 370
Z UMi | 250 | 3.4 | 646 | 0.13 | 97 | 0.37 | 1300 | 0.022 | 62
HV 12842 | 750 | 2.3 | 93 | 0.29 | 41 | 0.34 | 2500 | 0.015 | …
IRAS 05374-6355 | … | … | 104 | 1.00 | 42 | 1.00 | … | … | …
SP77 65-2 | 1363 | 1.9 | 119 | 0.07 | 40 | 0.16 | 2000 | 0.030 | …

###  5.2 Dip Frequency

A reasonable way to quantify the dip frequency is to determine the fraction of
the time that the star spends inside a dip. This is trickier than it sounds
because the frequency of light curve points is highly variable and clustered
in time, and we need to avoid duplications and correlations. So I have adopted
a simple assignment that a single observation shows a dip if the star is more
than 1.5 mag fainter than the average maximum magnitude. Further, I give a
lunation number for every observation, then I tallied up the number of
lunations for which I have data ($N_{\rm lun}$), and also the fraction of
those lunations during which the star is seen to be in a dip ($F_{\rm lun}$).
For another measure of dip frequency for a longer time-scale, I have assigned
each observation to a specific yearly observing season, then I have tallied
the number of observing seasons with observations for each RCB star ($N_{\rm
season}$), and calculated the fraction of those seasons that contain at least
one observed magnitude more than 1.5 mag below maximum ($F_{\rm season}$).
These numbers and fractions are in Table 6.

The $N_{\rm lun}$ value varies from 93–1877 lunations. With 12.37 lunations
per year, the cumulative coverage of the individual stars ranges from 7.5
years (for the faint HV 12842) to 152 years (for the bright R CrB). For R CrB,
we see near 84 per cent coverage from 1843–2023, and this usually goes through
the solar gaps (as R CrB is far above the ecliptic). We see large variations
in the dip fraction for lunations, ranging from 0.04 for V3795 Sgr, to 0.29
for R CrB, to 1.000 for IRAS 05374-6355. As for dip statistics by observing
season, I cover from 40 (for SP77 65-2) to 176 years (for R CrB). The dip
fractions for seasons are always larger than for lunations (by factors of up
to 2.8), because the seasons with a dip often contain lunations with no dip.
The fraction of observing seasons with dips ranges from 0.07 (V3795 Sgr) to
0.52 (for R CrB) to 1.00 (for IRAS 05374-6355).

###  5.3 Average Dip-to-Dip Time

What is the average time from dip-to-dip, $\langle\Delta T\rangle$? This is a
poorly defined question, even for a perfect light curve, because overlapping
dips cannot be easily distinguished, and it is unclear whether closely spaced
dips should have their small dip-to-dip times included in the average. And
$\langle\Delta T\rangle$ is not easy to measure because of the ubiquity of
gaps in all light curves. For the dips listed in Table 5, I can calculate a
problematic median value of $\Delta T$, as listed in Table 6.

To determine the underlying $\Delta T$ distribution with the presence of gaps
and overlaps, I have simulated a full observational history from 1889–2023,
complete with seasonal gaps, the Menzel gap, and other observational gaps. For
every lunation, the model decides whether a dip starts as based on a single
probability, $P_{\rm lun}$, then the simulation lays out a dip of some given
duration, and the availability of observations is checked. If two dips are
‘seen’ to overlap in some time interval, then the lunation is still counted
simply as being one dip. For each instance of randomly chosen event start
times, a list of ‘observed’ dips is tabulated, and a $\Delta T$ distribution
is created. The specific $\Delta T$ for each dip-to-dip interval is the number
of lunations between the two starts multiplied by 29.5 days. The underlying
average $\Delta T$ is just 29.37/$P_{\rm lun}$. I construct such distributions
for each single instance of randomly chosen event times, and I construct many
such random instances so as to see the typical variations in the model values
from instance-to-instance. This schematic model can give the $\Delta T$
distribution of dips while accounting for the effects of gaps and overlap, all
within a case for the completely random occurrence of dips.

Within this schematic model, I can look at the shape of the $\Delta T$
distribution. First, I model the case for R CrB alone, with no gaps, and a
mean dip duration of 4 lunations. I adopt $P_{\rm lun}$=0.030, which gives an
underlying $\langle\Delta T\rangle$=980 days. With this, the typical model
distribution gives a median $\Delta T$ around 820 days, while the range for
the central 68.3 per cent is from 300–1600 days. There is substantial
variation from instance-to-instance due to small number statistics. The full
range of $\Delta T$ is from 148 days (i.e., five lunations) to 3800 days.
These model numbers are identical (to within the instance-to-instance
variations suffered by reality) to those observed for R CrB. That is, my
simple model with random timing for dust ejection events is a good mimic of R
CrB’s dip times.

I have used the model to calculate $P_{\rm lun}$ values for each of my sample
of RCB stars (see Table 6). For all of the stars, the model (with random dip
times) finds consistency with the observed $\langle\Delta T\rangle$ as well as
with the observed range of $\Delta T$ values. I see no unexplained upper or
lower cutoffs in the observed $\Delta T$ distribution, I see no evidence of a
preferred inter-dip time interval, and I see consistency with random dip times
for time-scales longer than several dip durations.

###  5.4 Are the Dips Random in Time?

One question is whether the RCB dips occur randomly in time? In the past, much
research has gone into addressing this question, as reviewed by Clayton
(1996). The conclusion is that the dips occur at ‘irregular intervals’. This
should not be surprising because a randomness of the direction of ejection for
the dust would effectively mask any underlying periodicity. Further, there is
evidence that the dust ejection occurs only during a restricted phase of the
RCB pulsational cycle. Such would have the dust ejection tied to particular
conditions in the outer envelope of the star that vary with the pulsation. But
this evidence is weak, contradicted by other data for the same stars, and for
one RCB requires the simultaneous use of three periodicities.

The randomness-or-periodicity can be tested by looking at the distribution of
dip-to-dip intervals, $\Delta T$. A periodicity should appear with $\Delta T$
as a constant for each RCB star, or possibly as multiples of the period. A
quasi-periodicity would produce a broadened peak, possibly with further peaks
at multiples of the average period. Random dust ejections will produce a
Poisson distribution of $\Delta T$. In practice, for all my target stars, the
observed $\Delta T$ distributions do not display any singular value, nor a set
of integer multiples of some period, nor with any apparent broadened peaks.
That is, I am not seeing any evidence for periodicities or quasi-
periodicities. Rather, the distribution of $\Delta T$ is always consistent
with a Poisson distribution. With allowing for gaps and overlaps, my model
results (see the previous Section) shows that the dust-ejection times are
random in time. So this is all just confirmation that RCB stars have random
dust ejection times, at least on time-scales longer than the dip duration.

Is the dust ejection random for time-scales of less than the dip duration? Or
are the ejections clustered together, or are they periodic on the pulsation
period? I cannot test the connection to any pulsations, because ephemerides
for pulsation phase are generally not known and the pulsations change in
period and they come-and-go with no discernible pattern. For short-term
clustering, the problem is to recognize singular dips that overlap. In
practice, useable statistics can only be had for R CrB itself, because only
this star has a high-cadence long-term high-accuracy light curve. In the
wonderful R CrB visual light curve, successive overlapping dips can be
recognized by their local minima in brightness. For example, the long and
complex R CrB dip from 2007–2017 shows 11 local minima, indicating that this
is composed of 11-or-more singular dips.

The R CrB light curve distinctly shows that the singular dips are strongly
clustered. For example, the 2000.0–2020.0 (7305 days) light curve has 14 local
minima, of which 11 are in a central 3510 day interval. The probability of
random dip times being so clustered is of order 0.02. For the previous
interval of 1990.0–2000.0 (3652 days), there are long stretches of R CrB at
maximum light, punctuated by 4 episodes, lasting a total of 595 days, which
contain 15 dips. The probability of such clustering resulting from random
event times is vanishingly small. The point is that R CrB has its singular
dips clustered on time-scales of the dip duration. That is, if R CrB has a
single dust ejection event, then the probability is relatively high that it
will soon have another dust ejection event, and so on, making for multiple
overlapping singular dips that appear as one long and complex dip. This short-
term clustering is independent of the long-term randomness.

###  5.5 Well-observed Isolated Dips and the Depth/Duration Correlation

The durations in Table 5 have a variety of problems for purposes of looking at
the shapes of dips. The primary problem is that multiple dips get their
durations concatenated together. This could be either due to separate singular
dips running together so as to appear as one long dip, or poor sampling of the
light curve can make for large errors in the tabulated durations. So if we are
looking for the statistics involving the individual shapes of dips, we should
come up with a pristine sample of well-observed isolated singular dips. For
this, I have looked through all my light curves, and identified 18 such dips,
as tabulated in Table 7.

Table 10: Well-Observed Isolated Singular Dips RCB star | $JD_{\rm min}$ |  $\delta$ (mags) |  $T_{\rm HWHM}$ (days) |  $D$ (days)
---|---|---|---|---
UX Ant | 2431108 | 2.48 | 34 | 29
R CrB | 2399793 | 4.62 | 17 | 26
R CrB | 2420729 | 2.62 | 19 | 18
R CrB | 2422768 | 3.32 | 8 | 8
R CrB | 2422973 | 3.71 | 19 | 22
R CrB | 2424070 | 2.85 | 13 | 12
R CrB | 2442095 | 5.31 | 61 | 82
R CrB | 2446347 | 4.32 | 21 | 32
R CrB | 2446499 | 1.29 | 19 | 0
R CrB | 2447778 | 2.33 | 17 | 13
R CrB | 2451904 | 6.52 | 19 | 30
W Men | 2456900 | 2.30 | 120 | 70
VZ Sgr | 2446860 | 0.93 | 15 | 0
VZ Sgr | 2448534 | 1.20 | 5 | 0
V3795 Sgr | 2428360 | 1.23 | 370 | 0
V3795 Sgr | 2431800 | 1.39 | 800 | 0
V3795 Sgr | 2448798 | 2.23 | 220 | 240
Z UMi | 2455705 | 3.09 | 62 | 57

For each of the isolated dips, I tabulate four statistics. The second column
of Table 7 lists the JD of the minimum. The next column gives the depth of the
minimum, in magnitudes below $B_{\rm 1950}$. The next column gives the half-
width half-maximum (HWHM) of each dip, with $T_{\rm HWHM}$ being the time
interval from the observed minimum until the brightness has recovered to half
way between $B_{\rm min}$ and $B_{\rm 1950}$. Operationally, this is easy to
get from the light curves. The last column in Table 7 gives the duration, $D$,
with a slightly different definition from that used in Tables 5 and 6. Here, I
take $D$ to be the time from the minimum until the brightness is within 1.5
magnitudes of the average maximum magnitude. This changed definition is done
to isolate the physics of the light curve rise after the minimum (involving
the dispersion of the dust clouds) from the different physics of the light
curve decline before the minimum (involving the formation of the dust).

The median $T_{\rm HWHM}$ values in Table 7 varies substantially from star-to-
star, with $\langle T_{\rm HWHM}\rangle$ tabulated in the tenth column of
Table 6. The only RCB star with many well-observed isolated dips is R CrB,
with a median HWHM duration of 19 days. The values vary from 10 days (for VZ
Sgr) to 370 days (for V3795 Sgr), over a range with a factor of 37$\times$ in
size.

With sparse and imperfect data, it appears that the individual dips have
$T_{\rm HWHM}$ that are approximately a constant for each star, at least as
compared to the whole 37$\times$ range of star-to-star variation. The best set
of HWHM durations is for R CrB, where all-but-one have an RMS scatter of 24
per cent. The one outlier (at 61 days) has an easy explanation as possibly
being a composite of multiple dips that run together so as to not produce
multiple local minima in brightness. VZ Sgr has only two measures that can be
expressed as having an RMS scatter of 7 days, which is small, or expressed as
3$\times$, which is still small compared to the overall 37$\times$ range.
V3795 Sgr has three measures, all greatly longer than any others, extending
over a range of 3.6$\times$, which is small compared to the overall range of
37$\times$. With this weak evidence, it appears that RCB isolated dips have a
relatively constant duration, albeit with substantial scatter.

If dips for each RCB has constant $T_{\rm HWHM}$, then the dip shape must be
similar for each star. This is easily confirmed by eye when looking at many
individual dip light curves, where they all share the same fast rise, a
rounded minimum, followed by a fast brightening that changes to a slow
brightening, to asymptotically return to the pre-dip level. The time-scale
$T_{\rm HWHM}$ will be governed by the initial position, velocity, and
acceleration for the ejected dust, and this will be primarily a function of
each star’s radius and luminosity (i.e., its position on the H-R diagram).
With this, the dip shape will not change from dip-to-dip.

A second way to look at the dip shape is to seek a correlation between the dip
depth and duration for isolated dips. Only R CrB itself has useful data. For
this, there is a highly significant correlation, where $\delta$ is roughly a
linear function of $D$, with only modest scatter. The existence of such a
function implies that the shape for the R CrB isolated dips is roughly a
constant from dip-to-dip. That is, for dips light curves normalized by the
depth, the $\delta$ versus $D$ function requires some specific dip shape.

###  5.6 Correlations

I have measured intrinsic dip characteristics ($\langle D\rangle$,
$\langle\delta\rangle$, $P_{\rm lun}$, and $\langle T_{\rm HWHM}\rangle$) for
my ten RCB stars. And Table 1 has collected and calculated $M_{\rm B}$,
$T_{\rm eff}$, and $\log[\epsilon(H)]$, with these representing the position
on the HR diagram and the composition. I have sought significant correlations
between these various properties.

I have found no convincing correlations. I could point to
$\langle\delta\rangle$ versus $M_{\rm B}$, with the most luminous RCB stars
having low median dip-depth, but this correlation does not pass any of the
usual tests of significance.

I do not recover Jurcsik’s correlation between hydrogen abundance and the
frequency of dips. Unfortunately, all-but-one of my targets have middling
abundances covering only a small range of $\log[\epsilon(H)]$. The one
exception is V3795 Sgr, with very low hydrogen abundance and the lowest
$P_{\rm lun}$ value, as recognized by Jurcsik and supporting his correlation.
However, the close-second-lowest $P_{\rm lun}$ value is for UX Ant with a
middling hydrogen abundance. Further, the two highest $P_{\rm lun}$ values are
stars (R CrB and VZ Sgr) that are of middling hydrogen abundance. So the best
that I can say is that the correlation has large scatter.

Jurcsik’s correlation has the dip frequency dominated by the hydrogen content
of the atmosphere. However, this cannot be the entire determinant of the dip
frequency because DY Cen changed from an ordinary RCB star with frequent dips
in the year 1906, when the dips turned off as the star heated up to 7500 K in
1932, and the dips have remained off up until 2023 with DY Cen having a
surface temperature of 24,800 K; see Schaefer (2016). So the star’s surface
temperature must also play a dominant role in dip frequency. To follow this
up, I have sought a correlation between $T_{\rm eff}$ and $P_{\rm lun}$. The
stars with the two low $T_{\rm eff}$ values (V742 Lyr and Z UMi) have middling
$P_{\rm lun}$ values. The two stars with the highest $P_{\rm lun}$ values (R
CrB and VZ Sgr) have the same temperature as the two stars with the lowest
$P_{\rm lun}$ values (UX Ant and V3795 Sgr). So there is no correlation
between $T_{\rm eff}$ and $P_{\rm lun}$, at least for ordinary cool RCB stars.

I have also sought correlations with the measured properties of individual
dips on each RCB star, as taken from Table 5. For this, I do find a strong
correlation between duration and depth for the RCB stars with a substantial
number of useable dips. This correlation is highly significant, but also with
large scatter. This is always in the direction that deep dips have long
durations. Empirically, this can be represented as
$D\propto\exp(\delta/\langle\delta\rangle)$. Notably, this correlation is for
all dips, most of which are composite, being composed of many isolated dips
superposed to form a complex shape. The obvious interpretation is that
episodes with a long series of dust ejections will result in a long duration
for the complex dip, while the piling up of the dust from the many ejections
will result in a deep depth.

##  6 Light Curve Shape for Isolated Dips

A perusal of close-up light curves for RCB dips shows a wide range of dip
shapes. But it also shows that the long complex dips look to be composites of
multiple short dips. Indeed, when these contributing dips can be separated
out, the component singular dips appear to have the same shape as each other
and the same shape as isolated dips. The component dips all have widely
varying depths, but the shapes appear to be similar in all cases. The weak
depth/duration function for R CrB points to a fairly constant dip-shape. This
can be investigated further by collecting and normalizing the isolated dip
light curves from Table 7. Importantly, the dip shape can be derived from
standard theory, and then compared to the reality of the light curves.

###  6.1 Light Curve Shape for Isolated Dips

The intrinsic shape of dips should be best seen by overplotting normalized
isolated dip light curves. With this, we can see the shape of the minimum, to
determine whether the dip light curve is flat immediately following the time
of deepest minimum. We can also see how variable is the light curve shape from
dip-to-dip. And we can get an averaged dip shape for comparison with
theoretical models.

How should the dip light curves be normalized? First, the dip light curve
should be shifted in time so that the time of deepest observed magnitude is at
zero time. Second, the time axis should be scaled by the $\langle T_{\rm
HWHM}\rangle$ for each RCB star (as tabulated in Table 6). In practice, this
scaling will have $(t-t_{0})(19/T_{\rm HWHM})$ for the horizontal axis, with
$t-t_{0}$ being the time with respect to the time of the deepest minimum. I
have chosen the factor 19, in days, so that the scaling is unity for R CrB.
Third, the magnitude should be shifted so that the maximum light is set to
zero. Fourth, the magnitude scale should be normalized so that the minimum is
at 1.0 for that particular dip. The vertical axis will be $(B-B_{\rm
1950})/\delta$, stretching from near 1.0 at the bottom to 0.0 at the top.

I have taken the eleven dips for R CrB from Table 7, scaled the rises from
minimum, and overplotted these in Fig. 3. The scaling involved is to plot
$(m-6.063)/\delta$ versus $(t-t_{0})(19/T_{\rm HWHM})$ in days, with $T_{\rm
HWHM}$=19 days for R CrB. We see that all eleven scaled light curves closely
match each other. The accuracy of this shape can be quantified as being around
0.05 for times of 10, 30, and 40 days in Fig. 3. This is dominated by the
expected measurement error, so the underlying accuracy of the shape model must
be substantially smaller than 0.05. This is a direct demonstration that the R
CrB isolated dips all have the same shape.

Figure 3: Overplot of eleven well-observed singular isolated dips for R CrB.
The light curves after the time of minimum are scaled in time so they all have
a 19 day $T_{HWHM}$ and are scaled vertically to be for a dip with a depth of
one magnitude. Importantly, we see that all eleven dips follow the same shape.
We see that the recovery to maximum light starts out slowly (with a shallow
slope in the first few days), then the recovery speeds up, with a
substantially steeper slope after ten days or so. This acceleration in the
brightening is consistent with the dust cloud starting out with near-zero
velocity and being immediately accelerated to high velocity by the star’s
radiation pressure. Further, Equation 14 has been used to give a theoretical
model for the brightening due to the expansion and dilution of the dust shell,
and this model light curve is shown as the black curve. This black curve is
mostly covered by the individual data points along the centre of the light
curve, and this is to say that the simple physics provides an excellent
description of the ‘universal’ dip shape. Figure 4: Overplot of eight well-
observed singular isolated dips for UX Ant, W Men, VZ Sgr, V3795 Sgr, and Z
UMi. The light curves are scaled and overplotted light curves for the dips in
Table 7, other than for those of R CrB itself, which appear in Fig. 3. The
construction, scales, and model curve of this plot are the same as in Figs 3
and 5, so these figures can be directly compared. I am disappointed that these
are the best data for singular isolated dips (other than those for R CrB),
with the figure showing substantial scatter. Despite the scatter, we see that
the dips for the other RCB stars follows the shape of both the R CrB dips and
the physics model. While being poorer as a test than in Fig. 3, we still have
a confirmation that dips from many RCB stars all apparently have the same
shape.

I have constructed a similar plot (see Fig. 4) for the eight remaining dips in
Table 7. The scatter is much larger than for R CrB, emphasizing that even the
best data in the world for all other RCB stars are much poorer than for R CrB
itself. In Figure 4, I see a number of individual outlier points, and in all
cases these are significantly different from many other observations of the
star taken at the same time, so these are likely some form of observational
error. Part of the problem is also that half of these dips have small depth,
so ordinary measurement errors get magnified on my plot. This overall scatter
looks poor, making for a relatively poor test of the constancy of the dip
shapes. Fortunately, there are many magnitudes, so the average light curve
shape is accurately measured.

Importantly, the average dip shape is the same in Figs 3 and 4. With the dip
shape being the same for all the dips of all seven RCB stars, this shape
appears to be of a general nature, and perhaps even ‘universal’. This is the
shape that should be used for general application, and the shape that needs a
theory explanation.

Importantly, the light curve shape is flat for the first few days after the
deepest minimum. This will be critical for knowing the initial acceleration of
the newly formed dust.

###  6.2 Dip Shape from Theory

Let us derive the theoretical shape in the light curve of the recovery from a
dip. The rapid decline of the dip is separate physics, relating to how fast
the dust forms. At some time, the dust formation is essentially complete, and
that corresponds to the minimum. After that, the ordinary ejection of the dust
cloud makes for a geometric dilution of the extinction, as the dust is spread
over a larger area, extending outside the surface of the star. The primary
force ($F$) acting on each dust particle is from the radiation pressure, which
will have the form

$F=kLR^{-2},$ (1)

see Equation 3 of Clayton, Geballe, & Zhang (2013). Here, $L$ is the star’s
luminosity, $R$ is the distance of the dust from the star’s centre, and $k$ is
some constant depending on the dust size and composition. There will also be a
force from gravity, but this will always be negligibly small, so it only acts
to make the effective value of $k$ slightly smaller. Now, the $R$ for the dust
comes from the usual application of Newton’s Law, so we get a non-linear
differential equation for the motion;

$M_{\rm dust}\times d^{2}R/dt^{2}=kLR^{-2}.$ (2)

Here, $M_{\rm dust}$ is the mass of the dust particle. The initial conditions
are that the dust is at radius $R_{0}$ with outward velocity $V_{0}$ at time
$t_{0}$. With the solution for this equation, we can calculate the
characteristic changes in the density of the expanding dust cloud, the changes
in the extinction of the starlight, and thus the shape of the light curve.

The density of the dust in the cloud is $\rho_{0}$ at $t_{0}$ and $\rho$ at
time $t$. The usual geometric dilution from expansion makes for

$\rho=\rho_{0}(R/R_{0})^{-2}.$ (3)

The optical depth of the dust cloud covering the star is

$\tau=\rho/C,$ (4)

where $C$ is a constant depending of the dust size and properties, as well as
the thickness of the dust shell. In a form involving magnitudes, Bouguer’s Law
relates the un-extincted magnitude of the star, $m_{0}$, with the observed
magnitude as a function of time, $m_{\rm obs}$ as

$m_{\rm obs}=m_{0}+1.086\tau.$ (5)

With this equation, we can calculate the brightness of the RCB star as a
function of the radial position of the dust, and hence as a function of time.

Unfortunately, Equation 2 does not have any simple closed-form for its
solution. Instead, I offer three solutions. The first solution is for the case
where the acceleration is negligibly small (i.e., for $k$=0). The second
solution is for the case where the expansion is dominated by the acceleration
at the early times (i.e., for $V_{0}$=0 and the acceleration is effectively a
constant over the critical time interval). The first and second solutions
represent limiting cases that span the behavior. The third solution is where I
numerically solve Equation 2, optimizing the parameters to match the observed
light curve shape from Fig. 3.

The first solution is where the acceleration is negligible. The cloud will be
moving outward with its original ejection velocity of $V_{0}$, and will keep
coasting at this velocity. We can define a timescale for the zero-acceleration
case as

$T_{0}=R_{0}/V_{0}.$ (6)

Putting these equations together, we get a result describing the shape of the
recovery after the dip minimum;

$m_{\rm obs}=m_{0}+(1.086\rho_{0}/C)[1+(t-t_{0})/(R_{0}/V)]^{-2}.$ (7)

The radius of the dust cloud from the centre of the star at a time $t$ after
the minimum time $t_{0}$ is

$R=R_{0}[1+(t-t_{0})/T_{0}].$ (8)

With this, the shape of the recovery follows a very specific equation, which
depends only on the timescale $T_{0}$. The depth of the minimum, $\delta$, is
$m_{\rm obs}-m_{0}$ for $t=t_{0}$, and depends only on the original dust
density and properties. We then get the general equation for single isolated
dips (after the fast fading to minimum) of

$m_{\rm obs}=m_{0}+\delta[1+(t-t_{0})/T_{0}]^{-2}.$ (9)

So here is a simple equation giving the universal shape for the after-minimum
dip for the zero-acceleration case. We can relate the observed timescale
$T_{\rm HWHM}$ to the theoretical timescale $T_{0}$ by noting that $m_{\rm
obs}-m_{0}=\delta/2$ when $t-t_{0}=T_{\rm HWHM}$ to get

$T_{0}=2.41T_{\rm HWHM}.$ (10)

Further, we can derive the duration-depth relation by requiring that $m_{\rm
obs}-m_{0}=1.5$ mag when $t-t_{0}=D$,

$D=T_{0}[(\delta/1.5)^{0.5}-1].$ (11)

So the limiting case with no acceleration has a simple analytic solution for
the shape.

This is a very specific shape that should be universal (for the case of
negligible acceleration). The depths $\delta$ can change from dip-to-dip, and
the timescale $T_{0}$ can change star-to-star. Complicated dip shapes come
simply from the superpositions of multiple dips, each formed by ejecta with
different $t_{0}$ times.

This universal shape will have a variety of complications. For example, there
will be some small distortions if the dust cloud at minimum does not cover the
entire star. Further small effects will arise from the shifts of the star
color as the dust clears. And it is unclear at to when to take the time
$t_{0}$, with small variations in the model predictions. Further, the
underlying star has pulsations with typical amplitudes of a quarter of a
magnitude superposed on the dip shape. All these effects are much smaller than
the depth of the dips, with such distortions within the measurement errors.

The first solution (with $k$=0) corresponds to the scenario where the dust
forms far outside the star’s atmosphere, where the local gas has cooled down
to the dust formation temperature. The second solution (with $V_{0}$=0)
corresponds to the scenario where the dust forms in the atmosphere of the
star, whereupon the star’s radiation rapidly accelerates the dust to high
velocities. There had long been a tension between these two scenarios, but
this has been decided, with the dust formation from $V_{0}$=0 gas in the
star’s atmosphere (Clayton et al. 1992, Clayton et al. 2013).

Now let us take the other extreme limiting case, where the initial velocity is
zero. Further, let us take the acceleration to be a constant $A$, as is nearly
the case where it matters only for some relatively narrow range of $R$. We can
define a timescale as

$T_{A}=(2R_{0}/A)^{0.5}.$ (12)

In this limit, we have

$R/R_{0}=1+[(t-t_{0})/T_{A}]^{2}.$ (13)

Following the derivation for the first solution, we get

$m_{\rm obs}=m_{0}+\delta\\{1+[(t-t_{0})/T_{A}]^{2}\\}^{-2}.$ (14)

Like for Equation 10, we can derive the duration-depth relation as

$D=T_{A}[(\delta/1.5)^{0.5}-1]^{0.5}.$ (15)

This equation adequately fits the observed depth/duration data for R CrB. The
time-scales will be related as

$T_{A}=1.55T_{\rm HWHM}.$ (16)

So now we have a ‘universal’ shape for all singular dips after the minimum.
The two free shape parameters are the depth ($\delta$) and the timescale
($T_{A}$). So for the limiting case where the acceleration dominates, we have
a somewhat different equation that defined the shape of the dip recovery.

The model light curve shape for the first scenario (Equation 9) and for the
second scenario (Equation 14) are plotted in Fig. 5. Critically, the two
scenarios predict greatly different shapes. The first scenario predicts that
the light curve rapidly brightening immediately after the minimum, and this is
definitely rejected by the data in Figs 3 and 4. The second scenario predicts
a flat light curve in the first few days after the minimum, followed by a
steep brightening, and this is exactly what is seen in Figs 3 and 4. So we
have a confident empirical confirmation that the second scenario is correct,
while the first scenario is rejected.

Figure 5: Model dip shapes. The timescale for all three models have been
adjusted so that $T_{HWHM}$=19 days, and so these models are directly
comparable to the data in Figures 3 and 4. The green curve (that is the lowest
curve on the right side of the plot) is for the first scenario of zero
acceleration (Equation 9), and has no relatively-flat initial rise. The red
curve (highest on the right side) is for the second scenario (Equation 14)
where the initial velocity is zero and the acceleration does not change
throughout. This initial-low-velocity case has a relatively flat initial rise,
as the dust is accelerated from low to high velocity by the star’s radiation
pressure. The blue dotted curve (in the middle on the right, just below the
red curve) is for the full numerical solution of Equation 2. This case has the
parameters $V_{0}$=0 km/s and $R_{0}$=75 R⊙, as appropriate for the dust being
formed in the outer stellar atmosphere, only to be accelerated away from the
star by the radiation pressure. These particular model parameters were
selected so that the initial rise is fairly flat, as seen in the data (Figures
3 and 4). The two curves for the zero-initial-velocity cases are quite close,
only differing slightly in its late-time asymptotic approach to maximum. This
difference is effectively not observable due to the ordinary variations of the
RCB star, for example from ordinary pulsations that can reach up to half a
magnitude in amplitude. This lack of significant difference shows that the
drop of acceleration as the dust gets far away from the star has little
consequence for the light curve shape. This also means that the simple
Equation 14 is an adequate description of the light curve shape.

The first solution (with $k$=0) corresponds to the scenario where the dust
forms far outside the star’s atmosphere, where the local gas has cooled down
to the dust formation temperature. The second solution (with $V_{0}$=0)
corresponds to the scenario where the dust forms in the atmosphere of the
star, whereupon the star’s radiation rapidly accelerates the dust to high
velocities. There had long been a tension between these two scenarios, but
this has been decided, with the dust formation from $V_{0}$=0 gas in the
star’s atmosphere (Clayton et al. 1992, 2013).

Now let us take the third solution, where Equation 2 is solved numerically.
For this, we have to specify particular values within the model. The only
parameters that matter for the shape are $R_{0}$, $V_{0}$, and $A$. With R CrB
itself having a stellar radius of somewhat less than 100 R⊙. The initial
velocity might be near-zero if the dust simply forms in the atmosphere (with
$R_{0}$$\sim$75 R⊙), or it might be comparable to the RCB escape velocity
($V_{0}$$\lesssim$100 km/s) if the dust forms in some ejected gas (perhaps
around R⊙$\sim$200 R⊙). The acceleration $A$ is then varied until $T_{\rm
HWHM}$ matches the observations. With these initial conditions, the numerical
solution of of Equation 2 is easy, and the light curve is derived from the $R$
as a function of time with the use of Equations 3–5. Thus, for this exact
solution of Equation 2, we get a single unique function for the shape of the
dips.

I have optimized the initial parameters ($R_{0}$, $V_{0}$, and $A$) to match
the observed light curves in Figs 3 and 4. Critically, the flat light curve in
the first few days after minimum requires that $V_{0}$ must be near-zero. In
practice, the observed flat light curve immediately after the deepest minimum
is predicted only for cases with $V_{0}$$<$10 km s-1. The $R_{0}$ and $A$
values mainly matters only with their ratio, forming $T_{A}$, with this being
determined by the HWHM time-scale. So the third solution is set, with
$V_{0}$=0 and the other initial parameters set to reproduce the observed
$T_{\rm HWHM}$. This numerical solution to Equation 2 is presented as the blue
dotted curve in Fig. 5.

The primary difference amongst these three cases is that the initial rise can
be linear or parabolic. That is, the recovery to maximum can start out with
some relatively fast and unchanging rate of brightening, or the recovery can
start out slowly and accelerate over time to a steeper and steeper light
curve. From the physical models, the initial rise will be linear when the dust
starts out with some substantial velocity (like $>$10 km/s). The initial rise
will be fairly flat when the initial velocity is low, because the dust cloud
remains at nearly the same distance (and hence the same extinction) for a
length of time while the acceleration from radiation pressure accumulates
until the dust is moving outward fast and thinning out fast. We see that the
second and third case have very similar dip shapes. The cases only differ in
that Equation 14 makes the implicit assumption that the acceleration is a
constant throughout the dip, while the numerical solution correctly has the
acceleration falling as in Equation 2. That these two solutions give a similar
light curve just goes to show that the acceleration at late times is not
important for the light curve.

The close similarity of the zero-initial-velocity solutions also means that,
in practice, we can use the simple formula in Equation 14 to model the light
curve shape after the minimum, rather than use some numerical solution that
cannot be readily expressed except numerically in specific cases. That is,
Equation 14 provides an adequate physical model in a simple and convenient
form.

The observations in Fig. 3 show that the rise starts fairly flat and then
rapidly steepens. This is a consequence of the substantial acceleration of the
dust when it is first formed near the star. The initial flat section of the
light curve is not present if the initial velocity is much greater than 10
km/s, or if the acceleration is too low. Indeed, we take this result as
substantial evidence that the dust must start out with a very slow expansion
velocity and be rapidly accelerated by the starlight’s radiation pressure.

Equation 14 works for RCB stars of all luminosity and surface temperature, and
for all levels of dust formation. The end result is a realistic physical model
for the shape of the RCB dips after the initial fast decline, and the shape is
expected to be universal. The critical evidence for this model is that all the
observed dip shapes for isolated dips closely fits Equation 14 (see Figs 3 and
4).

##  7 Conclusions

I have constructed century-long light curves for ten RCB stars, using 323,464
magnitudes from the Harvard plates and from the AAVSO database. These are all
calibrated into a modern magnitude system, consistent throughout the century.
I have used my century-long light curves to search for secular variations in
the brightness at maximum light, as well as to get statistics on the
properties and shapes of the dips.

I extracted the magnitudes for the RCB stars for times when they were
certainly at maximum light, far from any dip. These were then binned in time
to get nine light curves of the stars at maximum. These light curves show no
significant secular evolution, with variations at a rate of typically $<$0.10
mag per century.

The lack of secular evolution is perhaps surprising, when seen from the point
of view that some fraction of the RCB stars must be arriving or departing from
the RCB region on the HR diagram over a century time. Recall that the Hot RCB
stars all show fast evolution in surface temperature resulting in brightness
changes around 1 magnitude per century (Schaefer 2016). And in particular, the
now-hot RCB star DY Cen was observed to be brightening at a rate of 2.9
magnitudes per century, back at a time when $T_{\rm eff}$=5800 K. The
confident example of DY Cen demonstrates that at least some cool-R CrB stars
are brightening very fast. However, my sample includes some of the hottest of
the cool-RCBs and some of the most luminous of the cool-RCBs, which are those
expected to have the fastest evolution, yet they show no brightness change on
a time-scale of over one century.

I have used the light curves to extract out the dips and their measured
properties. I find only one useful correlation, and that is that the well-
observed isolated dips of R CrB itself shows a highly significant relation
between the dip depth and duration.

The shapes of RCB dips are often long and complex, with these apparently
composed of multiple singular dips all run together. In many of these cases,
we can see that the complex dips are composites of singular dips of the same
shape, with this shape being the same as for isolated dips. So apparently,
each individual dust ejection leads to a dip of the same shape, with these
often superposed to appear as a long complex dip. The general dip shape can be
seen from the well-observed isolated events. With appropriate scaling, the dip
shape appears to be the same for all dips from all RCB stars with good light
curves. Critically, this shape is closely flat for the first several days
after the time of minimum light. Empirically, this shape is the same for all
the isolated dips to an accuracy of roughly 5 per cent.

Theoretically, the shape of RCB dips can be derived from the acceleration of
the dust and the thinning out of the dust cloud. Critically, the predicted
slope of the light curve immediately after minimum depends closely on the
initial velocity of the dust on formation, where an initially-flat light curve
implies an initial velocity near zero. When combined with the empirical result
that light curve is initially flat, I derive that the initial velocity is
$<$10 km s-1. This rejects the scenario where the dust formation occurs far
from the star from gas that is coasting at high velocity. This result confirms
the scenario where the dust formation occurs in zero-velocity gas that is the
outer atmosphere of the RCB star.

In the end, I have an empirical and theoretical shape for isolated singular
dips with the equation $m_{\rm obs}=m_{0}+\delta[1+(t-t_{0})/T_{0}]^{-2}$.
$T_{0}$ and $m_{0}$ should be a constant for a given RCB star, while $\delta$
and $t_{0}$ will vary from dip-to-dip. In principle, this equation should be
applicable to all RCB dips, regardless of the star’s luminosity, temperature,
or composition, and regardless of amount of dust formed. And empirically, this
equation applies to all well-measured isolated dips from many RCB stars. With
this, we can think that the dip shape is universal for singular dips.

##  8 ACKNOWLEDGEMENTS

I thank G. Clayton (Louisiana State University) for valuable discussions on
this paper. The American Association of Variable Star Observers (AAVSO) has
provided much that was required for all of the the observers for my program,
including finder charts, comparison star magnitudes (through the APASS
program), programatic advice, and a data archive. Funding for APASS has been
provided by the Robert Martin Ayers Sciences Fund. The DASCH data from the
Harvard archival plates was partially supported from National Science
Foundation grants AST-0407380, AST-0909073, and AST-1313370.

##  9 DATA AVAILABILITY

All data are publicly available, either in the cited literature, the cited
public databases, or in the tables.

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