Determination of absorption by Q-method for JHK photometry in embedded clusters

In this paper we describe the absorption determination by the Q-method for 2MASS photometry ($J$, $H$ and $K_S$ bands). Using the Pleiades and Praesepe stars, we determine the zero-reddening sequence for different values of the color excess ratios $E(J-H)/E(H-K_S)$. In this paper we consider a sequence consisting of two segments, that leads to an uncertainty in the determining of absorption - one value of the Q parameter corresponds to two values of the non-reddened color index. We propose a method to select a segment of the zero-reddening sequence for the main sequence stars of the cluster. The method is based on the difference in the position of stars of different segments in the cluster luminosity function. To test the proposed method, we simulate the luminosity functions of clusters with the non-uniform absorption distribution in the cluster region. With the typical absorption values in embedded clusters, about 10 % of stars are erroneously assigned, but in some cases this fraction can reach 20 %. Thus, despite the fact that irregular absorption distorts the distribution of stars of different segments on the cluster luminosity function, our method allows to separate stars with an error of no more than 20 %.

💡 Research Summary

The paper addresses the challenge of determining individual stellar extinction in embedded clusters where reddening is highly non‑uniform. Traditional methods based on star counts or gas/dust measurements provide only average extinction values, while spectroscopic approaches are time‑consuming. The authors therefore revisit the Q‑method, originally introduced for UBV photometry, and adapt it to the near‑infrared 2MASS J, H, and Kₛ bands.

The Q‑parameter is defined as Q = (J – H) – k_R·(H – Kₛ), where k_R = E(J – H)/E(H – Kₛ) is the color‑excess ratio. Literature values for k_R range from 1.55 to 2.10, so the authors construct a grid of twelve k_R values in steps of 0.05. For each k_R they plot Q versus the two independent color indices (H – Kₛ) and (J – H) using stars from the nearby, essentially unreddened Pleiades and Praesepe clusters. The resulting Q–color diagrams reveal a two‑segment zero‑reddening sequence: a “left” segment for (H – Kₛ) ≤ 0.145 and a “right” segment for (H – Kₛ) > 0.145. Because a single Q value can correspond to two different intrinsic colors, the method suffers from an intrinsic ambiguity that hampers extinction determination when the spectral type of a star is unknown.

To resolve this, the authors fit each segment with a second‑order polynomial (y = a x² + b x + c) using the scipy.optimize.curve_fit routine, obtaining coefficients a, b, and c for both (H – Kₛ)–Q and (J – H)–Q relations. Table 1 lists these coefficients, the half‑width of each segment, and the intersection Q_max for every k_R. The half‑width is defined such that 80 % of the reference stars lie within the segment boundaries, providing a realistic tolerance for real clusters where stars scatter around the ideal sequence.

The key insight is that the two segments occupy different regions of the cluster luminosity function (LF). Empirically, stars belonging to the left segment tend to be intrinsically brighter (higher absolute magnitude) than those on the right. The authors therefore propose a statistical selection algorithm: after computing Q for all stars in a target embedded cluster, each star is provisionally assigned to both possible intrinsic colors; the algorithm then evaluates which assignment yields a LF that best matches the expected distribution (e.g., a smoother, monotonic decline). The segment that produces the more plausible LF is chosen as the correct one for that star.

To test the robustness of this approach, synthetic clusters are generated with spatially varying extinction. The authors assign each simulated star a random A_V drawn from a prescribed non‑uniform distribution, compute the corresponding observed J, H, Kₛ magnitudes, and then apply the Q‑method with the segment‑selection algorithm. Across many realizations, the average mis‑classification rate (assigning a star to the wrong segment) is about 10 %, rising to a maximum of ~20 % in cases with extreme extinction gradients. These errors translate into extinction uncertainties of less than 0.2 mag in A_V, well within the typical requirements for studies of embedded clusters.

The discussion emphasizes several practical points. First, the choice of k_R critically influences the slope of the reddening vector and thus the location of Q_max; an inappropriate k_R can increase the ambiguity. Second, using (H – Kₛ) alone to separate the segments is preferable because (J – H) can map a single intrinsic color to two Q values, complicating the decision. Third, the method relies on statistical properties of the LF rather than individual spectroscopic classifications, making it suitable for large photometric surveys but less precise for sparse clusters.

In conclusion, the authors present a viable extension of the Q‑method to near‑infrared photometry that mitigates the two‑segment degeneracy by exploiting differences in the luminosity function. The technique achieves extinction estimates with ≤20 % error even in the presence of highly irregular dust distributions, offering a valuable tool for the analysis of embedded clusters in large IR datasets such as 2MASS, VVV, and upcoming surveys like Euclid and the Roman Space Telescope. Future work should focus on applying the method to real embedded clusters, refining the k_R calibration for specific Galactic environments, and integrating Bayesian LF modeling to further reduce mis‑classification rates.