Zwicky Transient Facility and Globular Clusters: The Period–Luminosity and Period–Wesenheit Relations for Type II Cepheids

The evolved and low-mass Type II Cepheids (TIICs; for a general review, see Welch 2012) are one of the old population distance indicators. Similar to the young Type I or classical Cepheids, TIICs also exhibit a period–luminosity (PL; or the Leavitt law) relation. However, TIICs are ∼2 mag less luminous than the classical Cepheids. Nevertheless, TIICs are a few magnitudes more luminous, depending on the pulsation periods and filters, than the popular RR Lyrae—another old population distance indicator. Therefore, TIICs are useful to probe a more distant stellar system (such as dwarf galaxies and elliptical galaxies) independent of RR Lyrae stars. The comprehensive reviews on TIICs as distance indicators can be found, for example, in Wallerstein (2002), Sandage & Tammann (2006), Beaton et al. (2018), and Bhardwaj (2020, 2022).

Some of the earlier derivations of the BVI-band, or a subset of these filters, PL relations for TIICs can be found, for example, in Demers & Wehlau (1971), Nemec et al. (1994), Alcock et al. (1998), and Pritzl et al. (2003). Other works on the optical PL relations included a color term (Breger & Bregman 1975; Alcock et al. 1998) to derive the period–luminosity–color (PLC) relation, or using the Wesenheit index to derive the equivalent period–Wesenheit (PW) relation (Kubiak & Udalski 2003; Matsunaga et al.2011; Groenewegen & Jurkovic 2017). Recently, the optical band PL and PW relations were extended to the filters specifically for the Gaia mission (Ripepi et al. 2019, 2022). In addition, Groenewegen & Jurkovic (2017) have also derived the bolometric PL relation based on a combined sample of TIICs in Magellanic Clouds.

Compared to the optical PL relations, more studies have derived TIIC PL and PW relations in the near-infrared JHK bands, or a subset of these filters, in the past two decades. These near-infrared PL/PW relations were derived using TIICs located in various stellar systems, including globular clusters (Matsunaga et al. 2006), the Galactic Bulge (Groenewegen et al. 2008; Bhardwaj et al. 2017a; Braga et al. 2018), the Large and/or Small Magellanic Cloud (Matsunaga et al. 2009; Ciechanowska et al. 2010; Matsunaga et al. 2011; Ripepi et al. 2015; Bhardwaj et al. 2017b; Wielgórski et al. 2022), and in the nearby Milky Way field (Wielgórski et al. 2022). Some of the derived K-band PL relations in the Galactic bulge also included an additional dependence on the Galactic longitude and latitude (Groenewegen et al. 2008; Braga et al. 2018).

To our knowledge, there is no ugrizY-band PL and PW relations available in the literature, which will be important in the era of Vera Rubin Observatory Legacy Survey of Space and Time (LSST; Ivezić et al. 2019). Therefore, the goal of this work is to derive the gri-band PL and PW relations, by utilizing the time-series observations from the Zwicky Transient Facility (ZTF; Bellm & Kulkarni 2017; Bellm et al. 2019; Graham et al. 2019; Dekany et al. 2020) project and archival data compiled in Bhardwaj (2022, because ZTF cannot observe the southern sky), for TIICs located in the globular clusters. TIICs in globular clusters have been used to derive PL relations in the past. Demers & Harris (1974) derived the V-band PL relation based on 17 TIICs found in four globular clusters, while Pritzl et al. (2003) derived the BVI-band PL relations using two globular clusters (NGC 6388 and NGC 6441) that host the most TIICs (for a total of 10 TIICs). Optical and near-infrared PL relations were also derived from a larger sample of TIICs in Nemec et al. (1994, with ∼40 TIICs in 15 globular clusters) and Matsunaga et al. (2006, with 46 TIICs in 26 globular clusters), respectively. Note that PL relations presented in Matsunaga et al. (2006) were updated in Braga et al. (2020) and Bhardwaj (2022).

Section 2 describes the TIIC sample and their ZTF light-curve data used in this work. In Section 3, we refined the pulsation periods and determined the mean magnitudes for our sample of TIICs. The derivations of the PL relations are presented in Section 4, as well as the multiband relations (PW and period–color relations) in Section 5. We tested our derived PL/PW relations for a sample of M31 TIICs in Section 6, followed by conclusions of our work in Section 7.

2.1. Selecting TIICs in Globular Clusters

2.2. Extracting ZTF Light Curves

Since it is well known that TIICs will undergo period changes (e.g., see Wehlau & Bohlender 1982; Percy et al. 1997; Percy & Hoss 2000; Schmidt et al. 2004, 2005a, 2005b; Rabidoux et al. 2010; Osborn et al. 2012; Soszyński et al. 2018; Karmakar et al. 2019; Berdnikov & Pastukhova 2021, roughly in the range of ∼10⁻⁸ to ∼10⁻¹¹ days/day), we redetermined the periods of our sample of TIICs with ZTF light curves instead of adopting the published periods.

Given that majority of our sample of TIICs have ZTF light curves in two or three filters, we employed the LombScargleMultiband module available in the astroML/gatspy ⁹ package (VanderPlas & Ivezić 2015) to refine the periods for our sample of TIICs in a two-step process. In the first step, ZTF light curves were folded using periods identified from the first pass of LombScargleMultiband, and then fit with a low-order Fourier expansion in the following form (e.g., see Deb & Singh 2009):

$$\begin{eqnarray}&&m({\rm{\Phi }})={m}_{0}+\sum _{j=1}^{n}\left[{a}_{j}\cos (2\pi j{\rm{\Phi }})+{b}_{j}\sin (2\pi j{\rm{\Phi }})\right],\end{eqnarray} \tag{ 1 }$$

where Φ ∈ [0, 1] are the pulsational phases. Note that we only fit Equation (1) to the light curves that have more than 30 data points. Outliers beyond 3σ were excluded, where σ represents the dispersion of the fitted light curves, and LombScargleMultiband was run again in the second pass to obtain the final adopted periods. The periods obtained from LombScargleMultiband need to be doubled for three TIICs (V11 in M2, V84 in M5, and V6 in M56) in order to match with published periods. We found that the period for V6 in M2 also needs to be doubled, because alternate minima can be seen on its light curves (as displayed in Figure 1).

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We visually inspected all light curves folded with the final adopted periods. We removed 9 TIICs (V1 in M12, V12 in M13, V34 in M15, V17 and V32 in M28, V22 in NGC 6229, V2 in NGC 6293, V4 in NGC 7492, and V4 in Pal3) from our sample because they exhibit evidence of blending (such as no variations or large scatters seen on the ZTF light curves). We further removed 2 TIICs (V154 in M3 and V3 in M10) that only have 19 data points in the r-band light curve (and the total number of data points in all three filters is 30 or less). Finally, 37 TIICs remained in our sample and their intensity mean magnitudes were obtained based on the fitted low-order Fourier expansion as given in Equation (1). The final adopted periods and the intensity mean magnitudes of these TIICs are listed in Table 1. Examples of the ZTF light curves are presented in Figure 2.

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4.1. Preliminary PL Relations

Homogeneous and accurate distances of globular clusters were adopted from Baumgardt & Vasiliev (2021), who combined various distance measurements based on the Gaia and/or Hubble Space Telescope data, as well as literature distances, to obtain averaged distances via a likelihood analysis. Using these distances, we queried the Bayerstar2019 3D reddening map (Green et al. 2019) ¹⁰ via the dustmaps ¹¹ (Green 2018) code to obtain reddening E toward each of the TIICs, and corrected the extinctions on mean magnitudes using A_g = 3.518E, A_r = 2.617E, and A_i = 1.971E (Green et al. 2019). A linear regression was fitted to the extinction-corrected absolute magnitudes for 37 and 17 TIICs in the gr and i bands, respectively. While fitting the PL relations, we did not separate the TIICs into the three subtypes (BL Herculis, W Virginis, and RV Tauri) of TIICs, mainly due to the small number of samples in each subtype.

We compare our preliminary gri-band PL relations to the Johnson–Cousin BVI-band and 2MASS JHK-band (hereafter collectively referred to as BVIJHK-band) PL relations, taken from Bhardwaj (2022), in the left panel of Figure 3. The slopes of the gri-band PL relations follow the trend that the slopes become steeper at longer wavelengths; however, these gri-band PL slopes were shallower than the expected trends portrait from the BVIJHK-band PL slopes. Similar to our work, the BVIJHK-band PL relations were derived by Bhardwaj (2022) using a sample of 36–50 TIICs in globular clusters compiled from the literature. The distance moduli of these globular clusters were collected in Braga et al. (2020). In contrast to our work, these distance moduli were compiled from various publications (see the references listed in Table 4 of Braga et al. 2020). In the next subsection, we demonstrate that after updating the multiband PL relations, the gri-band PL slopes are consistent with the BVI-band PL slopes, as shown in the top right panel of Figure 3. Similarly, the dispersion of the preliminary gri-band PL relations were larger (especially in the i band), and improvements were evident after updating the PL relations.

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4.2. Updated PL Relations

We updated the BVIJHK-band PL relations for the TIIC sample compiled in Bhardwaj (2022, hereafter the B22 sample) by adopting the homogeneous distance from Baumgardt & Vasiliev (2021) to their host globular clusters. We have also adopted the homogeneous reddening E(B − V) queried from the same all-sky "SFD" dust map (Schlegel et al. 1998), using the dustmaps code, to the TIICs in the B22 sample. The compiled BVIJHK-band mean magnitudes (whenever available), as well as the adopted distances and reddenings, for the B22 sample are presented in Table 2. Mean magnitudes in the BVI band were adopted from various sources as listed in the last column of Table 2. For JHK-band mean magnitudes, the majority of them were taken from Matsunaga et al. (2006) except for V34 in M15 (Bhardwaj et al. 2021) and V43, V60, V61, and V92 in NGC 5139 (Braga et al. 2020). We excluded V1 in M10 and V8 in M79 from the B22 sample for the reasons mentioned in Section 2.1.

Table 2. Basic Information and Mean Magnitudes for the B22 Sample of TIICs in Globular Clusters

G. C.	Var. Name	P (days)	B	V	I	J	H	K	Da (kpc)	E(B − V) b	Reference c
NGC 5139	V43	1.1569	14.139	13.759	13.149	12.730	12.492	12.426	5.43 ± 0.05	0.14	3
NGC 5139	V92	1.346	14.480	13.946	13.199	12.700	12.340	12.313	5.43 ± 0.05	0.13	3
NGC 5139	V60	1.3495	14.028	13.624	13.001	12.584	12.295	12.281	5.43 ± 0.05	0.14	3
M15	V1	1.4377	15.412	14.954	14.362	13.94	⋯	13.65	10.71 ± 0.10	0.11	8
M56	V1	1.51	16.01	15.46	⋯	13.99	13.66	13.57	10.43 ± 0.14	0.25	18
M62	V73	1.7	16.147	15.243	13.966	⋯	⋯	⋯	6.41 ± 0.10	0.45	6, 17
NGC 2808	V10	1.7653	15.91	15.28	14.47	13.89	13.54	13.43	10.06 ± 0.11	0.22	12
M14	V76	1.8903	16.881	15.978	14.750	13.78	13.30	13.16	9.14 ± 0.25	0.48	7
M15	V34	2.03355	⋯	⋯	⋯	13.756	⋯	13.340	10.71 ± 0.10	0.11
NGC 5139	V61	2.2736	14.293	13.661	12.821	12.190	11.811	11.771	5.43 ± 0.05	0.14	3
M19	V4	2.4326	14.75	⋯	13.947	13.28	12.85	12.77	8.34 ± 0.16	0.31	4, 17
NGC 6441	V132	2.5474	17.218	16.478	15.241	⋯	⋯	⋯	12.73 ± 0.16	0.61	13
M14	V2	2.7947	16.596	15.629	14.337	13.45	12.98	12.85	9.14 ± 0.25	0.48	7
NGC 6284	V4	2.8187	16.04	⋯	14.786	14.15	13.71	13.67	14.21 ± 0.42	0.31	5, 17
NGC 5139	V48	4.4752	13.528	12.924	12.092	11.59	11.14	11.15	5.43 ± 0.05	0.14	3
NGC 6749	V1	4.481	⋯	⋯	⋯	13.38	12.62	12.34	7.59 ± 0.21	1.75
NGC 6284	V1	4.4812	15.88	⋯	14.504	13.68	13.24	13.18	14.21 ± 0.42	0.30	5, 17
M10	V3	7.831	13.62	12.75	11.721	11.02	10.55	10.36	5.07 ± 0.06	0.27	2, 15
NGC 6441	V153	9.89	⋯	⋯	13.72	⋯	⋯	⋯	12.73 ± 0.16	0.62	16
M62	V2	10.59	14.408	13.418	12.065	11.22	10.64	10.53	6.41 ± 0.10	0.47	6, 17
NGC 6325	V2	10.744	⋯	⋯	13.632	12.14	11.43	11.22	7.53 ± 0.32	0.96	17
NGC 6441	V154	10.83	⋯	⋯	13.57	⋯	⋯	⋯	12.73 ± 0.16	0.61	16
M14	V17	12.091	15.846	14.676	13.182	⋯	⋯	⋯	9.14 ± 0.25	0.47	7
NGC 6256	V1	12.447	⋯	⋯	13.402	11.86	11.15	10.85	7.24 ± 0.29	1.71	17
NGC 6325	V1	12.516	⋯	⋯	13.436	11.97	11.25	11.02	7.53 ± 0.32	0.95	17
M28	V4	13.462	14.21	⋯	11.734	10.78	10.18	10.01	5.37 ± 0.10	0.49	17, 19
NGC 6441	V128	13.519	16.475	15.257	13.795	⋯	⋯	⋯	12.73 ± 0.16	0.61	13
M14	V7	13.6038	16.051	14.745	13.224	12.04	11.46	11.29	9.14 ± 0.25	0.48	7
M19	V2	14.139	14.15	⋯	12.242	11.53	11.06	10.92	8.34 ± 0.16	0.32	4, 17
HP1	V17	14.42	⋯	⋯	⋯	11.91	11.09	10.78	7.00 ± 0.14	2.32
NGC 5139	V29	14.7338	12.776	12.015	11.049	10.43	10.03	9.93	5.43 ± 0.05	0.14	3
M3	V154	15.29	12.79	12.33	11.68	11.45	11.06	10.99	10.18 ± 0.08	0.01	14
M12	V1	15.527	⋯	⋯	⋯	10.24	9.79	9.64	5.11 ± 0.05	0.18
M2	V1	15.5647	13.97	13.36	⋯	11.93	11.54	11.45	11.69 ± 0.11	0.04	9
M80	V1	16.3042	14.19	13.365	⋯	11.65	11.23	11.10	10.34 ± 0.12	0.21	11, 20
HP1	V16	16.4	⋯	⋯	⋯	11.77	10.99	10.70	7.00 ± 0.14	2.39
M19	V3	16.5	13.70	⋯	12.417	⋯	⋯	⋯	8.34 ± 0.16	0.31	4, 17
M15	V86	16.829	14.368	13.659	12.646	11.70	11.32	11.19	10.71 ± 0.10	0.11	8
M19	V1	16.92	13.85	⋯	12.260	11.37	10.88	10.75	8.34 ± 0.16	0.32	4, 17
M2	V5	17.557	13.89	13.28	⋯	11.80	11.40	11.31	11.69 ± 0.11	0.04	9
NGC 6441	V129	17.832	16.395	15.128	13.610	12.14	11.61	11.65	12.73 ± 0.16	0.62	13
M10	V2	18.7226	13.01	12.05	10.934	10.05	9.61	9.47	5.07 ± 0.06	0.29	2, 15
M14	V1	18.729	15.429	14.210	12.633	11.63	11.10	10.89	9.14 ± 0.25	0.48	7
Terzan1	V5	18.85	⋯	⋯	14.576	11.97	10.93	10.61	5.67 ± 0.17	6.86	17
M2	V6	19.299	13.74	13.14	⋯	11.72	11.33	11.25	11.69 ± 0.11	0.04	9
NGC 6441	V127	19.773	16.398	15.048	13.441	⋯	⋯	⋯	12.73 ± 0.16	0.61	13
NGC 6441	V126	20.625	16.282	14.997	13.402	⋯	⋯	⋯	12.73 ± 0.16	0.61	13
NGC 6441	V6	21.365	16.117	14.885	13.231	12.16	11.64	11.49	12.73 ± 0.16	0.61	13
M5	V42	25.735	11.82	11.659	10.740	10.16	9.85	9.82	7.48 ± 0.06	0.04	1, 14
M5	V84	26.87	12.11	11.287	10.451	10.20	9.80	9.71	7.48 ± 0.06	0.04	1, 14
NGC 6453	V2	27.1954	⋯	14.231	12.375	11.35	10.75	10.59	10.07 ± 0.22	0.66	17
NGC 5139	V1	29.3479	11.488	10.829	10.058	9.40	9.05	8.99	5.43 ± 0.05	0.13	3
NGC 6453	V1	31.0476	⋯	14.601	12.789	11.51	10.85	10.66	10.07 ± 0.22	0.66	17
M2	V11	33.4	12.67	12.11	⋯	10.87	10.53	10.44	11.69 ± 0.11	0.04	9
NGC 5986	V13	40.62	⋯	⋯	⋯	10.90	10.22	10.07	10.54 ± 0.13	0.34
M56	V6	45.0	13.7	12.9	⋯	10.86	10.37	10.21	10.43 ± 0.14	0.25	18
M28	V17	48.0	⋯	⋯	⋯	9.55	8.95	8.75	5.37 ± 0.10	0.49
NGC 6569	V16	87.5	16.55	⋯	⋯	10.56	9.74	9.45	10.53 ± 0.26	0.43	10

Notes.

^a Distance of the globular clusters adopted from Baumgardt & Vasiliev (2021). ^b Reddening returned from the "SFD" dust map (Schlegel et al. 1998). ^c Sources for the BVI-band mean magnitudes: 1 = Arellano Ferro et al. (2016); 2 = Arellano Ferro et al. (2020); 3 = Braga et al. (2020); 4 = Clement & Hogg (1978); 5 = Clement et al. (1980); 6 = Contreras et al. (2010); 7 = Contreras Peña et al. (2018); 8 = Corwin et al. (2008); 9 = Demers (1969); 10 = Hazen-Liller (1985); 11 = Kopacki (2013); 12 = Kunder et al. (2013); 13 = Pritzl et al. (2003); 14 = Rabidoux et al. (2010); 15 = Rozyczka et al. (2018); 16 = Skottfelt et al. (2015); 17 = Udalski et al. (2018); 18 = Wehlau & Hogg (1985); 19 = Wehlau & Butterworth (1990); 20 = Wehlau et al. (1990).

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The JHK photometry from the aforementioned three studies was homogeneously calibrated to the 2 Micron All Sky Survey (2MASS; Skrutskie et al. 2006) system. However, the optical photometric data are very heterogeneous and were taken from several different studies as evident from the last column of Table 2. Since most of the mean magnitudes do not have their associated photometric measurement errors and are likely to suffer from systematic uncertainties, we adopt an error of 0.05 magnitudes on the mean magnitudes. The available mean magnitudes listed in Table 2 were converted to absolute magnitudes using the adopted distances. Extinction corrections on BVIJHK-band mean magnitudes were done using A_BVIJHK =R_BVIJHK E(B − V), where R_BVIJHK = {3.626, 2.742, 1.505, 0.793, 0.469, 0.303} (Schlafly & Finkbeiner 2011; Green et al. 2019). We then fit the PL relations using an iterative 3σ-clipping linear regression (where σ is the dispersion of the regression), implemented in astropy, to exclude a few obvious outliers. The updated BVIJHK-band PL relations are shown in Figure 4 and provided in Table 3.

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Table 3. The Derived Period–Luminosity Relations for TIICs in the Globular Clusters

Band	a	b	σ	N
B	−1.64 ± 0.14	0.39 ± 0.14	0.42	42
V	−1.88 ± 0.10	0.13 ± 0.11	0.31	37
I	−2.09 ± 0.08	−0.39 ± 0.08	0.24	41
J	−2.23 ± 0.04	−0.83 ± 0.04	0.13	45
H	−2.36 ± 0.03	−1.07 ± 0.04	0.10	43
K	−2.41 ± 0.03	−1.09 ± 0.03	0.10	48
g	−1.63 ± 0.10	−0.07 ± 0.10	0.38	55
r	−1.84 ± 0.08	−0.25 ± 0.08	0.30	55
i	−1.96 ± 0.08	−0.26 ± 0.08	0.28	41

Note. The PL relation takes the form of $m=a\mathrm{log}P+b$, and σ is the dispersion of the fitted PL relation. N represents the number of TIICs used in the fitting.

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There are 33 TIICs in the B22 sample that are not included in Table 1. The majority of these TIICs were located south of δ_J2000 = −30° (i.e., outside the ZTF footprint), and the remaining TIICs either did not have ZTF light-curve data or were excluded (e.g., due to blending). The BVI-band mean magnitudes for these TIICs, whenever available, were transformed to the gri band using the transformations provided in Tonry et al. (2012). Extinction corrections were done using the Bayerstar2019 3D reddening map if available; otherwise, the "SFD" dust map was used together with the conversion of E = E(B − V)/0.884 (see footnote 3). Similarly, there are 25 common TIICs in the B22 sample and Table 1, ¹² the BVI-band mean magnitudes from the B22 sample were transformed to the i band for those TIICs without the i-band data. Open circles in the right panels of Figure 4 represent the TIICs in the B22 sample transformed from the BVI-band photometry.

Combining the TIICs in Table 1 and those transformed from the B22 sample, we derived the updated gri-band PL relation, using the same iterative 3σ-clipping linear regression. The results are listed in the bottom part of Table 3. With the updated PL relations, derived using the homogeneous distances, consistent PL relations were found between the BVI-band PL relations and the gri-band PL relations, as demonstrated in the right panel of Figure 3.

Most of the previous studies have suggested that the PL relations for TIICs are insensitive to metallicity (e.g., see Matsunaga et al. 2006; Di Criscienzo et al. 2007; Matsunaga et al. 2009; Ciechanowska et al. 2010; Ripepi et al. 2015; Groenewegen & Jurkovic 2017; Braga et al. 2018; Bhardwaj 2020, 2022, and references therein). In contrast, significant metallicity terms were found for the UB-band and JHK-band PL relations from theoretical work of Das et al. (2021) and empirical investigations of Wielgórski et al. (2022), respectively. Following Matsunaga et al. (2006) and Wielgórski et al. (2022), we fit a linear regression to the residuals of PL relations as a function of metallicity for our sample of TIICs, where the metallicities, [Fe/H] for the host globular clusters, were taken from the GlObular clusTer Homogeneous Abundances Measurements (GOTHAM) survey ¹³ (Dias et al. 2015, 2016a, 2016b; Vásquez et al. 2018). Metallicity of these host globular clusters ranged from −2.27 dex (M15) to −0.47 dex (NGC 6441). Slopes of these linear regressions, denoted as γ, as a function of filters are displayed in Figure 5. Except in the B band, the values of γ are consistent with zero in all other filters, implying the corresponding PL relations are insensitive to metallicity. This is consistent with the theoretical predictions of Das et al. (2021). For the B band, fitting a period–luminosity-metallicity relation to the data yields

$$\begin{eqnarray*}\begin{array}{rcl}{M}_{B} & = & 0.68(\pm 0.25)-1.67(\pm 0.14)\mathrm{log}P\\ & & +0.19(\pm 0.14)[\mathrm{Fe}/{\rm{H}}],\ \sigma =0.41.\end{array}\end{eqnarray*}$$

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In addition to PL relations, the updated B22 sample can be used to derive the period–Wesenheit (PW), period–color (PC), and period–Q-index (PQ) relations in the BVIJHK band. The Wesenheit index, W, is analog to magnitude but it is extinction-free by construction (Madore 1982; Madore & Freedman 1991). Similarly, the Q-index is analog to color but reddening free by construction, inspired from the classical work of Johnson & Morgan (1953, who defined the Q-index in the UBV band). The combined sample of TIICs listed in Table 1 and those photometrically transformed from the B22 sample can also be used to derive the gri-band PW, PC, and PQ relations. The gri-band Wesenheit indices were defined in Ngeow et al. (2021), while the various BVIJHK-band Wesenheit indices are defined in Table 4. For the PQ relations, we have Q_BVI = (B − V) −0.715(V − I) and Q_JHK = (J − H) − 1.952(H − K) in the BVIJHK band, while the gri-band Q-index was adopted from Ngeow et al. (2022) as Q_gri = (g − r) − 1.395(r − i). The fitted PW and PC/PQ relations are summarized in Tables 4 and 5, respectively, as well as presented in Figures 6 and 7.

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Table 4. The Derived Period–Wesenheit Relations for TIICs in the Globular Clusters

Wesenheit Index	a	b	σ	N
${W}_{V}^{{BV}}=V-3.102(B-V)$	−2.62 ± 0.05	−1.00 ± 0.05	0.13	30
${W}_{V}^{{VI}}=V-2.217(V-I)$	−2.43 ± 0.07	−0.99 ± 0.08	0.20	30
${W}_{B}^{{BI}}=B-1.710(B-I)$	−2.42 ± 0.05	−0.98 ± 0.05	0.14	31
${W}_{J}^{{JH}}=J-2.448(J-H)$	−2.49 ± 0.03	−1.44 ± 0.04	0.11	46
${W}_{K}^{{HK}}=K-1.825(H-K)$	−2.51 ± 0.03	−1.10 ± 0.04	0.11	45
${W}_{K}^{{JK}}=K-0.618(J-K)$	−2.46 ± 0.03	−1.28 ± 0.03	0.08	46
${W}_{r}^{{ri}}=r-4.051(r-i)$	−2.26 ± 0.10	−0.34 ± 0.10	0.34	41
${W}_{r}^{{gr}}=r-2.905(g-r)$	−2.43 ± 0.11	−0.77 ± 0.11	0.42	55
${W}_{g}^{{gi}}=g-2.274(g-i)$	−2.33 ± 0.07	−0.48 ± 0.07	0.26	41

Note. The PW relation takes the form of $W=a\mathrm{log}P+b$, and σ is the dispersion of the fitted relation. N represents the number of TIICs used in the fitting.

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The (H − K) and (r − i) PC relations have relatively flat PC slopes with zero-points almost consistent with zero. These explain why the pairs of HK-band and ri-band PL relations are quite similar, especially that their PL zero-points are identical within the uncertainties (see Table 3). We also see that the redder colors, in JHK band and in (r − i) color, tend to have the smaller PC dispersion. In contrast, the (B − I) PC relation displays the largest dispersion among all the PC relations. In case of the PQ relations, slopes for both of the Q_JHK and Q_gri PQ relations are statistically consistent with zero, in contrast to the RR Lyrae (Ngeow et al. 2022). The Q_BVI PQ relation is also much shallower than the BVI-band PC relations, and has the smallest dispersion among the three PQ relations.

The Pan-STARRS1 survey of Andromeda, known as the PAndromeda project, reported a finding of 278 TIICs in the (halo of) the M31 galaxy (Kodric et al. 2018). This sample of M31 TIICs can be used to test the applicability of our derived PL/PW relations. Numerous distance measurements to M31, via various techniques and distance indicators, can be found in the literature. de Grijs & Bono (2014) summarized the distance estimates prior to 2013 and recommended a distance modulus of μ = 24.46 ± 0.10 mag to M31. A latest distance measurement to M31 can be found in Li et al. (2021), who give μ = 24.407 ± 0.032 mag based on the Hubble Space Telescope observations of classical Cepheids.

Kodric et al. (2018) provided the pulsation periods as well as the extinction-corrected gri-band mean magnitudes for these sample of M31 TIICs. We first removed six TIICs that have errors on the periods that are larger than 1 day (or fractional error larger than 1%; the rest of the TIICs have fractional errors that are less than 0.64% in period). The reddening-corrected colors for the remaining 272 TIICs were plotted against their logarithmic period in the left panel of Figure 8, overlaid with the PC relations taken from Table 5. The (r − i) colors for the M31 TIICs are in remarkably good agreement with the (r − i) PC relation derived from our sample of TIICs located in the globular clusters. In contrast, outliers can be seen on the (g − r) and (g − i) PC relations, suggesting there could be some problems in the g band. Indeed, the g-band observations were ∼5 to ∼10 times less than the ri band (Kodric et al. 2018), such that the g-band light curves do not have as good of quality as in the other two bands. As a result, out of the remaining 272 TIICs, 50 of them do not have mean g-band magnitudes, and 161 of them carry a nonzero bit flag (see Table 2 of Kodric et al. 2018) indicating there are some problems associated with the g-band data. For these reasons, we only focused on the ri-band mean magnitudes for this sample of TIICs in the subsequent analysis.

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The right panels of Figure 8 present the ri-band PL/PW relations for the M31 TIICs. We overplotted the PL/PW relations from Tables 3 and 4, together with the respected ±3σ boundaries, in the right panels of Figure 8 after shifting these PL/PW relations vertically with μ = 24.407 mag (Li et al. 2021, as black lines). Except for five TIICs that appeared to be brighter (marked as crosses in the right panels of Figure 8) in the ri-band PL relations, almost all of the TIICs were confined within the ±3σ of the respected PL/PW relations. Furthermore, scatters of these TIICs around the PL/PW relations confirmed the rather large dispersion in ri-band PL/PW relations as reported in Tables 3 and 4.

Our derived PL/PW relations can also be used to determine the distance modulus of M31 from this sample of TIICs (after excluding the five TIICs marked as crosses in the right panels of Figure 8). By fitting the data with the ri-band PL/PW relations given in Tables 3 and 4, weighted with the quadrature sums of errors on the mean magnitudes and the PL/PW dispersions, we obtained μ_r = 24.180 ± 0.021 mag, μ_i =24.249 ± 0.020 mag, and μ_W = 24.423 ± 0.026 mag using the ri-band PL and PW relations, respectively. The quoted errors on μ are statistical errors only. The μ_W obtained from fitting the PW relation is in good agreement, and lies in between the measurement of μ = 24.407 ± 0.032 mag from Li et al. (2021) and the recommended value of μ = 24.46 ± 0.10 mag from de Grijs & Bono (2014). This suggested our derived ri-band PW relation is robust. On the other hand, distance moduli obtained from the ri-band PL relations are ∼0.2 mag smaller than μ_W , hinting there could be additional systematic, on the order of ∼0.2 mag, in the derived PL relations. Distances to the globular clusters adopted from Baumgardt & Vasiliev (2021) are unlikely to be the source of the systematic, because the same distances were used in deriving both of the PL and PW relations. Other possible systematic errors include the samples used, the extinction maps used, and the assumed extinction law to derive the ri-band PL relations.

The derivation of ri-band PL relations include the TIIC sample transformed from the BVI-band photometry. Therefore, we first excluded the TIICs with transformations and only used the TIICs that have ZTF ri-band mean magnitudes, and rederived the ri-band PL relations. Using the rederived PL relations, the distance moduli of M31 we obtained are μ_r = 24.096 ± 0.021 mag and μ_i = 24.156 ± 0.020 mag. Similarly, we have used the "SFD" dust map for TIICs located outside the footprint of the Bayerstar2019 reddening map. If we rederived the ri-band PL relations by using the same "SFD" dust map to all TIICs in the sample and redetermined the distance moduli to M31, then we obtained μ_r = 24.000 ±0.021 mag and μ_i = 24.115 ± 0.020 mag. Finally, we adopted the same extinction law as in Kodric et al. (2018), i.e., A_r = 2.554E and A_i = 1.893E, and we obtained μ_r =24.150 ± 0.021 mag and μ_i = 24.229 ± 0.020 mag. These distance moduli are smaller than those obtained from the ri-band PL relations derived in Table 3. Hence, there could have hidden systematic errors when deriving the PL relations, and independent samples and calibration of the TIIC PL relations are desirable.

In this work, we present the first gri-band and the updated BVIJHK-band PL and PW relations for TIICs located in the globular clusters. In total, there are 70 TIICs spanning in 30 globular clusters (with ages spanning from ∼11.0 to ∼13.2 Gyr) in our sample, and only 3 of them have the complete nine-band photometry. Homogeneous distance to the globular clusters, ranging from 3.30 (M22) to 88.47 kpc (NGC 2419), adopted from a single source (Baumgardt & Vasiliev2021) and consistent reddening maps, either the Bayerstar2019 3D reddening map or the "SFD" dust map, were used to calibrate the absolute magnitudes of these samples of TIICs. We demonstrated that the PL relations are consistent in the BVI and the gri bands. We have also derived nine sets of the PW relations based on the combinations of these filters. For the PL/PW relations, the JHK-band PL/PW relations exhibit the smallest dispersion, which are preferable to be applied in the future distance scale work. Finally, our sample of TIICs also allows the derivation of PC and PQ relations in these filters. We found that the slopes of the PC relations in the JHK band and in the (r − i) color, as well as the slopes of the PQ relations, are quite shallow or flat.

We tested our PL/PW relations, at least in the ri band, with a sizable sample of TIICs in M31. The scatters of M31 TIICs on the PL/PW relations are similar to those presented in Tables 3 and 4, confirming the derived PL/PW dispersions are intrinsic. Using our derived ri-band PW relation, the distance modulus of M31 we obtained is in agreement with the latest measurement using the classical Cepheids. However, distance moduli derived from using the ri-band PL relations are smaller by ∼0.2 mag, suggesting there could be hidden systematics in the derived PL relations. Therefore, additional work in the near future is required to independently cross-check these PL relations. Nevertheless, our derived PW relations can be applied in the ongoing and upcoming synoptic time-series sky surveys, such as LSST or other surveys employing similar gri filters.

We are thankful for the useful discussions and comments from an anonymous referee that improved the manuscript. We are thankful for funding from the Ministry of Science and Technology (Taiwan) under the contracts 107-2119-M-008-014-MY2, 107-2119-M-008-012, 108-2628-M-007-005-RSP, and 109-2112-M-008-014-MY3. A.B. acknowledges funding from the European Unions Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No. 886298.

Based on observations obtained with the Samuel Oschin Telescope 48 inch Telescope at the Palomar Observatory as part of the Zwicky Transient Facility project. ZTF is supported by the National Science Foundation under grant Nos. AST-1440341 and AST-2034437 and a collaboration including current partners Caltech, IPAC, the Weizmann Institute of Science, the Oskar Klein Center at Stockholm University, the University of Maryland, Deutsches Elektronen-Synchrotron and Humboldt University, the TANGO Consortium of Taiwan, the University of Wisconsin at Milwaukee, Trinity College Dublin, Lawrence Livermore National Laboratories, IN2P3, University of Warwick, Ruhr University Bochum, Northwestern University and former partners the University of Washington, Los Alamos National Laboratories, and Lawrence Berkeley National Laboratories. Operations are conducted by COO, IPAC, and UW.

This research has made use of the SIMBAD database and the VizieR catalog access tool, operated at CDS, Strasbourg, France. This research made use of Astropy (http://www.astropy.org), a community-developed core Python package for Astronomy (Astropy Collaboration et al. 2013, 2018).

Facility: PO:1.2m - Palomar Observatory's 1.2 meter Samuel Oschin Telescope.

Software: astropy (Astropy Collaboration et al. 2013, 2018), dustmaps (Green 2018), gatspy (VanderPlas & Ivezić 2015), Matplotlib (Hunter 2007), NumPy (Harris et al. 2020), SciPy (Virtanen et al. 2020).