RESCUER: Cosmological K-corrections for star clusters

The advent of JWST (the James Webb Space Telescope) now allows entire star cluster populations to be imaged in galaxies at cosmologically significant redshifts, bringing with it the need to apply K-corrections to their magnitudes and colour indices. Since the stellar populations within star clusters can be well approximated by a single age and metallicity, their spectral energy distributions are very different from those of galaxies or supernovae, and their K-corrections behave differently. We derive the photometric K-corrections versus redshift for model star clusters that cover a wide range of ages and metallicities, illustrating the results particularly for the broadband filters on the HST/ACS and the JWST/NIRCam cameras that are most commonly being used for imaging of populations of star clusters in distant galaxies. In an Appendix, we introduce a simple webtool called RESCUER that can generate K-values for any user-defined combination of cluster properties.


INTRODUCTION
The James Webb Space Telescope (JWST) has opened up the ability to observe entire populations of star clusters at distances and lookback times well beyond the Local Universe (Faisst et al. 2022;Lee et al. 2022;Harris & Reina-Campos 2023). But before the photometry of cosmologically distant systems can be compared with similar data for their zero-redshift counterparts, K-corrections need to be applied to account for the effects of the cosmological redshift on the measured magnitudes and colour indices.
K-corrections in their various forms have long been familiar in the literature for galaxies and supernovae (Hubble 1936;Humason et al. 1956;Oke & Sandage 1968;Hamuy et al. 1993;Kim et al. 1996;Lubin & Sandage 2001;Hogg et al. 2002;Blanton & Roweis 2007;Boldt et al. 2014, to cite only a few). But they are almost unknown for photometry of star clusters (see Kalirai et al. 2008;Alamo-Martínez et al. 2013;Harris & Reina-Campos 2023, for rare examples where globular clusters were observed in galaxies with significant redshifts). The essential problem is that the SEDs (spectral energy distributions) for star clusters are not the same as those of galaxies, and they vary with redshift in a different way. For the composite stellar populations that make up galaxies, SED shapes (and thus −corrections for any redshift) are determined by their morphological type, or more precisely their star formation history. By contrast, star clusters are close approximations to single-age SSPs (Simple Stellar Populations), and the major factors determining their SED shapes are instead metallicity and age. A separate treatment of the problem specifically for star clusters is therefore appropriate, and timely for upcoming JWST data. ★ E-mail: reinacampos@mcmaster.ca (MRC) † E-mail: harrisw@mcmaster.ca (WEH) In the following discussion, we adapt the general theory for −corrections to SEDs of star clusters and demonstrate how −values change with redshift up to = 1. We describe the formalism in Sect. 2, and introduce model SEDs from the E-MILES stellar library with a selected set of filters in Sect. 3. In Sect. 4, which has the main results of our paper, we start with a simple example for a blackbody spectrum, and then go on to calculate and discuss full K-corrections for more realistic star clusters, as observed through selected filters for JWST and HST (Hubble Space Telescope). We briefly summarize our findings in Sect. 5.

COSMOLOGICAL K-CORRECTIONS
The cosmological photometric K-correction quantifies the flux difference from a source at a redshift relative to its intrinsic luminosity. This correction is often needed in galaxy surveys to correct the observed apparent magnitudes into a uniform system of absolute magnitudes (e.g. Blanton & Roweis 2007), but it has not yet been calculated in a systematic way for star clusters.
By convention, we define the K-correction in terms of the absolute and apparent magnitudes (Hogg et al. 2002;Hogg 2022), where Q is the rest-frame absolute magnitude of the source (i.e. the magnitude if the source were to be observed at 10 pc) in the filter , and R the apparent magnitude of the source observed in the (possibly different) filter . The luminosity distance L to the source depends on the cosmology assumed, and in a flat Universe, it is proportional to the comoving distance to the source, L = (1 + ) C (e.g. Hogg 1999).
The K-correction can be defined in terms of either frequency or wavelength (Hogg et al. 2002), but in the present discussion, only the more common wavelength version is presented, and only in terms of photon-counting instruments. As will be seen below, the calculation of K is built from various integrals of the general form Here is the number of recorded photons per unit time per unit area, from a source with flux ( ), measured by a detector with overall throughput (i.e. transmission profile) ( ) for a given filter. The flux describing the SED of the source is assumed to be in units of energy/time/area/wavelength, so it is converted to counts/time/area/wavelength by dividing by the energy per photon (ℎ / ). The transmission profile represents the entire system throughput, which for JWST and HST includes the telescope optics, camera, filter, and detector efficiencies as a function of wavelength 1 . Thus ( ) essentially gives the probability that an incoming photon of wavelength will be recorded by the detector.
In terms of wavelength, the general expression for is (see Hogg et al. 2002, for derivation) QR = −2.5 log 10 The integrals cover the range of observed and rest-frame wavelengths, o and rf 2 , respectively, and the terms ( ) and ( ) describe the transmission curves of the two filters. For these we use the published throughput curves for JWST/NIRCam 3 and HST/ACS 4 . The terms and correspond to the spectral flux densities of the standard source in the filters and , respectively. In this work we use the AB magnitude system (Oke & Gunn 1983), where the magnitudes are defined in terms of a hypothetical constant source of flux density in frequency space, AB ≡ 3.631×10 −20 erg cm −2 s −1 Hz −1 at all frequencies. The spectral density of this source can be transformed to wavelength space by = ( / 2 ), using = and = .
The K-correction can also be described in terms of the intrinsic luminosity of the source, ( ) (i.e. the energy per unit time per unit wavelength). This expression can be related to its spectral flux density via the luminosity distance and redshift, . Replacing all terms in eq. (3), the K-correction is thus QR = −2.5 log 10 1 For ground-based instruments, the atmospheric transmission would also be included. 2 Note that previous studies write the rest-frame wavelength as or 'emitted'. We prefer to use rf to make the distinction with more general; see the discussion below. 3 https://jwst-docs.stsci.edu/jwst-near-infrared-camera/nircaminstrumentation/nircam-filters 4 https://www.stsci.edu/hst/instrumentation/acs/data-analysis/systemthroughputs The K-values written this way respond to the question of how much the observed apparent magnitude should be corrected in order to reflect the intrinsic luminosity. The calculation of K can go in two different directions that we refer to as homochromatic and heterochromatic: (i) Homochromatic ( = ): The correction is done within the same wavelength or frequency range as the observed (redshifted) measurement. For a homochromatic K-correction in the filter , the previous equations simplify to QR = −2.5 log 10 This version is by far the most frequently used one and intuitively clear: the observed flux at wavelength was emitted in the rest-frame spectrum at the shorter wavelength /(1 + ), and the energy of every photon is reduced by the same factor 1/(1 + ).
(ii) Heterochromatic ( ≠ ): This more general formulation allows the transformation from the observed wavelength o to be done to another wavelength rf on the rest-frame (emitted, un-redshifted) spectrum. For example, an obvious use of a heterochromatic conversion would be where filter is just the redshifted version of filter in which the flux was emitted, such that rf = e = o /(1 + ).
However, Eq. (4) is general enough to allow the transformation to be done into any arbitrary filter where the rest-frame rf does not have to equal /(1 + ). If we are given an accurate SED, then the rest-frame magnitude through any other filter at either shorter or longer wavelength can be predicted from the measured flux through (e.g. Kim et al. 1996;Blanton & Roweis 2007). This is the main reason why we use the term rf and not e to refer to the rest-frame wavelength.
In short, the K-correction can in principle be used to step from the observed flux at to any point on the rest-frame spectrum, but clearly the validity of the result will depend heavily on the accuracy of the assumed SED. In this sense, working with SEDs for star clusters, which closely resemble blackbody-like SSPs (see below), is more straightforward than for the far more complex parameter space needed to model galaxy spectra (cf. Blanton & Roweis 2007, for an extensive discussion of galaxy template spectra).
A simple way to build some further intuition is to assume that the filter transmission curves are described by delta functions, ; this is the monochromatic version of K as used in, e.g., Condon & Matthews (2018). With this simplifying assumption, eq. (4) reduces to QR = −2.5 log 10 1 (1 + ) This equation is only valid in the AB magnitude system because its spectral flux density in frequency space is constant. In the case of homochromatic K-corrections ( o = rf ), we recover the expression provided by Condon & Matthews (2018) in their equation (67).
The basic behaviour of the K-correction is illustrated in Fig. 1, where an intrinsic emitted SED for a simple stellar population (SSP) of 5 Gyr age and [M/H] = −0.96 is compared with its redshifted version at = 0.5 (left-hand panel). To show the intrinsic and emitted SEDs on the same scale, we assume for display purposes that the intrinsic curve (black line) represents the object at its observed distance , but sitting at rest, while the observed redshifted SED (shaded ⊙ , and the curves in the middle and right-hand panels are normalised by 2.2 × 10 37 Å ergs s −1 M −1 ⊙ . Positive K-values correspond to the observed flux being dimmer than the intrinsic one (blue arrows) and the absolute magnitude needing to increase, whereas negative K-values indicate that the observed flux is brighter (green arrows), and thus the magnitude has to decrease. line) is attenuated and displaced towards redder wavelengths. This way we isolate just the effect of redshift.
The middle and right-hand panels show the same pair of spectra now multiplied by 2 . Examples of homochromatic K-corrections are shown in the middle panel, while heterochromatic K-corrections are in the right-hand panel. The resulting sign of the K-correction contains information about the shape of the SEDs. A positive (K > 0) value (blue arrows) indicates that the observed flux is dimmer than the intrinsic flux at that wavelength, and thus the absolute magnitude should be brighter than expected from the inverse-square law (Eq. 1). In contrast, a negative (K < 0) value (green arrows) makes the absolute magnitude fainter because the observed flux is brighter than the intrinsic flux. When both curves are equal, the K-correction is null (orange arrows).

Stellar population synthesis models: E-MILES
To make the calculations for K, we need to work from a library of homogeneous SEDs that cover suitably large ranges in wavelength, metallicity, and age. For this study, we use SSPs calculated with the E-MILES models (Röck et al. 2016) 5 . The SEDs produced from these models cover the range 1680 Å-50000 Å at high resolution, can be generated for any desired age or metallicity spanning the observed ranges for star clusters, and are well tested against observed SEDs for stellar systems (e.g. Röck et al. 2016;Vazdekis et al. 2015Vazdekis et al. , 2016. More generally, several modern SSP codes are now available that accurately match the integrated spectra of real globular clusters in the Milky Way or M31 from the UV through the infrared (e.g. Barber et al. 2014;Conroy et al. 2018;Ashok et al. 2021;Martins et al. 2019;Boquien et al. 2019;Maraston et al. 2020, among others) and several of these would be similarly useful for the purposes of this study.
The E-MILES stellar population models can be generated for different choices of stellar initial mass functions (IMF) and theoretical stellar isochrones. In this work, we use the models derived in version v11.0 assuming a Chabrier 2003 IMF and the BaSTI isochrones from Pietrinferni et al. (2004). In Appendix A, we show that the results presented here are only mildly affected (∼ 0.02 mags by = 0.5, and ∼ 0.12 mags by = 1) when assuming Padova isochrones instead (Girardi et al. 2000), and remain unchanged for a Kroupa (2001) IMF.
An illustration of the SEDs from the E-MILES models is shown in Fig. 2. Regardless of the age or metallicity of the SSP, the shape of the SEDs is remarkably similar to that of a blackbody at = 5000 K; the SEDs are dominated by a single peak in the optical with a decay towards redder wavelengths. Comparing the SEDs of 5 Gyr old SSPs with different metallicities (top panel), the peaks of those with lower metallicity are more prominent by a factor of 2.9 than in those with super-solar abundances. In contrast, for a given metallicity, young SSPs ( = 2 Gyr) emit more radiation across their spectrum than old SSPs ( = 12 Gyr).

Filter selection: HST and JWST
For the selection of filters, we focus on three commonly used HST filters for GC systems in other galaxies ( 475 , 606 and 814 ), plus a set of eight broadband filters for JWST NIRCam. These include the SWC (short wavelength channel) filters ( 070 , 090 , 115 , 150 , 150 2, and 200 ) and the LWC (long wavelength channel) ( 277 and 356 ). We show their bandpasses in the bottom panel of Fig. 2. These NIRCam filters have already been used for GC photometry in high-systems (Faisst et al. 2022;Lee et al. 2022;Harris & Reina-Campos 2023) and more studies are in progress.
It is worth noting that the formulation presented here for the cosmological K-corrections is general and easily adaptable to any other set of filters for which the bandpass is known. The version of the webtool RESCUER 6 presented below in Appendix B is restricted to this set of HST and JWST filters, but the code is easily adaptable via the public repository on GitHub.

RESULTS
The K-corrections presented in this work are in the AB magnitude system (Oke & Gunn 1983).

Test case: blackbody spectrum
The SEDs of star clusters over the age range of interest here are approximated well to first order by Planck blackbody spectra (see Fig. 2), which can be used to give an initial impression of the behaviour of K versus redshift for the set of filters listed above. Consider the spectrum emitted by a blackbody at = 5000 K. The spectral radiance of the blackbody, i.e. the energy per unit time, per unit solid angle, and per unit of area normal to the propagation, in terms of wavelength is where ℎ is the Planck constant, and B is the Boltzmann constant. We calculate the homochromatic K-corrections using Eq. (3), and show them in Fig. 3. As a sanity check, all filters require a null Kcorrection at = 0. At higher , the value of the K-correction for most filters can increase up to a few magnitudes. The filters for which the correction would be the smallest at all redshifts are 090 and 115 . For both of these filters, the emission mostly comes from the region in the spectrum around the peak of the blackbody emission, where the curve is shallow. The peak of the blackbody emission crosses the wavelength range of 090 at ∼ 0.5, hence the sign change of the K-correction from negative to positive. In the case of the filter 115 , the peak would cross it at > 1, and the sign change is thus not visible in the figure.

The oldest SSPs of [M/H] = -2.27 as a function of redshift
An extremely interesting consequence of observing remote star clusters is that they have a well defined maximum observable age at a given redshift: that is, max ( ) = ( BB ( ) − form ), where BB is the time since the Big Bang and form is the time interval needed for galaxy and star cluster formation to start. Drawing from recent observations (e.g. Labbé et al. 2023), we adopt form ≃ 500 Myr for the present discussion. Under this assumption, the stellar population of a ∼ 13 Gyr old globular cluster at = 0 would (for example) have been 8 Gyr old at = 0.51, and 2 Gyr old at = 2.61. Despite the attenuation introduced by redshift, the displacement of the peak into redder wavelengths and the increase in luminosity from the younger populations (see top panel in Fig. 4) implies that high-star clusters remain bright in the infrared and should be detectable by JWST up to ∼ 1 and possibly beyond, even without the help of lensing. We now calculate the homochromatic K-corrections for the model SEDs of age max ( ) versus , and show them in the bottom panel of Fig. 4. By redshift = 1, the required K-corrections range from −2 to 2 mags, and there are three filters ( 814 , 090 and 115 ) for which they stay within [−0.5, 0.5] mags. As in the case of the blackbody spectrum, these three filters mostly capture emission coming from around the peak of the spectrum where it is shallower. Due to the number of uncertainties in the stellar population models, large K-values might be introducing large errors into the magnitudes, and therefore, filter transformations involving smaller K-corrections should be preferred. As seen in the top panel of Fig. 4, there are two competing factors on the displacement of observed SED: as redshift increases, the observed wavelengths get stretched and redder, but conversely the populations of the clusters are younger and their intrinsic SED is bluer. To explore these competing effects, we determine the wave-  length at which the SED peaks as a function of redshift for a variety of SSPs of different ages and metallicities (Fig. 5). Because the peaks of the intrinsic SEDs for a given metallicity are roughly the same (see top and middle panel in Fig. 2), then the displacement is mostly given by the stretching due to redshift. Interestingly, the peaks of the SEDs do not enter the infrared regime until > 1. This implies that one of the main advantages of JWST over HST for distant populations of star clusters is its higher resolution and much larger collecting area, rather than its infrared capability. The advantage gained from imaging in the near-infrared comes from the K-correction itself.

Homochromatic K-corrections
We show homochromatic K-corrections needed for a SSP of 5 Gyr observed with the JWST filters in Fig. 6. For the bluest filter ( 070 ) and regardless of the metallicity of the SSP, the K-corrections are always positive; that is the observed SED is dimmer than its intrinsic value, and thus require a brighter absolute magnitude. The well contained within the [−1, 1] magnitude range. Because the peak of the SED is much more prominent in SSPs of lower metallicities, the K-values at low metallicities tend to show more variation over redshift and tend to be more negative. In the filters 090 , 115 and 150 , the curves for some of the models lie near a null K = 0 until the peak of their SED finally shifts past the filter bandpass. For the filters of the JWST/NIRCam long wavelength channel, 277 and 356 , two effects are seen. Firstly, the variation of the SED with metallicity is so small that there is not much difference between their K-values (see top panel in Fig. 2). Secondly, the Kvalues are quite large, K ≥ −1.5. Because the intrinsic luminosities are low in these bands, any radiation coming from around the peak and redshifted results in a large change of brightness, and thus, the absolute magnitudes have to be dimmed.

Heterochromatic K-corrections
In the case of heterochromatic K-corrections, the transformation is performed from the observed filter towards any arbitrary filter. Thus, these corrections can allow comparisons between observations of star cluster populations in the JWST filters and those in the HST filter set.
We use the JWST/NIRCam filter 150 as our fiducial starting point for the observed wavelength range, and compute the corrections needed to transform it to any of the other filters in our set. We show the resulting K-corrections for a SSP of 5 Gyr emitting as a function of redshift in Fig. 7. Regardless of the metallicity of the SSP, the heterochromatic corrections are always negative. A trend is visible when correcting towards different parts of the spectrum: the corrections to the blue filters are negative and large (K ≤ −1 mags), become smaller and closer to zero when transforming towards wavelengths ∼ 0.9-2 m, and increase again when correcting to the redder part of the spectrum. This trend results from the different shapes between the observed and the intrinsic SEDs, and can be seen from the green arrow in the right-hand panel of Fig. 1 tracing the intrinsic emitted curve.
The correction towards filters redder than 070 and bluer than 200 leads to K-values within [0, −1] mags. An interesting effect occurs in the transformations into the filters 814 , 090 , and 115 : at a particular redshift, the correction is the same regardless of the metallicity of the SSP. This is caused by the wavelength range of the target filter roughly corresponding to the wavelength range from which the photons were originally emitted. In the case of 115 , its midpoint wavelength at a redshift of = 0.25 is 11500 Å ×(1+0.25) = 14375 Å, very close to the midpoint wavelength 15000 Å of the 150 filter. At this particular lookback time, the luminosity terms in Eq. (4) cancel out and the K-correction is equal for all the SSPs.

SUMMARY AND DISCUSSION
We present the photometric K-corrections versus redshift for model star clusters that cover a wide range of ages and metallicities, for several broadband filters on the HST/ACS and the JWST/NIRCam cameras. Since star clusters are well described by simple stellar populations of a single age and metallicity, we use the spectral energy distributions for them from the E-MILES stellar library.
The main effect of observing objects backwards in time is that their SEDs are attenuated and shifted towards redder wavelengths by the redshift. The K-correction characterizes how much brighter or dimmer the observed SED is relative to the intrinsic luminosity of the object (Fig. 1). In this work, we adapt the formalism developed for galaxy studies to describe the corrections needed for observations star clusters as a function of redshift. We have calculated these corrections both within the same wavelength range (i.e. homochromatic corrections, Sect. 4.3) and across filters (i.e. heterochromatic corrections, Sect. 4.4).
Observing remote star clusters has an interesting limitation: they have a well defined maximum observable age that decreases with increasing redshift. Despite the attenuation due to redshift, the displacement of the SED peak into redder wavelengths and the increase in luminosity from the younger populations implies that high-star clusters remain bright in the infrared and should be detectable by JWST up to ∼ 1.
All the K-corrections are publicly available in a Zenodo repository, and we have developed an interactive webtool called RESCUER to generate the K-values for any user-defined combination of cluster properties (App. B).

DATA AVAILABILITY
The spectral models are publicly available in the E-MILES website: http://research.iac.es/proyecto/miles/pages/ssp-models.php. The RESCUER interactive webtool is hosted by Streamlit in https://rescuer.streamlit.app/, and we have deposited the tables with all the K-corrections derived in this work in a Zenodo repository with DOI: 10.5281/zenodo.8387817. Here we briefly explore the effect of the assumed stellar isochrone and IMF on the K-corrections for a SSP of ∼ 5 Gyr and [M/H] ∼ −1 from the E-MILES stellar library. Changing the IMF to a Kroupa (2001) IMF leads to small changes in the intrinsic luminosity (≪ 10 28 ergs s −1 Å −1 M −1 ⊙ ), whereas changing the stellar isochrones to the Padova models (Girardi et al. 2000) produces SEDs that are ∼ 10 28 ergs s −1 Å −1 M −1 ⊙ brighter. When propagating those changes to the K-corrections, we find that for the JWST SWC filter 150 the homochromatic correction is 0.02 mags higher at = 0.5 and 0.12 mags higher at = 0.1 when assuming the Padova models relative to the BaSTI isochrones. Similar offsets are found in the heterochromatic K-correction between the JWST SWC filters 150 and 115 .

APPENDIX B: RESCUER: K-CORRECTIONS VIA A WEBTOOL
To facilitate the calculation of the photometric K-corrections for star clusters (i.e. SSPs of a given age and metallicity), we have developed a webtool that allows users to do so interactively. The code is called RESCUER (for REdshifted Star ClUstERs). The current version of the webtool calculates the cosmological K-corrections for SSPs from the E-MILES stellar library assuming BaSTI isochrones and a