Spectral Analysis of Star-Forming Galaxies at z < 0.4 with FADO: Impact of Nebular Continuum on Galaxy Properties

Abstract

The star formation rate (SFR) is a crucial astrophysical characteristic for understanding the formation and evolution of galaxies, determining the interplay between the interstellar medium and stellar activity. The mainstream approach to studying stellar properties in galaxies relies on stellar population synthesis models. However, these methods neglect nebular emission, which can bias SFR estimates. Recent studies have indicated that nebular emission is non-negligible in strongly star-forming regions. However, targeted research is currently limited, particularly regarding galaxies at slightly higher redshifts ($z<0.4$). In this work, 696 star-formation galaxies with stellar mass in ${10}^9-{10}^{11}{M}_{\odot }$ are selected from the SDSS-DR18 and their spectra are fitted via the fitting analysis using differential evolution optimization (FADO) technique. FADO self-consistently fits both stellar and nebular emissions in galaxy spectra. The results show that the median $H_{\alpha }$ flux from FADO fitting differs from that of qsofitmore by approximately 0.028 dex. Considering the stellar mass effect, we found that although the nebular emission contribution (Nebular Ratio hereafter) is minimal, it increases modestly with redshift. We advocate explicitly accounting for nebular emission in the spectral fitting of higher-redshift galaxies, as its inclusion is essential to obtaining higher precision in future analyses.

1. Introduction

The star formation rate (SFR), particularly the instantaneous SFR (SFR(t)), is a fundamental tracer of galactic star formation activity, evolutionary history, and chemical enrichment [1]. Combining SFR measurements with simulations of galaxy evolution helps to identify the dominant physical processes that drive galaxy growth. These processes include quenching, gas accretion, star formation, and feedback from supernovae and active galactic nuclei (AGN) [2,3]. Measuring SFR across cosmic time constrains the contributions of different galaxy populations to the cosmic background radiation, refines cosmological ionization models, and guides searches for Population III (Pop III) stars [4]. To date, extensive efforts have focused on SFR, including environmental variations formation of clusters and massive stars [5,6,7,8], enhanced SFR or black hole growth triggered by galaxy interactions [9,10], SFR corrections relevant to cosmological re-ionization [4,11,12,13], and re-evaluation of star formation within individual molecular clouds and gas relations [14]. Several major surveys underpin studies of star-forming galaxies, including the Sloan Digital Sky Survey (SDSS) [15], Hubble Ultra Deep Field (HUDF) [16], and Atacama Large Millimeter Array (ALMA) [17].

Mainstream methods for SFR estimation typically begin with spectral fitting of galaxy spectra. This involves representing each spectrum as a linear combination of stellar population templates, using stellar-only models as the primary fitting basis (e.g., STARLIGHT [18], pPXF [19]). Historically, many analyses of SDSS galaxy spectra have omitted explicit modeling of nebular gas, thereby neglecting nebular emission [20,21]. The limitation of these stellar models lies in their inability to solve different processes of gas emission within galaxies, as they incorporate assumptions that combine stellar mass, gas mass, and dust mass within a single framework. Studies by Krueger et al. [22] and Salim et al. [23] indicate that vigorous star forming activities can lead to the nebular continuum contributing approximately 30–70% to optical and near-infrared emissions, and that nebular emission in star-forming regions can account for about 60% of the total emissions [2,24]. Pacifici et al. [25] demonstrated that neglecting nebular emission in spectral modeling can overestimate SFR by approximately 0.12 dex. Ignoring the nebular continuum also biases fits toward larger stellar masses and shallower ultraviolet slopes [26]. Galaxy evolution is driven by star formation, gas accretion, interactions/collisions, and mergers [27]. FADO is an independent and self-consistent spectral fitting tool that simultaneously models stellar and nebular components, dynamically decomposing their contributions across the full galaxy spectrum [28]. Therefore, we employ FADO to explicitly model nebular emission in this work.

Nebular emission arises from ionized gas in galaxies, primarily excited by ultraviolet ionizing photons from young massive stars. It comprises the nebular line emission and nebular continuum, including free–free, free–bound, and two-photon processes [28]. The nebular continuum modifies the overall spectral shape because it is generally flatter than the stellar continuum. If the nebular continuum is ignored during spectral fitting, it may be misattributed to the stellar component. This misattribution can overestimate the stellar luminosity contribution and bias the derived stellar mass and age [29,30]. Moreover, the nebular continuum dilutes stellar absorption features, making them appear weaker than their intrinsic strengths [29].

In this work, we select spectra of actively star-forming galaxies at low redshift ($z<0.4$) from SDSS-DR18. We fit these spectra with FADO, which simultaneously models stellar and nebular components, and with qsofitmore, which employs stellar population templates with fixed nebular emissions. The aim is to test whether, at $z<0.4$, star-forming galaxies (SFGs) require self-consistent modeling of nebular emission rather than treating it as fixed. This allows us to assess how different treatments affect SFR estimates and preliminarily examine impacts on other derived properties. $\mathrm{H}\alpha$ and $\mathrm{H}\beta$ are widely used recombination-line tracers of SFR [31]. For $z>0.5$, the [O II]$\lambda$3728 line is often used as an SFR indicator out to $z<1.4$ [32]. For $z<0.5$, SFR is commonly traced by $\mathrm{H}\alpha$ [33], as well as by [O II]$\lambda$3728 [32] at intermediate redshifts; at higher redshifts, [O III]$\lambda$5007 can also serve as an SFR diagnostic [34].

Several recent studies have demonstrated the robust performance of FADO in spectral fitting and emission line measurement. Breda et al. [30] used simulated data to compare FADO with STARLIGHT, finding that STARLIGHT overestimates the mass-weighted and light-weighted mean stellar ages as well as the metallicity. These results indicate that FADO recovers stellar mass, age, and mean metallicity with accuracy of ∼0.2 dex. Furthermore, Cardoso et al. [29] showed that for galaxies with EW $\left(\mathrm{H}\alpha \right)>3\AA$, galaxy-integrated stellar mass and metallicity can differ by up to ∼0.06 dex. Most recently, Miranda et al. [20] compared SFRs for the MPA–JHU sample derived with both FADO and STARLIGHT, reporting a difference of 0.01 dex in H$\alpha$ flux between the methods. They also argued that nebular emission is crucial for star-formation activity and impacts SFR estimates, particularly at high redshift. Additionally, Izotov et al. [26] showed that neglecting the nebular continuum biases fits toward larger stellar masses and shallower ultraviolet slopes.

This work compares the spectral fits of qsofitmore and FADO in order to explore the possible impacts of their different treatment of nebular emission on the resulting SFR estimation. In addition, we explore the relationship between nebular emission and other galaxy properties. Section 2 outlines the criteria for selecting sources in galaxy spectroscopic analysis and the process of spectral fitting employed in this work. In Section 3, we present our statistical analysis of the selected sample and the main findings. Section 4 discusses the limitations of fitting techniques and samples with respect to the spectral data used in this work. Finally, Section 5 summarizes this preliminary research. Throughout the paper, we assume a standard cosmological model with Hubble constant $H_0$ = 72 km s⁻¹ Mpc⁻¹, matter density ${\Omega }_M$ = 0.3, and cosmological constant ${\Omega }_{\Lambda }$ = 0.7.

2. Sample Selection and Spectral Fitting

The galaxy sample and its spectra used in this work are selected from the Sloan Digital Sky Survey Data Release 18 (SDSS-DR18) [35]. SDSS-DR18 integrates data from the previous seventeen releases and includes new observations and pipeline improvements. The galaxy sample was obtained by SQL query, and the corresponding spectroscopic data were retrieved from the SDSS Science Archive Server (SAS) (https://dr18.sdss.org/optical/spectrum/search) (accessed on 6 October 2024).

2.1. SQL Query

To guarantee the subsequent spectral fitting, it is necessary to select spectra with sufficient signal-to-noise ratio (S/N). Because FADO performs reliably for spectra with median S/N> 5 [36], the SQL query was adopted to select high S/N spectra. The query obtained 91,252 spectra with 5 < S/N < 20 for 0 < z < 0.2 galaxies and 6604 with S/N > 5 for 0.2 < z < 0.4 galaxies. Additionally, we required S/N> 5 for the principal emission lines in each spectrum. The following conditions were added to the query in order to ensure the reliability of the redshift measurements: zwarning = 0, veldisperr > 0, veldisp > 3 * veldisperr, and 80 < veldisp < 350 (see Appendix A for detailed SQL query).

2.2. BPT Selection

After SQL selection, the BPT diagram [37] was adopted to identify SFGs in the sample. The BPT diagram employs the line ratios [N II]$\lambda$6583/$\mathrm{H}\alpha$ and [O III]$\lambda$5007/$\mathrm{H}\beta$. Two empirical boundaries divide the diagram into three regions: star-forming, composite, and AGN. In Figure 1, the dashed boundary Equation (1) is derived from Kewley et al. [38], while the dotted one Equation (2) is derived from Kauffmann et al. [39]. Galaxies in the star-forming region were selected as SFGs (blue points in Figure 1).

$$log\left({\displaystyle \frac{\left[\mathrm{O}\mathrm{III}\right]\lambda 5007}{{\mathrm{H}}_{\beta }}}\right)={\displaystyle \frac{0.61}{log\left(\left[\mathrm{N}\mathrm{II}\right]\lambda 6538/{\mathrm{H}}_{\alpha }\right)-0.47}}+1.19,$$

(1)

$$log\left({\displaystyle \frac{\left[\mathrm{O}\mathrm{III}\right]\lambda 5007}{{\mathrm{H}}_{\beta }}}\right)>{\displaystyle \frac{0.61}{log(\left[\mathrm{N}\mathrm{II}\right]\lambda 6538/{\mathrm{H}}_{\alpha })-0.05}}+1.3.$$

(2)

A total of 21,455 galaxies were classified as SFGs, comprising 18,176 galaxies with 0 < z < 0.2 (lower-redshift sample) and 3279 galaxies with 0.2 < z < 0.4 (higher-redshift sample). To avoid sample size bias, we randomly selected 3279 low-redshift galaxies to match the higher-redshift sample size. Ultimately, we fit 6558 galaxies with qsofitmore and FADO to derive accurate emission line fluxes. Figure 2 shows the redshift distributions of the two subsamples.

2.3. qsofitmore vs. FADO

qsofitmore is a Python module for fitting and analyzing quasar and galaxy spectra [40]. The module has demonstrated good performance in spectral fitting and analysis, and has become increasingly popular in quasar and galaxy spectroscopic studies [41,42,43,44]. qsofitmore estimates component uncertainties via Monte Carlo (MC) resampling. It uses principal component analysis (PCA)-derived host galaxy templates from large observational samples (e.g., SDSS) to model the host galaxy continuum. These templates contain a fixed average level of nebular emission from the training sample, as described in Section 5 of Yip et al. [45]. For a detailed description of the fitting, refer to [46,47,48].

FADO is a self-consistent spectral fitting tool that dynamically estimates galaxy spectral parameters. It infers nebular contributions from emission line luminosities. FADO operates in Full-Consistency (FC), Nebular-Continuum (NC), and STellar (ST) modes [28]. The main difference between FADO and qsofitmore is whether nebular emission is modeled self-consistently. FADO self-consistently models nebular emission based on the Lyman-continuum photon output of the stellar populations and the inferred physical conditions (e.g., electron temperature, electron density, metallicity of the ionized gas) rather than fixing nebular emission at a constant level. In this work, we adopted base template spectra from the BC03 model for both qsofitmore and FADO [49]. In addition, both were used to perform extinction correction, rest framing (de-redshifting), and K-correction given the sky coordinates and redshifts. These settings ensure the comparability of the spectral fits. A comparison of spectral fitting by qsofitmore and FADO for the same galaxy is shown in Appendix B.

In general, the differences between qsofitmore and FADO can be summarized in terms of the following three aspects:

(i):

Spectral Fitting Models: FADO employs population spectral synthesis (PSS) to decompose a galaxy spectrum into a linear combination of single stellar population (SSP) templates of different ages and metallicities, thereby inferring the fractional contributions of stellar populations. Additionally, FADO implements physically self-consistent coupling between stellar and nebular emission; it computes nebular continuum and Balmer emission line intensities based on the Lyman continuum (LyC) photon output from the fitted stellar populations and simultaneously fits these components with the observed spectrum. In contrast, qsofitmore performs linear fitting with PCA-based host galaxy templates [41]. These eigenspectra are derived from large observational samples, and implicitly include an average level of nebular emission inherited from the training set; however, they do not distinguish between the stellar and nebular continua of the target galaxy or adapt the nebular component to the specific physical conditions of the target. For emission line fitting, qsofitmore employs high-order Balmer series and complex Fe ii templates.

(ii):

Fitting Algorithms: FADO uses a differential evolution optimization (DEO) algorithm, which is a population-based evolutionary method that employs mutation, crossover, and selection operators to iteratively approach a global optimum. This approach enables multi-objective fitting, allowing several objectives to be optimized simultaneously and promoting physically meaningful solutions. Conversely, qsofitmore performs linear combination fitting using predefined spectral templates (SSPs, emission lines, and Fe ii complexes), minimizing the residuals via least-squares or similar linear optimization techniques.

(iii):

Treatment of Nebular Emissions: FADO implements self-consistent modeling of both the nebular continuum and Balmer emission lines, ensuring consistency between the nebular emission and the properties of the underlying stellar populations. On the other hand, qsofitmore does not compute nebular emission in a self-consistent manner. Instead, it employs PCA-derived galaxy templates in which the nebular emission is fixed at an invariant average level, which does not reflect the actual nebular emission of the target galaxy [40,41,45].

Many studies have compared results from FADO and STARLIGHT, consistently reporting minor differences [20,29,30]. In this work, our comparison between FADO and qsofitmore for low-redshift SFGs tests whether a self-consistent treatment of nebular emission is necessary and whether spectral fits that include nebular emission self-consistently differ from those that treat it as fixed. For details of the two spectral fitting packages, see [28,40].

2.4. Spectral Fitting

Following the comparison above, we performed spectral fitting of the 6558 galaxies identified in Section 2.1 using qsofitmore and FADO. In order to consider intrinsic extinction correction, both methods must return physically meaningful $\mathrm{H}\alpha$ and $\mathrm{H}\beta$ fluxes along with their uncertainties. Accordingly, qsofitmore successfully fit 6546 galaxies and FADO fit 6523. Finally, 6511 galaxies were retained for which the spectra were successfully fit by both FADO and qsofitmore. However, the 6511 successfully fitted galaxies do not meet the requirements of this work due to the number of sources per redshift bin being inconsistent, with a sharp decrease at $0.3<z<0.4$. This interval lies beyond the median redshift of the SDSS survey (around $\sim 0.1$) [50], leading to a noticeable drop in the sample counts. In addition, $\mathrm{H}\alpha$ approaches the upper wavelength limit of SDSS at $z\sim 0.4$, which can prevent a reliable fit. Therefore, we imposed additional sample restrictions.

To place the sample within the typical stellar mass range for SFGs, we restricted the SFG stellar mass ($M_{★}$) to ${10}^9⩽M_{★}⩽{10}^{11}{M}_{\odot }$. This range spans intermediate-mass high-specific star formation rate (sSFR) systems. Stellar masses were derived from the FADO spectral fits. This restriction enables systematic assessment of how nebular emission affects inferred galaxy properties while mitigating biases from extreme or peculiar systems [51]. Furthermore, random sampling was applied to ensure the consistency of the samples within each redshift bin size. We set a redshift bin width of 0.01 to ensure near-equal counts per bin. Finally, all samples were selected via SQL queries and BPT diagrams before fitting with FADO and qsofitmore, ensuring a redshift range of $0<z<0.4$ and ${10}^9⩽M_{★}⩽{10}^{11}{M}_{\odot }$. Ultimately, we selected 696 high-quality samples (referred to hereafter as the Golden Sample), the redshifts and number count distributions of which are displayed in Figure 3. The absence of samples at $0⩽z<0.02$ results from the lower mass threshold at ${10}^9{M}_{\odot }$, as discussed in Section 4.

2.5. Intrinsic Extinction Correction

Young massive stars in star-forming regions produce sufficient ionizing photons to ionize hydrogen. This ionized gas captures photons and undergoes recombination transitions, leading to simultaneous emission of $\mathrm{H}\alpha$ and $\mathrm{H}\beta$. However, dust in the interstellar medium attenuates this emission. Because dust attenuation is more efficient at shorter wavelengths, $\mathrm{H}\beta$ is more suppressed than $\mathrm{H}\alpha$, increasing the observed Balmer decrement ($\mathrm{H}\alpha$/$\mathrm{H}\beta$) [52]. The intrinsic Balmer decrement is typically $(\mathrm{H}\alpha /\mathrm{H}\beta )=2.86$ under Case B recombination, which we adopt for intrinsic extinction correction [53,54,55]. In this work, we define intrinsic extinction as the dust attenuation internal to the galaxy, estimated from the deviation of the observed $\mathrm{H}\alpha$/$\mathrm{H}\beta$ from its intrinsic value. Therefore, the intrinsic extinction of Golden Sample$\mathrm{H}\alpha$ was corrected using the Balmer decrement $(\mathrm{H}\alpha /\mathrm{H}\beta )$.

3. Analysis and Results

This section compares the $\mathrm{H}\alpha$ flux of Golden Sample fitted by qsofitmore and FADO to assess their mutual consistency. We calculate the nebular contribution in the $\mathrm{H}\alpha$ wavelength to examine its redshift dependence. In addition, this section examines the influence of stellar mass and SFR on the nebular contribution to disentangle stellar mass and SFR-related effects.

3.1. Comparison of $H_{\alpha }$ Flux

As shown in Figure 4, the $\mathrm{H}\alpha$ flux from FADO is statistically similar to that from qsofitmore. A Kolmogorov–Smirnov (KS) test was performed on the qsofitmore- and FADO-fitted $\mathrm{H}\alpha$ fluxes to test for distributional equivalence. The KS statistic is 0.059 and the p-value is 0.179, suggesting statistical uniformity between the two fitted $\mathrm{H}\alpha$ subsets. Furthermore, we binned the fluxes by redshift with a bin width of 0.1 in order to test for redshift-dependent offsets between the two fittings. The results in Figure 5 indicate no clear redshift dependent differences in $\mathrm{H}\alpha$ up to $z\sim 0.4$ and stellar mass in ${10}^9$–${10}^{11}{M}_{\odot }$. Generally, the $\mathrm{H}\alpha$ fluxes fitted by qsofitmore and FADO do not exhibit noticeable differences. The median difference is $0.028\mathrm{dex}\approx 6.68\%$.

For SFGs with stellar mass ${10}^9$–${10}^{11}{M}_{\odot }$ at $z<0.4$, these results suggest that $\mathrm{H}\alpha$ fluxes are only weakly sensitive to whether nebular emission is modeled self-consistently or approximated by an average level. However, this does not preclude larger impacts on other diagnostics (e.g., continuum, age, metallicities).

3.2. Nebular Ratio and Other Physical Parameters

We extracted the fitted nebular spectra from the FADO results to further analyze and quantify the nebular contribution. All the parameters provided by FADO are detailed in Appendix Figure A2. Despite known limitations (e.g., dust attenuation and redshift), the $\mathrm{H}\alpha$ flux remains a widely used SFR tracer for nearby galaxies and is the primary diagnostic in this analysis. Equation (3) defines the nebular ratio, while the SFRs of the Golden Sample were calculated with Equations (4) and (5):

$$\mathrm{Nebular}\mathrm{Ratio}={\displaystyle \frac{F_{\mathrm{Nebular}}}{F_{\mathrm{Nebular}}+F_{\mathrm{Stellar}}}}\times 100\%$$

(3)

$$L\left({\mathrm{H}}_{\alpha }\right)=F\left({\mathrm{H}}_{\alpha }\right)\times 4\pi d_L^2$$

(4)

$$\mathrm{SFR}={\displaystyle \frac{L\left({\mathrm{H}}_{\alpha }\right)}{\eta \left({\mathrm{H}}_{\alpha }\right)}}$$

(5)

where $F_{\mathrm{Nebular}}$ is the nebular model flux, $F_{\mathrm{Stellar}}$ is the stellar model flux, $F_{\mathrm{Nebular}}+F_{\mathrm{Stellar}}=F_{\mathrm{Total}}$, $d_L$ is the luminosity distance, $F\left({\mathrm{H}}_{\alpha }\right)$ is the best fitted flux of extinction-corrected $\mathrm{H}\alpha$, $L\left({\mathrm{H}}_{\alpha }\right)$ is the luminosity of $\mathrm{H}\alpha$, and $\eta \left({\mathrm{H}}_{\alpha }\right)$ is the multiplication factor provided by Miranda et al. [20]. We adopted $\eta \left({\mathrm{H}}_{\alpha }\right)={10}^{41.28}\mathrm{erg}{\mathrm{s}}^{-1}{M}_{\odot }^{-1}\mathrm{yr}$ in this work, which is calibrated to the IMF of Chabrier [56].

Figure 6 displays the nebular ratio (from FADO) across other parameters, which include the FADO-fitted $\mathrm{H}\alpha$ flux, extinction-corrected SFR, stellar mass derived from FADO, and galaxy redshift. The corresponding Spearman’s rank correlation coefficient ($\rho$) is marked in each panel. Here, $\rho >0$, $\rho <0$, and $\rho \approx 0$ indicate positive, negative, and no significant monotonic association, respectively. As shown in Figure 6, the nebular ratio is strongly correlated with $\mathrm{H}\alpha$, which is consistent with Miranda et al. [51]. The SFR derived from $\mathrm{H}\alpha$ also exhibits a significant positive correlation with the nebular ratio, as expected given their shared $\mathrm{H}\alpha$ dependence. Despite a substantial sample, the nebular ratio shows a weak but monotonic increase, with increasing stellar mass and redshift overall. In addition, the relationship between the two parameters (black solid line) in each panel was fitted and the uncertainty in each analytic function was estimated from the covariance matrix. The fitted relationships are listed as follows:

$$\text{Nebular}\text{Ratio}=8.799\times {10}^8\times e^{(1.340\pm 0.034)\times logF{\left(H_{\alpha }\right)}_{\text{FADO}}}\left(\text{top}\text{left}\right)$$

(6)

$$\text{Nebular}\text{Ratio}=2.634\times e^{(0.421\pm 0.018)\times log\text{SFR}}\left(\text{top}\text{right}\right)$$

(7)

$$\text{Nebular}\text{Ratio}=(0.431\pm 0.104)log{M}_{*}-(1.448\pm 1.059)\left(\text{bottom}\text{left}\right)$$

(8)

$$\text{Nebular}\text{Ratio}=(4.801\pm 0.524)z+(1.992\pm 0.118)\left(\text{bottom}\text{right}\right)$$

(9)

where $F{\left(H_{\alpha }\right)}_{\text{FADO}}$ is the H$\alpha$ flux fitted by FADO, SFR is calculated by Equation (5), $M_{*}$ is the stellar mass derived from FADO, and z is galaxy redshift.

Equation (6) reveals an exponential growth relationship. This trend is consistent with $\mathrm{H}\alpha$ tracing nebular emission from gas ionized by ultraviolet photons from young massive stars in star-forming regions [33]. Accordingly, larger $\mathrm{H}\alpha$ fluxes generally imply stronger nebular emission, yielding an expected strong correlation with the nebular ratio. Equation (7) shows a strong correlation between nebular ratio and SFR; because SFR is $F\left({\mathrm{H}}_{\alpha }\right)$-based, part of this trend reflects their shared dependence on $\mathrm{H}\alpha$. Galaxies with higher SFR tend to exhibit larger nebular contributions, and consequently larger nebular ratios. Equation (8) shows a weak dependence of the nebular ratio on stellar mass. This is consistent with massive galaxies having lower cold gas fractions and smaller proportions of young stars [57]. This result aligns with the “downsizing” proposed by Mannucci et al. [58], where more massive galaxies exhibit fewer SFRs and higher gas metallicity. Moreover, the large diffusion and small $\rho$ indicate that within ${10}^9-{10}^{11}{M}_{\odot }$, stellar mass is not an ideal predictor of the nebular ratio. Nevertheless, we report the fitted relation in Equation (8), noting that verification with broader mass coverage is required. Equation (9) shows a moderate positive correlation between the nebular ratio and redshift; because higher redshift corresponds to earlier cosmic times with higher sSFRs and gas fractions, such an increase is plausible within $z<0.4$. This trend is consistent with the broader picture in which star formation activity peaks at z∼2 [59]. This result is preliminarily consistent with the expectation put forward in Miranda et al. [20].

3.3. Redshift vs. Nebular Ratio

Figure 7 quantifies the dependence of the median nebular ratio on redshift for the SFG Golden Sample. We divide the sample into four redshift bins and compute the median nebular ratio in each bin; these medians serve as the summary statistics. Yellow points represent median nebular ratio in each redshift bin; a cubic polynomial provided the lowest chi-square and highest $R^2$ among the tested forms, and is used descriptively. The corresponding analytic expressions are presented in Equation (10).

$$MedianNebularRatio=-24.5z^3+2.35z^2+7.72z+1.123.$$

(10)

The results in Figure 7 are consistent with Figure 12 in Miranda et al. [51]; the y-axis has been calibrated to match their convention. Unlike their results, which combined theoretical considerations with EW estimates and extended to $z\sim 6$, our results are derived from direct spectral fitting of the SFG sample at $z<0.4$. We provide preliminary evidence that the predicted redshift dependence of the nebular ratio agrees with that derived from our data within $z<0.4$. This suggests that the nebular contribution to spectral fitting increases with redshift within our probed range, although higher-z behavior requires more observations.

4. Discussion

This section discusses three main issues which will serve as a foundation for future research in this field. First, we compare our results with Miranda et al. [20] to explain the differences between our main conclusions and the relevant parts of their work. Second, we examine other factors that could affect SFR estimation. In particular, we discuss how recent findings on the initial mass function (IMF) may affect SFR estimates and their redshift trends. Finally, we discuss some limitations of this work in terms of sample redshift and stellar mass.

4.1. Comparison with Miranda’s Results and Uncertainty

Comparing the results of Miranda et al. [20], they used the MPA-JHU integrated sample and reported a 0.1 dex difference between the $\mathrm{H}\alpha$ fluxes from FADO and MPA-JHU, while our results show that the median difference between the $\mathrm{H}\alpha$ fluxes fitted by FADO and qsofitmore (with PCA-derived galaxy templates) is 0.028 dex. This indicates that for SFGs at $z<0.4$, the impact of different nebular emission treatments on $\mathrm{H}\alpha$ is small. Unlike [20], this work uses SDSS-DR18 galaxy data and applies strict selection via SQL queries and the BPT diagram to ensure a pure SFG sample. Furthermore, our redshift range of $0<z<0.4$ exceeds the range of $0<z<0.3$ used in Miranda et al. [20], where data above $z\sim 0.25$ were sparse. Although many lower-quality spectra at $0.3<z<0.4$ were excluded during fitting, this interval approaches the upper limit for reliable $\mathrm{H}\alpha$-based SFR in SDSS, as $\mathrm{H}\alpha$ exits the spectral range near $z\sim 0.4$. Finally, we find a modest monotonic increase of the nebular ratio with redshift, and quantify a relation that is only weakly dependent on stellar mass.

4.2. Other Factors Affecting SFR Estimation

The $\mathrm{H}\alpha$ flux obtained by FADO is generally similar to that from qsofitmore, with a median difference of ∼0.028 dex. In FADO’s self-consistent spectral-fitting framework, gas conditions such as electron density, extinction, temperature, and nebular kinematics enter the nebular component, with electron density and temperature often playing leading roles [28]. Therefore, any apparent $\mathrm{H}\alpha$-based SFR underestimation could partially reflect assumed electron densities and temperatures, which we did not independently constrain. Accurate SFR estimates benefit from multi-band observations, as UV, optical, and IR emission arise in different regions and are differentially obscured by dust [60]. AGN can influence host galaxy star formation via feedback. Gas in the host disk can interact with AGN-driven outflows, potentially suppressing or enhancing the overall SFR. Finally, AGN activity in some starbursts can trigger, enhance, or suppress star formation [60]. Such effects can bias single-band SFR estimates regardless of the chosen band. Thus, robust exploration of SFRs requires multi-wavelength data.

Spectral fitting (e.g., FADO and qsofitmore) typically assumes a fixed IMF, yet Li et al. [61] reported IMF variations with metallicity and redshift. Such variations affect $\mathrm{H}\alpha$ luminosities and SFR calibrations [28,62,63]. Although both methods used here adopt the same BC03 IMF [49], ensuring comparability, IMF-dependent calibrations could modulate the observed correlation between the SFR and nebular ratio. We provide preliminary evidence that $\mathrm{H}\alpha$-based SFRs are affected by nebular contamination, with the apparent impact increasing with redshift within $z<0.4$. Accurate SFR measurements require accounting for nebular emission and combining optical, ultraviolet, and infrared data [14,64,65].

4.3. Sample Variation

In this work, all results are based on the conditions $0⩽z⩽0.4$ and ${10}^9⩽M_{★}⩽{10}^{11}{M}_{\odot }$; therefore our characterization of the nebular ratio applies only within this regime. The apparent lack of sources at $z=0-0.02$, shown in the lower-left panel of Figure 3, primarily reflects our stellar mass selection of ${10}^9-{10}^{11}{M}_{\odot }$. This sample incompleteness can be attributed to the Malmquist bias [66]. Due to the detection limits of the SDSS survey, progressively more massive (or intrinsically brighter) galaxies dominate at higher redshifts. Consequently, lower-mass galaxies fall below the flux limit at higher redshifts, biasing the sample toward more massive systems. Conversely, at very low redshifts ($z\sim 0.02$), the survey volume is small, meaning that galaxies with stellar masses exceeding ${10}^9{M}_{\odot }$ are relatively sparse in our selection. Previous studies have also shown that low-redshift field samples contain many systems with ${10}^7$–${10}^8{M}_{\odot }$ [67,68,69,70]. Although this selection approach excludes galaxies at very low redshifts ($z<0.02$) with extremely low masses and rare very-high-mass systems, the interval ${10}^9$–${10}^{11}{M}_{\odot }$ is the most uniformly populated within our sample. Breda et al. [30] investigated the impact of nebular emission on extreme-emission-line galaxies (EELGs) using FADO. However, how the nebular ratio responds in extreme mass systems (e.g., green peas, dwarf ellipticals, and giant ellipticals) remains an open question.

The redshift limitation primarily reflects SDSS sensitivity and wavelength coverage for $\mathrm{H}\alpha$. High-quality $\mathrm{H}\alpha$ detections are scare at $z=0.3-0.4$, and leave the SDSS spectral range beyond $z\sim 0.4$$\mathrm{H}\alpha$. Moreover, a nebular ratio rising with redshift is consistent with higher specific SFRs and gas fractions at earlier cosmic times. Numerous observations have demonstrated that the equivalent width (EW) of emission lines in galaxies, such as the $\mathrm{H}\alpha$ EW, shows a clear increasing trend with redshift, predicting that nebular emission will become a significant spectral component in the redshift interval $z\sim 2-6$ [51,71,72,73,74]. Miranda et al. [51] further pointed out that low-redshift samples, such as the SDSS main sample with an average redshift of $z\approx 0.07$, predominantly contain galaxies with weak nebular emission, leading to insufficient statistics of galaxies with high nebular contributions and thereby limiting our understanding of the nebular ratio’s impact.

In recent years, facilities such as the JWST have greatly expanded studies of high-redshift galaxies, revealing strong nebular contributions and their impact on inferred galaxy properties [75]. Therefore, limited redshift coverage can underestimate the importance of nebular emission in galaxy analyses, especially when studying galaxy evolution in the early universe. Future investigations using other high-quality spectroscopic surveys such as the Large Sky Area Multi-Object Fiber Spectroscopic Telescope Survey (LAMOST) [76] and DESI [77] remain necessary to further explore the nebular ratio. Because H$\alpha$ is inaccessible beyond $z>0.4$, whether other lines (e.g., [O ii], [O iii]) show similar sensitivity to the nebular ratio remains an open question. The effectiveness of FADO for spectra beyond the optical regime also remains to be explored.

5. Conclusions

This work aims to investigate the differences in SFR estimation arising from various spectral fitting packages, with particular emphasis on the combined effects of stellar and nebular emission. To achieve this goal, we have utilized the latest spectroscopic data from the SDSS-DR18 large-scale survey, applying SQL queries and BPT diagrams to select high-quality SFGs within the redshift range $z<0.4$ and stellar mass range ${10}^9$–${10}^{11}{M}_{\odot }$. Employing FADO, a spectral fitting tool capable of self-consistently modeling stellar and nebular emission, along with the recent stellar population-based fitting code qsofitmore, we performed spectral analysis and SFR calculations on the selected sample. This comprehensive approach shows a positive correlation between the nebular ratio and redshift within our sample ($z<0.4$) and stellar mass range in ${10}^9$–${10}^{11}{M}_{\odot }$. This constitutes a quantitative characterization based on direct spectral fits in this low-redshift regime, and is consistent with prior theoretical expectations. This finding has implications for future SFR studies of higher-redshift galaxies, particularly those with non-negligible nebular contributions such as starburst and actively star-forming galaxies. Based on our results, we summarize the following three key conclusions:

(1)

This work compares the recently developed qsofitmore stellar population-based code with PCA-derived galaxy templates to the nebular-inclusive FADO code, aiming to evaluate their differences in fitting the $\mathrm{H}\alpha$ emission. This comparison adds diversity to the evaluation between FADO and other consider nebula emission codes, supporting the robustness of FADO’s self-consistent approach for our sample. We find that FADO produces stable galaxy spectral fits within the low-redshift range ($z<0.4$), showing only minor differences compared to models that treat nebular emission as a constant. Specifically, the median difference in $\mathrm{H}\alpha$ flux between the two codes is approximately $0.028$ dex (corresponding to a linear difference of about $6.68\%$). This result is similar to the findings of Miranda et al. [20], who performed $\mathrm{H}\alpha$ fitting using FADO and MAP-JHU. We infer that for $\mathrm{H}\alpha$-based SFR estimates at $z<0.4$, the choice between self-consistent nebular modeling and an average nebular level has only a modest impact on the $\mathrm{H}\alpha$ flux.

(2)

Statistical analysis indicates that for galaxies with $z<0.4$ and stellar masses in the range ${10}^9-{10}^{11}{M}_{\odot }$, both $\mathrm{H}\alpha$ and SFR serve as reliable tracers of the nebular ratio, which itself shows weak sensitivity to stellar mass within this mass range. In particular, we observe an increasing trend of the nebular ratio with redshift, which is especially pronounced in the interval $z=0.2$ to $0.4$. This finding underscores the necessity of adequately accounting for the nebular contribution in future spectroscopic analyses of higher-redshift star-forming galaxies. Such effects are expected to become increasingly significant under these conditions, particularly when investigating the active star-forming environments of the early universe.

(3)

This work presents a quantitative characterization of the increasing trend of the median nebular ratio with redshift in SFGs, which is largely insensitive (within uncertainties) to variations in stellar mass over ${10}^9$–${10}^{11}{M}_{\odot }$. Moreover, a quantitative model describing the dependence of the average nebular emission fraction on the redshift is established. This empirical relation provides a practical tool for estimating nebular contributions in low-redshift SDSS-like samples, and with appropriate calibration may inform assessments at higher redshifts. This framework provides a basis for quantitatively assessing nebular contributions in future studies of high-redshift galaxies, thereby supporting improved SFR inference and laying a solid foundation for future galaxy evolution studies utilizing large spectroscopic surveys.

In the future, high-quality spectroscopic data from DESI will enable nebular emission samples spanning a broader redshift range. Thanks to its spectral resolution and depth, DESI will deliver precise high-S/N measurements to constrain nebular emissions, substantially increasing the size and fidelity of observational samples [77,78]. Using this dataset, we will be able to perform systematic analyses of the relative contributions and evolution of nebular emission across a wider span of cosmic time. These investigations should refine our understanding of star formation processes and their environmental interactions, thereby informing models of galaxy evolution. Finally, by integrating multi-wavelength spectroscopy, consistent modeling and subtraction of nebular emission across bands should enable more accurate recovery of intrinsic stellar continua. This approach will mitigate continuum flattening and absorption-line dilution caused by nebular emission, improving the precision of inferred ages, metallicities, and star-formation histories to support a more comprehensive view of galaxy formation and evolution.