As the X-ray luminosity, calculated in erg s−1 in the band energy of a range of 2–10 keV, is one of the outputs of X-CIGALE code, Fig. 20 presents it in comparison with that of adapted input data (observations) shown in Fig. 1, in the same band energy. This comparison shows a good agreement between calculated and observed values of this luminosity of the U/LIRGs where a tight linear fit is found with a slope of 0.896 ± 0.065.

Diffraction causes images to lose sharpness at high f-numbers (i.e., narrow aperture stop opening sizes), and hence limits the potential depth of field.[27] (This effect is not considered in the above formula giving approximate DOF values.) In general photography this is rarely an issue; because large f-numbers typically require long exposure times to acquire acceptable image brightness, motion blur may cause greater loss of sharpness than the loss from diffraction. However, diffraction is a greater issue in close-up photography, and the overall image sharpness can be degraded as photographers are trying to maximize depth of field with very small apertures.[28][29]

In relevance to the variation of the luminosity components as a function of the intrinsic luminosity, the evolution of their trend profiles shown in Fig. 12 can be expressed by the following relationships:

Over the considerations of the stellar mass and the SFR variations of our selected galaxy sample of U/LIRG, each Lst, LAGN and Lx−ray can be separately characterized with SFR. In the logarithmic SFR-luminosity frame, dependencies of Lst, LAGN and Lx−ray are shown in Figs. 9, 10 and 11, respectively. For Lst - SFR relation shown in Fig. 9, the 2nd ordered polynomial fit shows a nearly flat relation at log SFR < 0 for quiescent sources below MS, presented in Fig. 8, and an exponential increase at log SFR > 0 for sources within or above this MS. Both LAGN – SFR and Lx−ray - SFR relations as shown in Figs. 10 and 11, respectively, show increasingly linear variations with a Pearson-r coefficient of 0.36 and 0.32, respectively.

Across the multiwavelength of the SED, observational data were extracted from the NED in mJy. In the X-ray band, flux densities (FX−ray) within the 2–10 keV energy band are considered. In UV/optical/IR ranges, flux measurements were sourced from photometric instruments on various telescopes, including GALEX, SDSS, 2MASS, Spitzer, IRAS, and Herschel where each is equipped with different filter. The X-CIGALE code has prepared to configure with the set of filters. Flux densities in the UV bands were obtained from GALEX, while those in the ugriz bands were sourced from SDSS. The NIR bands (J, H, and K) fluxes were extracted from 2MASS, and the Infrared Array Camera (IRAC) bands at 3.6 and 4.5 μm were derived from Spitzer. In the MIR range, fluxes at 5.8 and 8 μm, as well as the Multiband Imaging Photometer for Spitzer (MIPS) band at 24 μm, were obtained from Spitzer. The 24 μm band data were also taken from IRAS. For the FIR bands, fluxes were gathered from IRAS for the 24 μm band, and from Herschel, using the Photodetector Array Camera and Spectrometer (PACS), for bands at 70, 100, and 160 μm. Additionally, data for bands at 450 and 850 μm were sourced from SCUBA. For all selected sources, each one has photometric measurements across the optical, NIR, MIR, and far-IR regimes, in addition to the X-ray band.

Efstathiou, A. et al. A new look at local ultraluminous infrared galaxies: the atlas and radiative transfer models of their complex physics. MNRAS 512, (2022).

As distance or the size of the acceptable circle of confusion increases, the depth of field increases; however, increasing the size of the aperture (i.e., reducing f-number) or increasing the focal length reduces the depth of field. Depth of field changes linearly with f-number and circle of confusion, but changes in proportion to the square of the distance to the subject and inversely in proportion to the square of the focal length. As a result, photos taken at extremely close range (i.e., so small u) have a proportionally much smaller depth of field.

Other authors such as Ansel Adams have taken the opposite position, maintaining that slight unsharpness in foreground objects is usually more disturbing than slight unsharpness in distant parts of a scene.[20]

Ali, A.A.M., Gadallah, K.A.K., Shalabiea, O.M. et al. Characterizing the luminosity components of luminous infrared galaxies in multi-wavelength from the X-ray to the far-infrared. Sci Rep 14, 25648 (2024). https://doi.org/10.1038/s41598-024-76203-5

Torbaniuk, O. et al. The connection between star formation and supermassive black hole activity in the local Universe. MNRAS 506, (2021).

Verma, A., Rowan-Robinson, M., McMahon, R. & Efstathiou, A. Observations of hyperluminous infrared galaxies with the Infrared Space Observatory: implications for the origin of their extreme luminosities. MNRAS. 335, 574–592 (2002).

According to the mass-luminosity characterization mentioned above, we can also induce the SFR-luminosity dependence of U/LIRGs which is linked to mass variation. Figure 8 illustrates that the trend variation of the SFR generally increases by increasing the stellar mass from ∼5.58 × 109M☉ to 1.6 × 1013M☉, with a mean value of 〈Mst〉 = 6.43 × 1011M☉. Some samples at intermediate stellar masses with low SFR have a remarkable deviation. The SFR varies from ∼ 0.09 M☉yr−1 for IRASF14544-4255 (with small redshift of 0.01573) to 260.95 M☉yr−1 for IRASF10214 + 4724 (with high redshift of 2.2856), with mean value of 〈SFR〉 = 32 M☉yr−1. For this variation, the linear fit reveals that the averaged SFR increases with Mst, especially for high values of the SFR.

Da Cunha, E., Charlot, S. & Elbaz, D. A simple model to interpret the ultraviolet, optical and infrared emission from galaxies. MNRAS 388, (2008).

DOF ≈ 2 N c ( u f ) 2 = 2 N c ( 1 − 1 M T ) 2 {\displaystyle {\text{DOF}}\approx 2Nc\left({\frac {u}{f}}\right)^{2}=2Nc\left(1-{\frac {1}{M_{T}}}\right)^{2}}

Many lenses include scales that indicate the DOF for a given focus distance and f-number; the 35 mm lens in the image is typical. That lens includes distance scales in feet and meters; when a marked distance is set opposite the large white index mark, the focus is set to that distance. The DOF scale below the distance scales includes markings on either side of the index that correspond to f-numbers. When the lens is set to a given f-number, the DOF extends between the distances that align with the f-number markings.

Depth offield calculator

Jones, L. R. et al. The WARPS Survey. II. The log N –log S Relation and the X-Ray Evolution of Low‐Luminosity Clusters of Galaxies. ApJ 495, (1998).

Comparison of the X-ray luminosity in the 2–10 keV band estimated by the SED fitting as an output of X-CIGALE code with those of the observed values of the sample as input data of this code, where the dotted line shows a linear fit.

Image sensor size affects DOF in counterintuitive ways. Because the circle of confusion is directly tied to the sensor size, decreasing the size of the sensor while holding focal length and aperture constant will decrease the depth of field (by the crop factor). The resulting image however will have a different field of view. If the focal length is altered to maintain the field of view, while holding the f-number constant, the change in focal length will counter the decrease of DOF from the smaller sensor and increase the depth of field (also by the crop factor). However, if the focal length is altered to maintain the field of view, while holding the aperture diameter constant, the DOF will remain constant. [6][7][8][9]

LIRGs and ULIRGs are a unique class of galaxies that exhibit extremely high infrared luminosities (LIR) of > 1011L☉ for LIRG and > 1012L☉ for ULIRG1,2,3. These galaxies are often associated with intense star formation and the existence of Active Galactic Nucleus (AGN), which significantly represent their energy sources. AGN multiwavelength properties, as summarized in overviews of4, are induced from various physical processes. Observations, utilizing the James Webb Space Telescope (JWST) and Near Infrared Spectrograph (NIRSpec), provide an evidence of high ionization rates in U/LIRGs, indicating a more complex interplay between star formation and AGN activity5. Understanding the characteristics and the physical properties when they host AGN is crucial for unraveling the complex interplay between star formation and AGN activity in these systems. U/LIRGs serve as valuable laboratories for investigating intense star formation and AGN activity that is heavily veiled by dust. These galaxies provide a unique opportunity to understand and interpret observations from analogous occurrences in the most luminous galaxies such as hyper luminous infrared galaxies6,7, submillimeter galaxies7,8,9and quasars10,11,12,13. Due to these various galaxies, U/LIRGs have garnered significant attention ever since their detection by the Infrared Astronomical Satellite (IRAS) in the 1980s14,15.

Precise focus is only possible at an exact distance from a lens;[a] at that distance, a point object will produce a small spot image. Otherwise, a point object will produce a larger or blur spot image that is typically and approximately a circle. When this circular spot is sufficiently small, it is visually indistinguishable from a point, and appears to be in focus. The diameter of the largest circle that is indistinguishable from a point is known as the acceptable circle of confusion, or informally, simply as the circle of confusion.

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Hickox, R. C. & Alexander, D. M. Obscured Active Galactic Nuclei. Annual Review of Astronomy and Astrophysics vol. 56 Preprint at (2018). https://doi.org/10.1146/annurev-astro-081817-051803

Ueda, Y., Akiyama, M., Hasinger, G., Miyaji, T. & Watson, M. G. Toward the standard population synthesis model of the X-ray background: evolution of X-ray luminosity and absorption functions of active galactic nuclei including compton-thick populations. ApJ 786, 104 (2014).

In comparison with other selected U/LIRG sample (67 galaxies) collected from GOALS, recent study by91 reproduced the SED fittings in a multi-wavelength range from the UV to the sub-millimetre. In which, the values of SFR have higher range from 2.4 to 410 M☉yr−1 with 〈SFR〉 = 81 M☉yr−1 than those of our sample (63 galaxies) which varies from ∼ 0.09 M☉yr−1 to 260.95 M☉yr−1 with 〈SFR〉 = 32 M☉yr−1. For our sample presented in Fig. 8, it is obvious that 15 galaxies laying below the dotted horizontal line have values of SFR less than 1 as listed bottom in Appendix A. For these sources, their corresponding AGN fractions are very low (< 0.06) with an exception for one source (IRASF14544-4255; referred by an arrow in Fig. 8) where its AGN fraction is high (0.3) giving high LX−ray of 1.2 × 1044 erg s−1. This source has the lowest value of the SFR limits with low mass of ≈ 1.63 × 1010M☉.

Traditional depth-of-field formulas can be hard to use in practice. As an alternative, the same effective calculation can be done without regard to the focal length and f-number.[b] Moritz von Rohr and later Merklinger observe that the effective absolute aperture diameter can be used for similar formula in certain circumstances.[19]

Thomas Sutton and George Dawson first wrote about hyperfocal distance (or "focal range") in 1867.[42] Louis Derr in 1906 may have been the first to derive a formula for hyperfocal distance. Rudolf Kingslake wrote in 1951 about the two methods of measuring hyperfocal distance.

Bridge, C. R. et al. A new population of high-z, dusty Lyα emitters and blobs discovered by wise: Feedback caught in the act? ApJ 769 (2013).

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

The SED curves of AGNs are shaped by a power-law distribution across multiple wavelengths, encompassing absorptions and emissions from various sources within the galaxy37. The UV/optical/NIR ranges are primarily dominated by stellar emissions, offering insights into the star-formation history and attenuation caused by dust. AGNs exhibit a distinctive X-ray emission which is widely utilized as a tracer of the black hole accretion rate38. The FUV-FIR range in the galaxy SED contains nebular lines and continuum emissions, arising from Lyman continuum photons, particularly by hydrogen lines, and is vital in tracing recent star formation39,40. The MIR range contributes fractionally to AGN emission where the MIR-FIR range presents dust processing due to absorption/emission of starlight. In this study, a combination of toroidal and polar dust with smooth and clumpy phases is assumed to reproduce SEDs in a broad range of emissions from X-ray to Far-Infrared (FIR). In multi-wavelengths from hard X-Ray to Radio, the SED physical properties of U/LIRGs have been comprehensively studied by41 considering their merger evolution, where their radio band is dominated by the starburst emission. For the radio-to-far-ultraviolet SEDs of ULIRGs, recent study by23 provides new insights into the global properties of infrared-bright galaxies.

This section covers some additional formula for evaluating depth of field; however they are all subject to significant simplifying assumptions: for example, they assume the paraxial approximation of Gaussian optics. They are suitable for practical photography, lens designers would use significantly more complex ones.

Hopkins,[33] Stokseth,[34] and Williams and Becklund[35] have discussed the combined effects using the modulation transfer function.[36][37]

For a given size of the subject's image in the focal plane, the same f-number on any focal length lens will give the same depth of field.[11] This is evident from the above DOF equation by noting that the ratio u/f is constant for constant image size. For example, if the focal length is doubled, the subject distance is also doubled to keep the subject image size the same. This observation contrasts with the common notion that "focal length is twice as important to defocus as f/stop",[12] which applies to a constant subject distance, as opposed to constant image size.

To generate the intrinsic stellar spectrum for young and old stars with various metallicities and initial mass functions, the bc03 module utilizes the stellar populations of46. Besides calculating stellar luminosity (Lst) and mass, this module also estimates the gas mass (Mg) parameter47,48. The gas mass specifically pertains to material ejected from stars, including stellar winds and supernovae, signifying the total gas mass returned to the interstellar medium through stellar evolution. Multicomponent photoionization models matching the emission lines can be generated by the nebular module relying on the adopted cloudy model49,50. The impact of attenuation laws in galaxies can be modeled using the dustatt_modified_starburst module proposed by51 which is better suited for applying the starburst curve. This introduces a power-law slope to modify the attenuation, along with a UV bump at 217.5 nm20,52. The dl2014 module computes of the dust emission contribution to the SEDs based on the model proposed by30,53. Dust grains are assumed to be a combination of amorphous silicate, graphite, and PAHs where they are heated primarily from stellar populations by a minimum single radiation field (Umin) in diffuse regions to its maximum value (Umax) star-forming regions. The dust emission curve is defined through the utilization of a power-law index (α) and the dust mass-fraction (γ) when the dust is exposed to the maximum radiation Umax. For the torus containing smooth and clumpy phases, the skirtor2016 module54 is used to fit the SEDs of AGN emissions, considering approximately 97% of the dust mass (Md) and the remaining of the smooth phase43. adapted this module considering both the old and new essential luminosities, normalized to unity. The parametrization of this model is based on the physical and geometrical properties of the galactic disc, reproducing the galaxy SED for various lines of sight, and viewing angles. For which, type 2 AGN (edge-on galaxy) is obscured by the dusty torus along the equatorial direction whereas type 1 AGN (face-on galaxy) is directly visible but slightly obscured by polar dust clumps. To estimate the luminosity as a function of wavelength in the FUV to FIR range43, updated the SKIRTOR model using luminosities supported by observations55:

As we are concern in characterizing the SED luminosity components of the U/LIRG galaxies, we compare their variations versus the intrinsic luminosity with those of the obscured AGN galaxies which is similar in highly emitted IR emissions due to the existence of the dusty torus. Both U/LIRG and obscured AGN galaxies are similarly fueled by an energetically active nucleus which is surrounded by a dusty torus. The decomposed SED luminosity components can be used to differentiate between them, Figs. 15, 16 and 17 compare the variation of the luminosity components of U/LIRGs to those of obscured AGN galaxies clarified by24 in relative to the intrinsic AGN power. In this comparison, it is clear that the variation of both the stellar and X-ray luminosities with increasing the intrinsic power of the nucleus is somehow faster of obscured AGN than those of U/LIRG, showing that this luminosity is highest of obscured AGN at high values of the intrinsic power. But for the variation of the AGN luminosity of both, they have a similar trend variation. For X-ray luminosity of our sample of U/LIRG compared to other similar sample of U/LIRG by41 as presented in Fig. 17, there is a god agreement in their variation with the intrinsic luminosity.

Montero, F. R., Davé, R., Wild, V., Anglés-Alcázar, D. & Narayanan, D. Mergers, starbursts, and quenching in the SIMBA simulation. MNRAS 490, (2019).

A. Ali and K. Gadallah wrote the main manuscript text and prepared the figures. O. Shalabiea and M. Beheary discussed the results and wrote the discussion. All authors reviewed the manuscript.

Just, D. W. et al. The X-Ray Properties of the Most Luminous Quasars from the Sloan Digital Sky Survey. ApJ 665, (2007).

Booth, C. M. & Schaye, J. The interaction between feedback from active galactic nuclei and supernovae. Sci. Rep. 3, 1738 (2013).

Another approach is focus sweep. The focal plane is swept across the entire relevant range during a single exposure. This creates a blurred image, but with a convolution kernel that is nearly independent of object depth, so that the blur is almost entirely removed after computational deconvolution. This has the added benefit of dramatically reducing motion blur.[22]

Within the host galaxy disk, the luminosity, partially absorbed by dust, dominates the SED spectrum in the IR ranges. This is mainly attributed to the re-emitted AGN dust luminosity where the U/LIRGs are characterized by their luminous IR emissions. For the dust-to-gas mass ratio (Md/Mg) of the galaxy sample of U/LIRGs, Fig. 5 illustrates that this ratio decreases with increasing MT, varying from ∼ 0.06 at the highest mass to ∼ 5.6 × 10−6 at the lowest one. Only one object (IRASF11095-0238) deviates from this pattern, displaying a lower value at low mass.

The SFR–Mst relation generally represents the galaxy main sequence (MS) of star formation in galaxies80,81,82. It is shown in Fig. 8 as line-open symbols, calculated using Eq. 5 in80with free parameters of the maximum log SFR (S0), the turnover mass (M0) in log Mst, and the power-law slopes of α and β at low and high stellar masses, respectively. The adaptation values of these parameters are given in Table A2 in Appendix A. For sources localized within the prediction band of the fit in Fig. 8, there are an agreement with two profiles of the MS provided with ranges of redshifts of 0.005 < z < 0.3 (dotted red line-symbol) and another of 0.3 < z < 2.3 (dashed blue line-symbol), respectively. On the other hand, there are two groups of sources showing deviation from the fit area. One is above the fit at low masses (Mst < 1011M☉) while another is low the fit with intermediate masses of Mst ∼ 1010 − 1012M☉. As our sample is for the U/LIRGs, it may contain sources with high SFR and others with low SFR. Sources, which appear above the prediction band of the fit with very high SFR, are probably starburst galaxies due to the high activity of their SFR. Other sources, which appear below the prediction band of the fit with very low SFR, are probably quiescent. The latter are expected to be exposed to galaxy quenching mechanisms such as: i) A strong jet, from an active supermassive black holes (SMBH), heats up the cold gas within the galaxy causing a reduction in its star formation activity83,84,85; ii) Although the galaxy mergers cause rapid bursts of star formation, hence they produce a high rate of supernovae which hold up the cold gas, leading to quenching86,87; iii) Environmental quenching occurs because a galaxy is affected by ram pressure force in dense intergalactic medium, leading to gas depletion88.

Photographers can use the lens scales to work backwards from the desired depth of field to find the necessary focus distance and aperture.[38] For the 35 mm lens shown, if it were desired for the DOF to extend from 1 m to 2 m, focus would be set so that index mark was centered between the marks for those distances, and the aperture would be set to f/11.[f]

Pérez-Torres, M., Mattila, S., Alonso-Herrero, A., Aalto, S. & Efstathiou, A. Star formation and nuclear activity in luminous infrared galaxies: an infrared through radio review. A&ARv 29, 2 (2021).

Toba, Y. et al. Does the mid-infrared–hard X-ray luminosity relation for active galactic nuclei depend on Eddington ratio? MNRAS 484, (2019).

Toba, Y. et al. How Does the Polar Dust Affect the Correlation between Dust Covering Factor and Eddington Ratio in Type 1 Quasars Selected from the Sloan Digital Sky Survey Data Release 16? ApJ 912, (2021).

Dubois, Y., Gavazzi, R., Peirani, S. & Silk, J. AGN-driven quenching of star formation: Morphological and dynamical implications for early-type galaxies. MNRAS 433, (2013).

We characterize luminosity components of Ultra/Luminous Infrared Galaxies (U/LIRGs) in multi-wavelength from the X-ray to far-infrared. A set of 63 AGN U/LIRGs was selected where these galaxies are powered by a central active galactic nucleus (AGN). Utilizing the X-CIGALE code, SEDs for these galaxies are carried out where their SEDs are fitted with observations. Accordingly, the physical parameters such as the stellar mass, the dust-to-gas mass ratio, and the star formation rate are calculated. The total luminosity and its decomposed components (stellar, AGN, X-ray) are also calculated. We characterized these luminosities in relative to the intrinsic luminosity and in relative to each other. As a function of the stellar mass, these luminosities reveal an increase with different correlation coefficients, showing a strong correlation. In correlation with the intrinsic AGN power, the stellar, AGN, and X-ray luminosities are strongly correlates in their variation to the intrinsic AGN luminosity, showing stronger correlations of AGN, and X-ray luminosities than those of the stellar one. In relationships between various luminosity components, both the stellar and X-ray luminosities reveal strong correlations with the AGN luminosity. On the other hand, the X-ray luminosity varies strongly with the stellar luminosity and moderately with IR luminosity. Compared to obscured AGN galaxies, both the stellar and AGN luminosities similarly vary with increasing the intrinsic power of the active nucleus but for obscured AGN they are faster in their variation than that of U/LIRG. These correlations may offer valuable insights to understand the physical properties and their relationships through the evolution of U/LIRGs.

Sanders, D. B., Mazzarella, J. M., Kim, D. C., Surace, J. A. & Soifer, B. T. The IRAS revised Bright Galaxy Sample. Astron. J. 126, 1607 (2003).

On a view camera, the focus and f-number can be obtained by measuring the depth of field and performing simple calculations. Some view cameras include DOF calculators that indicate focus and f-number without the need for any calculations by the photographer.[39][40]

The luminosity components, represented by the AGN luminosity (blue stars), stellar luminosity (red circles), and X-ray luminosity (black triangles) versus the intrinsic AGN luminosity.

Ramos Padilla, A. F. et al. The AGN contribution to the UV-FIR luminosities of interacting galaxies and its role in identifying the main sequence. MNRAS 499, (2020).

Dey, S., Goyal, A., Małek, K. & Díaz-Santos, T. Radio-only and radio-to-far-ultraviolet spectral energy distribution modeling of 14 ULIRGs: insights into the Global properties of Infrared Bright galaxies. ApJ. 966, 61 (2024).

Schulz, J., Fritze-Alvensleben, U., Möller, C. S. & Fricke, K. J. Spectral and photometric evolution of simple stellar populations at various metallicities. A&A 392, (2002).

The term "camera movements" refers to swivel (swing and tilt, in modern terminology) and shift adjustments of the lens holder and the film holder. These features have been in use since the 1800s and are still in use today on view cameras, technical cameras, cameras with tilt/shift or perspective control lenses, etc. Swiveling the lens or sensor causes the plane of focus (POF) to swivel, and also causes the field of acceptable focus to swivel with the POF; and depending on the DOF criteria, to also change the shape of the field of acceptable focus. While calculations for DOF of cameras with swivel set to zero have been discussed, formulated, and documented since before the 1940s, documenting calculations for cameras with non-zero swivel seem to have begun in 1990.

Ho, S. H., Serjeant, S., Myung, G. L., Kang, H. L. & White, G. J. The ultraluminous and hyperluminous infrared galaxies in the Sloan Digital Sky Survey, 2dF Galaxy Redshift Survey and 6dF Galaxy Survey. Monthly Notices of the Royal Astronomical Society vol. 375 Preprint at (2007). https://doi.org/10.1111/j.1365-2966.2006.11279.x

Motion pictures make limited use of aperture control; to produce a consistent image quality from shot to shot, cinematographers usually choose a single aperture setting for interiors (e.g., scenes inside a building) and another for exteriors (e.g., scenes in an area outside a building), and adjust exposure through the use of camera filters or light levels. Aperture settings are adjusted more frequently in still photography, where variations in depth of field are used to produce a variety of special effects.

Logarithmic relationships between the luminosity components including Lst and Lx−rayversus the LAGN (top panels), and Lx−ray versus Lst and LIR (bottom panel), where values of LIR were collected from the literature as listed in Table A1 in Appendix A. dashed-lines represents the linear fit with 95% confidence band (red-shaded areas).

where \(\:\varGamma\:\), \(\:E\) and \(\:{E}_{cut}\)are the photon index, the energy, and the cut-off energy, respecti vely. Other three sources contribute to X-ray emissions: low-mass X-ray binaries (LMXBs), high-mass X-ray binaries (HMXBs), and hot gas. In Eq. (3), X-ray emission arises from the AGN, LMXBs/HMXBs, and hot gas at cut-off energies (\(\:{E}_{cut}\)) of 300 keV, 100 keV, and 1 keV, respectively. Moreover, the estimates of the luminosities are considered to be viewing-angle-dependent where UV/optical emissions are assumed to be anisotropic and X-ray emissions to be isotropic. The photon index \(\:{\Gamma\:}\) was taken as 1.8 for X-ray AGN emission. To represent the fitting between X-ray and other wavelength ranges, the deviation of the maximum deviation (\(\:{\left|{{\Delta\:}}_{ox}\right|}_{max}\)) was used based on the αox − L2500Å relationship56. Here, L2500Å refers to the de-reddened AGN luminosity at 2500 Å, and αox represents the SED slope between the UV at 2500 Å and X-ray at 2 keV.

For a model of a U/LIRG radiatively fueled by an active nucleus and surrounded by a dusty torus, we are motivated to describe the variation of the SED luminosity components with the intrinsic AGN luminosity and the stellar mass of the host galaxy. For which, decomposing the SED spectrum is considered where the various emissions are released across the X-ray to the FIR. Both SED model and the galaxy sample are given in Sects. 2 and 3, respectively. In Sects. 4 and 5, we present the results and discussion of the SED model outputs, respectively, while the conclusion is summarized in Sect. 6.

The blur disk diameter b of a detail at distance xd from the subject can be expressed as a function of the subject magnification ms, focal length f, f-number N, or alternatively the aperture d, according to

Ferland, G. J. et al. The 2013 release of CLOUDY. Revista Mexicana de Astronomia y Astrofisica vol. 49 Preprint at (2013).

Ciesla, L., Elbaz, D. & Fensch, J. The SFR-M main sequence archetypal star-formation history and analytical models. A&A 608, (2017).

s = 2 D N D F D N + D F , {\displaystyle s={\frac {2D_{\mathrm {N} }D_{\mathrm {F} }}{D_{\mathrm {N} }+D_{\mathrm {F} }}},}

The depth of field can be determined by focal length, distance to subject (object to be imaged), the acceptable circle of confusion size, and aperture.[2] Limitations of depth of field can sometimes be overcome with various techniques and equipment. The approximate depth of field can be given by:

AGN activity in LIRGs or ULIRGs plays a pivotal role in shaping their Spectral Energy Distributions (SEDs). The unified model of AGN16,17,18 suggests that AGNs consist of a central supermassive black hole surrounded by an accretion disk and a torus of obscuring dust and gas. The interaction between the AGN and the surrounding environment, including the torus, influences the observed SED properties of LIRGs and ULIRGs. Observations further enhance our understanding of outflows and AGN activities in ULIRGs providing new insights into AGN-driven feedback mechanisms19.

Moreover, traditional depth-of-field formulas assume equal acceptable circles of confusion for near and far objects. Merklinger[c] suggested that distant objects often need to be much sharper to be clearly recognizable, whereas closer objects, being larger on the film, do not need to be so sharp.[19] The loss of detail in distant objects may be particularly noticeable with extreme enlargements. Achieving this additional sharpness in distant objects usually requires focusing beyond the hyperfocal distance, sometimes almost at infinity. For example, if photographing a cityscape with a traffic bollard in the foreground, this approach, termed the object field method by Merklinger, would recommend focusing very close to infinity, and stopping down to make the bollard sharp enough. With this approach, foreground objects cannot always be made perfectly sharp, but the loss of sharpness in near objects may be acceptable if recognizability of distant objects is paramount.

Yamada, S. et al. Comprehensive Broadband X-Ray and Multiwavelength study of active Galactic nuclei in 57 local luminous and Ultraluminous Infrared galaxies observed with NuSTAR and/or Swift/BAT. ApJ Suppl. Ser. 257, 61 (2021).

Inoue, A. K. Rest-frame ultraviolet-to-optical spectral characteristics of extremely metal-poor and metal-free galaxies. MNRAS 415, (2011).

Lonsdale, C. J., Farrah, D. & Smith, H. E. Ultraluminous Infrared Galaxies. in Astrophysics Update 2 285–336Springer Berlin Heidelberg, doi: (2006). https://doi.org/10.1007/3-540-30313-8_9

To represent the contributions of the galactic disc components to the galaxy SED, several modules were selected, covering aspects such as star-formation history, stellar populations, nebular emission, attenuation law, dust emission, and active nucleus. Detailed descriptions of these modules and their physical parameters based on the assumptions of42,43] and44, are given in (e.g24).

In luminosity-luminosity dependence, the variation of stellar, AGN, and X-ray luminosities with increasing the intrinsic AGN luminosity, shows upward trends but with various rates of variation. For which, these luminosities of U/LIRGs shows a strong correlation with the intrinsic AGN luminosity where both of AGN and X-ray have strong correlations than that of Stellar luminosity. This agrees with that of obscured AGN. Finally, relationships between different luminosity components are depicted, revealing strong correlations of stellar and X-ray luminosities with the AGN luminosity. On the other hand, the X-ray luminosity strongly corelates to the stellar luminosity but it has an intermediate variation with increasing IR luminosities. In summary, this analysis can offer valuable insights into the physical properties and their relationships of U/LIRGs.

where, \(\:{t}_{0}\) represents the time at a significant increase or decrease in SFH, and \(\:{r}_{SFR}\) denotes the ratio of the \(\:\text{S}\text{F}\text{R}\) at \(\:{t=t}_{0}\) to that at \(\:t\:>\:0\).

These relationships indicate that both AGN and X-ray (in the interval of 2–10 KeV) emissions are strongly corelated to the intrinsic accretion power that fuels the host galaxy disk than those of the stellar emissions from stars.

For the results shown in Figs. 4, 6 and 7 in log-log frame, the values of the correlation coefficient of the luminosity-mass dependences are listed in Table 2. Under the considerations assumed for the strength of each relationship in this table, we accordingly clarify its correlation coefficient. Versus the total mass shown in Fig. 4, the total luminosity shows a strong correlation while the intrinsic luminosity shows an intermediate one. For the luminosity components shown in Fig. 6 versus Mst, they have a strong correlation with increasing the stellar mass. On the other hand, as shown in Fig. 7, both of Lst, LAGN and LX−ray have weak correlations with increasing the dust contribution where they slightly decrease.

For LIRG/ULIRG galaxies, the parameters of selected modules were adapted to fit them with observational data from the X-ray to the FIR range. To produce their SED, a module (sfhdelayedbq) of delayed star formation history was used for flexibility by considering recent quenching of the star formation rate (SFR). This module enables the SFR to increase from the start of time (t) reaching a peak at t = τmain, which represents the e-folding times of the stellar populations. The SFR was estimated following the assumptions of45,

The X-CIGALE code was conducted to reproduce SED models for all galaxies of the sample. The best fit of these models, controlled by the free parameters given in Table 1, is evaluated by getting best values of the reduced chi-square (χ2) where its values are listed in the Table A1 (Appendix A).

At the extreme, a plenoptic camera captures 4D light field information about a scene, so the focus and depth of field can be altered after the photo is taken.

Numerical aperture

For some selected sources from the galaxy sample, Fig. 3 shows the SED fittings, in the rest frame, performed using the X-CIGALE code. The SED spectra of U/LIRGs, characterized by their IR emissions, clearly highlight differently the contribution of emissions overall multiwavelength spectra from the X-ray to FIR range.

Fritz, J., Franceschini, A. & Hatziminaoglou, E. Revisiting the infrared spectra of active galactic nuclei with a new torus emission model. MNRAS. 366, 767–786 (2006).

Lee, N. et al. A turnover in the galaxy main sequence of star formation at M ∗ ∼ 1010 M ⊙ for redshifts z < 1.3. ApJ 801, (2015).

For a given subject framing and camera position, the DOF is controlled by the lens aperture diameter, which is usually specified as the f-number (the ratio of lens focal length to aperture diameter). Reducing the aperture diameter (increasing the f-number) increases the DOF because only the light travelling at shallower angles passes through the aperture so only cones of rays with shallower angles reach the image plane. In other words, the circles of confusion are reduced or increasing the DOF.[10]

Gadallah, K. A. K. & Ali, A. A. M. Impacts of the PAH size and the radiation intensity on the IR features of illuminated dust within the reflection nebulae. Adv. Space Res. 67, 4222-4237 (2021).

Leslie, S. K. et al. The VLA-COSMOS 3 GHz Large Project: Evolution of Specific Star Formation Rates out to z ∼ 5. ApJ 899, (2020).

the harmonic mean of the near and far distances. In practice, this is equivalent to the arithmetic mean for shallow depths of field.[44] Sometimes, view camera users refer to the difference vN − vF as the focus spread.[45]

depth offield中文

In a logarithmic frame of luminosity-luminosity relations as shown in Fig. 12, variations of both of Lst, LAGN and Lx−ray are illustrated as a function of the intrinsic luminosity of the AGN’s disk accretion power. These variations demonstrate upward trends but with different variation rates as the intrinsic AGN luminosity increases. The correlation coefficients of their variations with intrinsic luminosity are also listed in Table 2. It is obviously that these correlation coefficients are strong for U/LIRGs in agreement with those of obscured AGN galaxies24 which is characterized by a highly silicate absorption band at 9.7 μm. What is only different is that the Lst - Lint correlation is slightly lower for U/LIRG than that of for obscured AGN galaxies.

The sample of galaxies was selected considering the availability of observed X-ray fluxes available from the NED (NASA/IPAC Extragalactic Database) across different energy bands where they carried out using diverse X-ray satellites. Since the sample selection is flux-limited, the sample was tested to the Malmquist bias in relevance to the redshift as shown in Fig. 1 where the rest-frame X-ray luminosity (L2–10keV) was adapted to the 2–10 keV range. The luminosity values were calculated relying on X-ray flux densities (FX−ray) within the 2–10 keV energy band. These values were corrected for the galactic absorption by approximately a factor of 2 using PIMMS (Portable Interactive Multi-Mission Simulator;78 which is related to the hydrogen column density. The rest-frame L2–10keV was estimated by L2–10keV= 4π(dL)2F2–10keV, where dL is the luminosity distance. The K-correction for this luminosity was carried out as the same way in Appendix B in79.

The AGN luminosity of different type of sources versus the SFR. Values of AGN (open circles), SB (closed circles), SIGS (open squares), and LSM (closed trianles) are taken from92.

Lensfocus

If a subject is at distance s and the foreground or background is at distance D, let the distance between the subject and the foreground or background be indicated by

The depth of field (DOF) is the distance between the nearest and the farthest objects that are in acceptably sharp focus in an image captured with a camera. See also the closely related depth of focus.

Some methods and equipment allow altering the apparent DOF, and some even allow the DOF to be determined after the image is made. These are based or supported by computational imaging processes. For example, focus stacking combines multiple images focused on different planes, resulting in an image with a greater (or less, if so desired) apparent depth of field than any of the individual source images. Similarly, in order to reconstruct the 3-dimensional shape of an object, a depth map can be generated from multiple photographs with different depths of field. Xiong and Shafer concluded, in part, "... the improvements on precisions of focus ranging and defocus ranging can lead to efficient shape recovery methods."[21]

For cameras that can only focus on one object distance at a time, depth of field is the distance between the nearest and the farthest objects that are in acceptably sharp focus in the image.[1] "Acceptably sharp focus" is defined using a property called the "circle of confusion".

When the POF is rotated, the near and far limits of DOF may be thought of as wedge-shaped, with the apex of the wedge nearest the camera; or they may be thought of as parallel to the POF.[17][18]

In Table A1 (Appendix A), listed a sample of 63 LIRG/UlIRG galaxies selected to perform SEDs for the current work. This sample includes a group of 53 galaxies taken from the Great Observatories All-sky LIRG Survey (GOALS59) which totally contains 180 LIRGs and 22 ULIRGs located in the local universe with redshifts z < 0.088. These galaxies are part of the IRAS Revised Bright Galaxy Sample (RBGS60), which is an extensive collection of 629 extragalactic objects showing 60 μm fluxes above 5.24 Jy at Galactic latitudes |b| > 5. These U/LIRGs have been extensively studied across multiple wavelengths, particularly, infrared observations carried out with the IR telescopes of Spitzer61,62,63, AKARI64,65, and Herschel66,67,68. Additionally, X-ray observations have been performed using the Chandra X-ray telescope69,70 and the NuSTAR X-ray telescope71,72. Another group consisting of 10 sources, with redshift z > 0.088, was collected from the literature73,74,75,76,77.

焦深

More so than in the case of the zero swivel camera, there are various methods to form criteria and set up calculations for DOF when swivel is non-zero. There is a gradual reduction of clarity in objects as they move away from the POF, and at some virtual flat or curved surface the reduced clarity becomes unacceptable. Some photographers do calculations or use tables, some use markings on their equipment, some judge by previewing the image.

Bendo, G. J. et al. Tests of star formation metrics in the low-metallicity galaxy NGC5253 using ALMA observations of H30α line emission. MNRAS 472, (2017).

Paspaliaris, E. D. et al. The physical properties of local (U)LIRGs: A comparison with nearby early- And late-type galaxies. A&A 649, (2021).

Hansma and Peterson have discussed determining the combined effects of defocus and diffraction using a root-square combination of the individual blur spots.[30][31] Hansma's approach determines the f-number that will give the maximum possible sharpness; Peterson's approach determines the minimum f-number that will give the desired sharpness in the final image and yields a maximum depth of field for which the desired sharpness can be achieved.[d] In combination, the two methods can be regarded as giving a maximum and minimum f-number for a given situation, with the photographer free to choose any value within the range, as conditions (e.g., potential motion blur) permit. Gibson gives a similar discussion, additionally considering blurring effects of camera lens aberrations, enlarging lens diffraction and aberrations, the negative emulsion, and the printing paper.[27][e] Couzin gave a formula essentially the same as Hansma's for optimal f-number, but did not discuss its derivation.[32]

The lens design can be changed even more: in colour apodization the lens is modified such that each colour channel has a different lens aperture. For example, the red channel may be f/2.4, green may be f/2.4, whilst the blue channel may be f/5.6. Therefore, the blue channel will have a greater depth of field than the other colours. The image processing identifies blurred regions in the red and green channels and in these regions copies the sharper edge data from the blue channel. The result is an image that combines the best features from the different f-numbers.[26]

With decomposing the total luminosity over its SED curve from the X-ray to the FIR, essential components of Lst, LAGN and Lx−ray are produced. Figures 6 and 7 display the variation of these luminosity components versus Mst and Md/Mg, respectively. For which, the Lst largely dominates the luminosity emissions while both of LAGN and LX−ray have a smaller contribution. In Fig. 6, the linear fit Lst, LAGN and Lx−ray with 95% confidence band of their logarithmic values shows positive various variations with increasing the stellar mass, giving a Pearson-r coefficient of 0.43, 0.44, and 0.42, respectively. On the other hand, as shown in Fig. 7, the variations these luminosities versus the logarithm of dust-to-gas mass ratio have a passive variation with Pearson-r coefficient of -0.37, -0.29, and − 0.26, respectively.

Light Scanning Photomacrography (LSP) is another technique used to overcome depth of field limitations in macro and micro photography. This method allows for high-magnification imaging with exceptional depth of field. LSP involves scanning a thin light plane across the subject that is mounted on a moving stage perpendicular to the light plane. This ensures the entire subject remains in sharp focus from the nearest to the farthest details, providing comprehensive depth of field in a single image. Initially developed in the 1960s and further refined in the 1980s and 1990s, LSP was particularly valuable in scientific and biomedical photography before digital focus stacking became prevalent.[23][24]

The relationship between estimated and true values (mock results) is depicted for Mst, Mg, and Md, SFR in the top panels (from left to right) and for the luminosity components Lst, LAGN, LX-ray, and Lint in the bottom panels (from left to right), and vertical lines indicate the 1σ error bars.

Figure 13 displays the relationships between the luminosity components themselves in log-log frame. From which, the variation of Lst and LX−ray versus LAGN (top panels) have strong correlation coefficients. As listed in Table 2, they have correlation coefficients with 0.867 ± 0.001 and 0.982 ± 0.001, respectively. On the other hand, those of LX−ray versus Lst has a strong correlation while it versus LIR has an intermediate correlation. For LX−ray component, its values (in erg s−1) and the corresponding AGN fractions (fAGN) are given in Table A1 in Appendix A. For these parameters, we found that 19 sources from our sample have LX−ray < 1042 erg s−1 where their SEDs fit with very low AGN fraction (< 0.02) with an exception for only one source (IRASF13197-1627) which its SED fits with high AGN fraction (0.35). This source may exhibit a strong AGN power originating from the SMBH.

Open access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB).

A new X-ray module incorporates the X-ray spectrum of the AGN as well as of the galaxy components. It is designed to fit the X-ray data covering the range of 10−6 to 5 × 10−3 μm in the rest frame. The intrinsic AGN X-ray spectrum’s flux (fν) is computed using a power-law form with a high-energy exponential cut-off as follows:

Kim, D. C. & Sanders, D. B. The IRAS 1 Jy Survey of Ultraluminous Infrared galaxies. I. The sample and luminosity function. ApJ Suppl. Ser. 119, 41 (1998).

b = f m s N x d s ± x d = d m s x d D . {\displaystyle b={\frac {fm_{\mathrm {s} }}{N}}{\frac {x_{\mathrm {d} }}{s\pm x_{\mathrm {d} }}}=dm_{\mathrm {s} }{\frac {x_{\mathrm {d} }}{D}}.}

For the sample of U/LIRGs, the SED curves were fitted with observational data to estimate various galactic physical parameters. The findings, generated by X-CIGALE code, are possibly affirmed by the analysis of a mock catalogue which serves as a consistent method giving the reliability of the derived physical properties. In this approach, the code determines the best fit for each object, constructing a mock catalogue. For each best-fit flux, noise is introduced by adding random values drawn from a Gaussian distribution with the same standard deviation as the observed flux. This mock data is analyzed in the same manner as the original observed data, allowing the accuracy of the parameter estimation to be evaluated by comparing observed values with the outputs of estimated values. The mock results that compare the estimated to observed physical parameters are shown in Fig. 2. The upper panels display stellar, gas, and dust masses (Mst, Mg, and Md, respectively) and the star formation rate (SFR). The lower panels illustrate the luminosity components which include stellar luminosity (Lst), AGN luminosity (LAGN), X-ray AGN luminosity (LX−ray), in addition to the intrinsic (unextinct) AGN luminosity (Lint) which represents the AGN accretion power. All these parameters demonstrate strong consistency between the estimated and observed values, as evidenced by their correlated relationship. The reliability of these estimates is indicated by the slope’s value in the insets of these panels where the slope approaches the unity.

For U/LIRG galaxies, SED fittings spanning a wide range of wavelengths from X-ray to FIR can produce many important physical properties. From which, both the stellar, gas and dust masses and SFR are considered in addition to the luminosity and its components. Both total luminosity and its components (stellar, AGN, and X-ray luminosities) are characterized versus the galaxy mass. Generally, it is obvious that the stellar luminosity has the dominate contributions relative to the total luminosity. For this total luminosity, it has a strong correlation versus the galaxy mass while the intrinsic luminosity has an intermediate one. Regarding the luminosity components (stellar, AGN and X-ray), they have strong correlations versus both of stellar and gas masses. On the other hand, the variations of these components versus dust mass have weak correlations.

Gadallah, K. A. K., Mutschke, H. & Jäger, C. Mid-infrared spectroscopy of UV irradiated hydrogenated amorphous carbon materials. A&A 544, A107 (2012).

Ricci, C. et al. A hard X-ray view of luminous and ultra-luminous infrared galaxies in GOALS - I. AGN obscuration along the merger sequence. MNRAS 506, (2021).

Chu, J. K. et al. The Great observatories All-Sky LIRG Survey: Herschel Image Atlas and aperture photometry. ApJ Suppl. Ser. 229, 25 (2017).

Thorne, J. E. et al. Deep Extragalactic VIsible Legacy Survey (DEVILS): SED fitting in the D10-COSMOS field and the evolution of the stellar mass function and SFR-M relation. MNRAS 505, (2021).

Inami, H. et al. Mid-infrared atomic fine-structure emission-line spectra of luminous infrared galaxies: Spitzer/IRS spectra of the goals sample. ApJ 777, 156 (2013).

Hou, L. G., Wu, X. B. & Han, J. L. Ultra-luminous infrared galaxies in sloan digital sky survey data release 6. ApJ 704, (2009).

Cardelli, J. A., Clayton, G. C. & Mathis, J. S. The relationship between infrared, optical, and ultraviolet extinction. ApJ 345, 245–256 (1989).

Padovani, P. et al. Active galactic nuclei: what’s in a name? A&ARv vol. 25 Preprint at (2017). https://doi.org/10.1007/s00159-017-0102-9

The DOF beyond the subject is always greater than the DOF in front of the subject. When the subject is at the hyperfocal distance or beyond, the far DOF is infinite, so the ratio is 1:∞; as the subject distance decreases, near:far DOF ratio increases, approaching unity at high magnification. For large apertures at typical portrait distances, the ratio is still close to 1:1.

Vazdekis, A. Evolutionary stellar population synthesis at 2 Å spectral resolution. Astrophys. Space Sci. 276, 921–929 (2001).

Circleofconfusion

Gadallah, K. A. K. redshift–luminosity characterization of active galactic nucleus galaxies having obscuring dusty material using the spectral energy distribution from the X-ray to far-infrared. MNRAS. 520, 2351–2366 (2023).

The luminosity components of stellar luminosity (blue stars), AGN luminosity (red circles), and x-ray luminosity (green triangles) versus dust to gas mass ratio.

For the decomposed AGN and X-ray luminosities in erg s−1, Figs. 18 and 19, respectively, compare them with those from literatures for similar and different galaxies with variation of the SFR. Figure 18 presents the AGN luminosity of our sample compared with those for various types of galaxies92. These galaxies include AGN galaxies characterized by their strong neon emission lines of high ionizing flux, Starburst galaxies (SB) dominated by star formation, the Spitzer Interacting Galaxies Sample (SIGS) being relatively bright and identified as spiral galaxies with a companion seen in close projection, and Late-Stage Merging (LSM) galaxies being in or approaching their final coalescence. In this comparison, it is shown that all samples of92 have similar ranges of the AGN luminosity agreed with those of U/LIRG. Compared to the SFR of U/LIRG, SB, SIGS, and LSM have noticeable different ranges of their SFR. SIGS sample agrees with U/LIRGs at low SFR while both SB and LSM appear at high SFR. For AGN sample, their SFR is almost like that of U/LIRG. Versus the SFR, Fig. 19 presents the X-ray luminosity of U/LIRGs compared to those of similar sample of LIRGs in the energy band 2–10 keV93. It is shown that the 2–10 keV luminosity of LIRG appear slightly lower than that of U/LIRG but LIRG is characterized by its high SFR in agreement with high SFR values of U/LIRG.

The variation SFR versus the stellar mass of U/LIRG galaxies. Below the dotted horizontal line there are 15 galaxies having values of SFR less than 1.

The rest-frame luminosity of 63 sources: the X-ray luminosity in the range of 2–10 keV. The dotted line represents the flux-limited cut-off line considering the Malmquist bias where the minimum fluxes are 7.0 × 10−15 erg s−1 cm−2 for X-ray luminosity.

SED models of IRASF04118-3207, IRASF23254 + 0830 and IRASF04315-0840 as examples of LIRG (right panels), and IRASF09320 + 6134, IRASF12540 + 5708 and SDSS_J015950.25 + 002340.8 as examples of ULIRG galaxies (left panels). The SED model spectrum is displayed with various components: attenuated stellar emission (yellow line) and unattenuated stellar emission (dashed blue line), nebular emission (green line), dust emission (red line), and AGN emission (orange line). The estimated model flux densities are represented by red dots, while the observed flux densities, along with their error bars, are denoted by open circles in light magenta.

Lu, N. et al. A Herschel Space Observatory Spectral Line Survey of local luminous Infrared galaxies from 194 to 671 microns. ApJ Suppl. Ser. 230, 1 (2017).

Díaz-Santos, T. et al. The spatial extent of (U)LIRGs in the mid-infrared. I. The continuum emission. ApJ 723, 993 (2010).

For selected samples classified as early and late mergers (Appendix A) based on the classification by41, Fig. 14 also displays a remarkable increase of their total luminosity with increasing the stellar mass. Obviously, both early and late mergers are scattered and distributed through the mass variation scale from ∼1010M☉ to ∼1012M☉, confirming that their classification is independent of the stellar mass. This is consistent with41 where the averaged values of logarithmic Mst doesn’t change with the merger stages. The linear fits show that the averaged total luminosity of late mergers is higher than that of early ones by a factor of 1.7 at the lowest stellar mass (∼1010M☉) and 2.5 at the highest one (∼1012M☉). On the other hand, these authors found that the SFR is effective where it increases with the merger stages.

Soifer, B. T. et al. The luminosity function and space density of the most luminous galaxies in the IRAS survey. ApJ. 303, L41 (1986).

For 35 mm motion pictures, the image area on the film is roughly 22 mm by 16 mm. The limit of tolerable error was traditionally set at 0.05 mm (0.0020 in) diameter, while for 16 mm film, where the size is about half as large, the tolerance is stricter, 0.025 mm (0.00098 in).[15] More modern practice for 35 mm productions set the circle of confusion limit at 0.025 mm (0.00098 in).[16]

Pereira-Santaella et al. H3 + absorption and emission in local (U)LIRGs with JWST/NIRSpec: Evidence for high H2 ionization rates. A&A 689, L12 (2024).

The acceptable circle of confusion depends on how the final image will be used. The circle of confusion as 0.25 mm for an image viewed from 25 cm away is generally accepted.[14]

Wardlow, J. L. et al. The LABOCA survey of the Extended Chandra Deep Field-South: A photometric redshift survey of submillimetre galaxies. MNRAS 415, (2011).

Draine, B. T. & Li, A. Infrared Emission from Interstellar Dust. IV. The Silicate-Graphite-PAH Model in the Post-Spitzer Era. The ApJ, Volume 657, Issue 2, pp. 810–837. 657, 810–837 (2007).

DOF simulator

Kim, W. J., Urquhart, J. S., Wyrowski, F., Menten, K. M. & Csengeri, T. New detections of (sub)millimeter hydrogen radio recombination lines towards high-mass star-forming clumps. A&A 616, (2018).

The total luminosity versus the stellar mass of samples classified as early (blue squares) and late mergers (red circles) as classified by41.

In this subsection, we first present the resulted SED physical properties including the mass components and SFR, in relations with the decomposed (stellar, AGN, X-ray) luminosities of U/LIRGs. For which, the variation of these luminosities is generally described versus the total and stellar masses, and the SFR. For the host galaxy disk linked with its accretion disk’s power, Fig. 4 presents, in a logarithmic scale, the variations of the total luminosity including stellar, AGN and X-ray luminosities of the galaxy, and those of the intrinsic AGN luminosity with increasing the total mass (MT) including star, gas and dust masses of that galaxy. These variations display a notable increase in their general trend profiles with averaged values of their slopes of 0.51216 ± 0.07655 and 0.54003 ± 0.20896 for LT and Lint, respectively. This indicates that unabsorbed total luminosity and the intrinsic AGN luminosity are strongly associated with increasing the total mass.

To generate SED spectra of LIRG/ULIRG galaxies in ranges from X-ray to FIR band, the PYTHON Code Investigating GALaxy Emission (CIGALE) developed by42 was used, considering the modification of the last version (X-CIGALE) by43,44. This version incorporated the X-ray band by adding an X-ray photometry module and a module for the polar dust. This code can calculate the galaxy physical properties such as stellar, gas, and dust masses, star formation rate, as well as the luminosity components of stellar, AGN, and X-ray emissions.

The hyperfocal distance has a property called "consecutive depths of field", where a lens focused at an object whose distance from the lens is at the hyperfocal distance H will hold a depth of field from H/2 to infinity, if the lens is focused to H/2, the depth of field will be from H/3 to H; if the lens is then focused to H/3, the depth of field will be from H/4 to H/2, etc.

Iwasawa, K. et al. C-GOALS: Chandra observations of a complete sample of luminous infrared galaxies from the IRAS Revised Bright Galaxy Survey. A&A 529, (2011).

SED analysis provides a comprehensive understanding of the energy output and physical properties of galaxies across a wide range of wavelengths20. In recent years, the investigation of SEDs has played a crucial role in studying U/LIRGs hosting AGN (e.g21). These galaxies exhibit intense infrared emission, indicating the presence of powerful AGN and/or vigorous star formation activities22. The analysis of radio spectral characteristics, as explored in23, provides additional insights into the star formation history and AGN activity within ULIRGs, further refining their classification.

Lee, J. C., Hwang, H. S., Lee, M. G., Kim, M. & Lee, J. H. Akari near-infrared spectroscopy of luminous infrared galaxies. ApJ 756, 95 (2012).

The free parameters of the modules described above are summarized in Table 1 to build up the X-CIGALE code. This code performs these modules simultaneously to fit the SED spectrum. It estimates the galaxy’s physical parameters such as the luminosity components, the stellar mass of the host galaxy and dust-to-gas mass ratio by the SED models.

Chabrier, G. The Galactic Disk Mass Function: Reconciliation of the [ITAL]Hubble Space Telescope[/ITAL] and Nearby Determinations. ApJ 586, (2003).

Depth of focus

Numerous efforts have been made to elucidate the relationship between the AGN and their host galaxies, particularly in the context of X-ray background radiation32 and on the degree of obscuration linked to the geometric structure of the dusty torus surrounding the AGN33. Several recent investigations34,35,36 have highlighted the correlation between X-ray and mid-infrared (MIR) emissions.

In optics and photography, hyperfocal distance is a distance from a lens beyond which all objects can be brought into an "acceptable" focus. As the hyperfocal distance is the focus distance giving the maximum depth of field, it is the most desirable distance to set the focus of a fixed-focus camera.[41] The hyperfocal distance is entirely dependent upon what level of sharpness is considered to be acceptable.

Zang, R. X., Peeters, E. & Boersma, C. Tying Spatial Variations in Polycyclic Aromatic Hydrocarbon (PAH) Emission to a Changing PAH Population in the Reflection Nebula NGC 2023. ApJ 887, (2019).

Yamada, S. et al. Hard X-Ray to Radio Multiwavelength SED Analysis of Local U/LIRGs in the GOALS Sample with a self-consistent AGN model including a Polar-dust component. ApJ Suppl. Ser. 265, 37 (2023).

The X-ray luminosity versus the intrinsic luminosity of U/LIRGs compared with those of other U/LIRGs41 and obscured AGN galaxies24.

Note that M T = − f u − f {\textstyle M_{T}=-{\frac {f}{u-f}}} is the transverse magnification which is the ratio of the lateral image size to the lateral subject size.[5]

The AGN of a galaxy having a dusty torus plays a crucial role in the AGN emissions from certain angles, particularly in the X-ray/UV/optical16 and in ranges from X-ray to Far Infrared range (e.g24). The absorbed energy by the torus is re-emitted as thermal radiation in the infrared (IR) band, leading to the characteristic IR emission observed in AGNs25. The SED spectra of U/LIRGs are strongly characterized by their MIR emissions due to the existence of dust. This dust mainly consists of small grains of polycyclic aromatic hydrocarbons (PAHs) and large grains of silicate and carbonaceous materials26,27,28,29 where the UV radiation heats these grains30,31.

Zou, F., Yang, G., Brandt, W. N. & Xue, Y. The host-galaxy properties of Type 1 versus type 2 active Galactic nuclei. ApJ. 878, 11 (2019).

Stierwalt, S. et al. Mid-infrared properties of nearby luminous infrared galaxies. I. Spitzer Infrared Spectrograph Spectra for the goals sample. ApJ Supplement Ser. 206, 1 (2013).

For our results, the correlation coefficient (r) of the relationships among the physical properties is calculated. An estimate of a linear correlation has been done in log–log space with x-, y-uncertainties of luminosity-mass and luminosity-luminosity correlations. For these correlations, the Bayesian maximum likelihood method89 was used where uniform prior distributions are assumed for the regression parameters. Considering the same way in35,36,90, the Markov chain Monte Carlo (MCMC) simulation was conducted using the mean and standard deviation from the posterior probability distributions with 10,000 iterations.

Other technologies use a combination of lens design and post-processing: Wavefront coding is a method by which controlled aberrations are added to the optical system so that the focus and depth of field can be improved later in the process.[25]

Gadallah, K. A. K. Hydrocarbon analogs of cosmic dust to trace the solid carbon abundance in the interstellar medium. Adv. Space Res. 55, 705-715 (2015).

Sajina, A. et al. Spitzer Mid-Infrared Spectroscopy of Infrared Luminous Galaxies at z ∼ 2. III. Far‐IR to Radio Properties and Optical Spectral Diagnostics. ApJ 683, (2008).

Asmus, D., Gandhi, P., Hönig, S. F., Smette, A. & Duschl, W. J. The subarcsecond mid-infrared view of local active galactic nuclei - II. The mid-infrared-X-ray correlation. MNRAS 454, (2015).

The blur increases with the distance from the subject; when b is less than the circle of confusion, the detail is within the depth of field.

Hernán-Caballero, A. & Hatziminaoglou, E. An atlas of mid-infrared spectra of star-forming and active galaxies. MNRAS 414, (2011).

Torres-Albà, N. et al. C-GOALS: II. Chandra observations of the lower luminosity sample of nearby luminous infrared galaxies in GOALS. A&A 620, (2018).

The luminosity components of stellar luminosity (blue stars), AGN luminosity (red circles), and x-ray luminosity (green triangles) versus stellar mass.

Noll, S. et al. Analysis of galaxy spectral energy distributions from far-UV to far-IR with CIGALE: studying a SINGS test sample. A&A. 507, 1793–1813 (2009).