Sean D Mcguire

Carolyn Jacobs

Pierre Bruno Mariotto

Corentin H C Grimaldi

Augustin Tibère-Inglesse

Christophe O Laux

Measurements and Modeling of Air Plasma Radiation in the VUV

teaching and research institutions in France or abroad, or from public or private research centers.

Nomenclature

𝐴 𝜈 𝜈 = Einstein coefficient for spontaneous emission between vibrational states 𝜈 and 𝜈 𝐸 𝐻 = ionization energy of the hydrogen atom 𝐸 ∞ = ionization energy of the radiating atom 𝐸 𝑖 = excited state energy of radiating state 𝑅 = average radius of electron orbit as defined and used by Griem (Ref. [START_REF] Griem | Principles of Plasma Spectroscopy[END_REF])

𝑅

I. Introduction

High temperature air radiation is important for the design of spacecraft heat shields. Radiative codes are required to accurately estimate the radiative heat flux. As a general rule, the radiative heat flux becomes important at high entry velocities and for large capsule sizes. Johnston et al estimate that, for a 1-m radius sphere entering at 15 km/s (Mars return case), the radiative heat flux accounts for more than 85% of the total heat flux (see Fig. 1 of their paper) [START_REF] Johnston | Radiative Heating Uncertainty for Hyperbolic Earth Entry, Part 1: Flight Simulation Modeling and Uncertainty[END_REF].

High entry velocities lead to high shock layer temperatures which lead to high levels of atomic and photoionization radiation. Furthermore, these high temperatures lead to large contributions from the UV and VUV spectral regions to the total radiative heat flux [START_REF] Johnston | Radiative Heating Uncertainty for Hyperbolic Earth Entry, Part 1: Flight Simulation Modeling and Uncertainty[END_REF][START_REF] Laux | Influence of Ablation Products on the Radiation at the Surface of a Blunt Hypersonic Vehicle at 10 km/s[END_REF]. At lower entry velocities, the convective heat flux generally dominates. Emission from molecular species becomes more important for the calculation of the radiative heat flux. Beyond calculation of the radiative heat flux for reentry missions, radiative codes are necessary for analyzing emission and absorption spectra collected in ground testing facilities. Such an analysis provides information about the temperature and chemical composition of the gas, which can then be used to validate hydrodynamic codes or chemical kinetic models. For these reasons, a lot of work has focused on the development and validation of radiative codes. An article by Zammit et al provides a recent example looking at modeling of high temperature air radiation [START_REF] Zammit | A comprehensive study of the radiative properties of NO-a first step toward a complete air opacity[END_REF].

The work in this paper addresses two goals. The first goal is to measure equilibrium air emission in the VUV for conditions relevant for atmospheric entry. The second goal is to compare with radiative code predictions in order to assess and improve model performance in this spectral zone. The radiative code SPECAIR will be used in this paper [START_REF] Laux | Optical diagnostics of atmospheric pressure air plasmas[END_REF][START_REF]SPECAIR, Software Package, Ver. 3.0[END_REF]. The work presented in this article was done with the CentraleSupélec plasma torch facility, which has the advantage of producing air plasmas in local Thermodynamic Equilibrium (LTE) at temperatures up to 7000 K and atmospheric pressure. The measured spectra provide an excellent means for validating radiative models because the thermochemical state of the plasma is well known.

II. Measurements

The facility used to produce the plasma for the emission studies is a TAFA Model 66 inductively coupled plasma (ICP) torch powered by a 120 kVA radio frequency LEPEL Model T-50-3 power supply. The power supply operates at 4

MHz and provides a maximum of 12 kV DC and 7.5 A to the oscillator plates. Details of the plasma torch facility may be found in previous publications [START_REF] Laux | Optical Diagnostics and Radiative Emission of Air Plasmas[END_REF][START_REF] Macdonald | Measurements of Air Plasma/Ablator Interactions in an Inductively Coupled Plasma Torch[END_REF]. The plasma at the exit of the torch is at atmospheric pressure. Figure 1 shows a schematic of the facility. For the experiments presented here, a 5-cm diameter exit nozzle was used. The temperature profile of the plasma jet was obtained by measuring the absolute emission from the oxygen triplet at 777 nm. This is done using a visible spectrometer (Acton SpectraPro 500i), imaging setup and a calibrated Tungsten ribbon lamp (Optronics Laboratories OL550). A filter is installed in the optical path to suppress higher order interferences within the spectrometer. The imaging system for the visible wavelength spectrometer makes use of parabolic mirrors and a periscope to image the spectrometer slit across the jet profile. Intensity measurements are Abel-inverted to provide spatially resolved intensity profiles. The procedure for these emission-based temperature measurements is documented in several references [START_REF] Laux | Optical diagnostics of atmospheric pressure air plasmas[END_REF][START_REF] Laux | Optical Diagnostics and Radiative Emission of Air Plasmas[END_REF][START_REF] Mcbride | Computer Program for Calculating and Fitting Thermodynamic Functions[END_REF]. The analysis relies upon the assumption of thermochemical equilibrium -an assumption that has been previously verified using this plasma torch facility with air injection [START_REF] Laux | Optical diagnostics of atmospheric pressure air plasmas[END_REF][START_REF] Laux | Optical Diagnostics and Radiative Emission of Air Plasmas[END_REF].

Figure 2a shows the temperature profile for the conditions studied here. Given this temperature profile, the composition of the air plasma may be calculated using NASA CEA [START_REF] Mcbride | Computer Program for Calculating and Fitting Thermodynamic Functions[END_REF] and is shown in Fig. 2b. Based on previous work, it is known that the NO bands are responsible for a large portion of the VUV/UV air emission at these conditions [START_REF] Laux | Optical diagnostics of atmospheric pressure air plasmas[END_REF][START_REF] Laux | Optical Diagnostics and Radiative Emission of Air Plasmas[END_REF]. The VUV measurements are a continuation of work begun several years back using a nitrogen purged VUV spectrometer [START_REF] Jacobs | Vacuum ultraviolet radiation studies in a plasma torch facility[END_REF]. Then, measurements were obtained down to 170 nm, but oxygen absorption was affecting the measurements below a wavelength of about 180 nm, making it somewhat difficult to compare with radiative model predictions. For this work, a McPherson vacuum spectrometer is used that is capable of making measurements down to 120 nm. The setup is identical to the system used in Ref. [START_REF] Mcguire | Measurements and modeling of CO 4th positive (A-X) radiation[END_REF], where VUV/UV emission from carbon monoxide was studied. A slightly modified setup was used to make VUV measurements in a recombining nitrogen plasma [START_REF] Tibère-Inglesse | Experimental study of recombining nitrogen plasmas: II. Electronic population distributions and nonequilibrium radiation of atoms[END_REF].

Figure 3 shows a diagram of the VUV system. An imaging box with two mirrors (Acton optics with a #1200 coating for 120 nm reflection) is attached to the spectrometer for imaging the spectrometer slit onto the plasma. An adapting tube protrudes from this imaging box and terminates at the plasma boundary to avoid absorption by the 𝑂 2 Schumann-Runge in ambient air (Fig. 3b). This tube is specially designed to withstand the thermal heat flux coming from the hot plasma.

The final piece in contact with the plasma is a water-cooled copper piece. Teflon and PEEK plastics are used in the portion of tube separating this copper adaptor from the imaging box. These plastics were chosen for electrical isolation Fig. 3 given that the plasma is at a 10 kV floating potential. Most of the system is placed under vacuum (< 10 -3 Torr).

However, the final portion of the optical setup which is in contact with the torch is placed under an argon purge (see Fig. 3b for a detail of this part). A continuous argon purge was run at a pressure slightly above 1 atm to prevent oxygen from leaking into the system. Pressure relief ports were drilled in the teflon piece to permit a portion of the argon to exit via these holes, rather than entirely into the plasma. The remaining portion exited through the final viewing hole, directly entering the plasma jet. Our measurements indicate that this purge did not affect the measured radiation. All measurements above 200 nm agreed very well with SPECAIR predictions and SPECAIR has already been validated in this spectral region. Ref. [START_REF] Mcguire | Measurements and modeling of CO 4th positive (A-X) radiation[END_REF] details additional tests performed on an alternate 𝐶𝑂 2 /𝐴𝑟 mixture.

For calibration, an argon discharge supplies radiation traceable to NIST standards in the VUV/UV spectral regions [START_REF] Klose | Radiometric Calibrations of Portable Sources in the Vacuum Ultraviolet[END_REF]. The adapting arm linking the imaging box to the plasma was replaced with a separate adaptor for these measurements so that the argon discharge was located at the system focus. A diaphragm was used to limit the aperture of the imaging setup and to ensure that the same aperture was used for both the calibration and plasma torch measurements.

Finally, due to high levels of stray light during the calibration measurement, a VUV transmission filter (eSource Optics 50150FBB) was installed for the calibration measurements. The spectrally resolved transmission of this filter was known and verified by performing calibrations both with and without the filter. For spectral regions where it was possible to compare the two calibrations, the measurements reproduced the quoted transmission profile. This transmission filter was not used in the plasma torch measurements where stray light was not a problem. An identical procedure was used in McGuire et al [START_REF] Mcguire | Measurements and modeling of CO 4th positive (A-X) radiation[END_REF].

For all emission measurements, a VUV compatible PI-MAX2 camera from Princeton Instruments was used. The raw measurements were therefore images, with one axis corresponding to wavelength and the second axis corresponding to plasma jet radius. To produce the final spectra, the zone along the spatial axis corresponding to the center of the plasma jet was first identified. The images were then binned across this zone to produce a spectrum. The same binning was also applied to stray light measurements, which were obtained by tuning the spectrometer to 120 nm where no signal was observed. The resulting stray light signal was subtracted from all spectra. Finally, any residual background due to dark current was also subtracted. This dark current was determined by registering the signal in zones where no light is incident on the camera due to the spectrometer design. This same procedure was applied to obtain calibration spectra. Multiplying the emission spectra in the torch by the calibration spectra produced the final spectra, calibrated in absolute intensity.

The intensity of the measured VUV/UV emission is a product of two measurements: the signal recorded in the torch and the calibration factor. Each of these factors has a corresponding uncertainty. 𝛿𝑈 𝑐𝑎𝑙𝑖𝑏 and 𝛿𝑈 𝑠𝑖𝑔𝑛𝑎𝑙 denote the relative uncertainty in the calibration and the signal recorded in the torch, respectively. Assuming these two errors to be uncorrelated, the relative uncertainty in the intensity is given by

𝛿𝑈 𝑖𝑛𝑡𝑒𝑛𝑠𝑖𝑡 𝑦 = √ 𝛿𝑈 2 𝑐𝑎𝑙𝑖𝑏 + 𝛿𝑈 2

𝑠𝑖𝑔𝑛𝑎𝑙

(where 𝛿𝑈 𝑖 = Δ𝑈 𝑖 /𝑈 𝑖 and Δ𝑈 𝑖 is the absolute uncertainty in 𝑈 𝑖 ). The dominant source of uncertainty comes from the calibration signal. The reported intensity curve is a smooth function. However, the measurements indicate that this intensity fluctuates slightly with wavelength. These fluctuations are less than 5% of the measured amplitude across the measured wavelength range. The only other known source of uncertainty is the camera dark current signal, but this is negligible in comparison. Therefore, the relative uncertainty in the absolute intensity of the final measurements is taken to be less than 5% across the entire wavelength range.

The slit function for the VUV/UV measurements is shown in Fig. 4. The VUV system provides line-of-sight measurements along the chord intersecting the plasma centerline, 2.5 cm downstream of the nozzle exit. The aforementioned temperature measurements are performed at the same vertical/axial location. Figure 5 shows the measured VUV spectrum calibrated in absolute intensity.

III. Modeling -SPECAIR radiation code

Radiation modeling for comparison with measurements was done using the radiation code SPECAIR [START_REF] Laux | Optical diagnostics of atmospheric pressure air plasmas[END_REF][START_REF]SPECAIR, Software Package, Ver. 3.0[END_REF]. The initial SPECAIR model for the nitric oxide bands was developed in Refs. [START_REF] Laux | Optical Diagnostics and Radiative Emission of Air Plasmas[END_REF][START_REF] Laux | Arrays of radiative transition probabilities for the N 2 first and second positive, NO beta and gamma, N + 2 first negative, and O 2 Schumann-Runge band systems[END_REF][START_REF] Laux | Modeling the UV and VUV radiative emission of high-temperature air[END_REF] to include important transitions in the VUV such as the delta and epsilon systems that had not been previously considered in high temperature air radiation models. The model was then updated in Ref. [START_REF] Laux | Optical diagnostics of atmospheric pressure air plasmas[END_REF] to account for line position perturbations in transitions involving the 2 Π states. This SPECAIR model was found to give very good agreement with measurements for wavelengths above 200 nm. This code was modified for this work to improve agreement at lower wavelengths. The next two sub-sections discuss the modeling of molecular and atomic emission, respectively. For each molecular system or atomic feature, the original SPECAIR model is presented, followed by any updates that were incorporated into the model as a result of this work.

A. Molecular Emission -nitric oxide bands

The measured molecular emission comes primarily from nitric oxide systems. For a generic band system, the Einstein coefficients included in SPECAIR are calculated using the method outlined in Laux and Kruger [START_REF] Laux | Optical Diagnostics and Radiative Emission of Air Plasmas[END_REF][START_REF] Laux | Arrays of radiative transition probabilities for the N 2 first and second positive, NO beta and gamma, N + 2 first negative, and O 2 Schumann-Runge band systems[END_REF], which yields the following equation:

𝐴 𝜈 𝜈 = 64𝜋 4 𝜈 3 3ℎ𝑐 3 (2 -𝛿 0,Λ +Λ ) (2 -𝛿 0,Λ ) (𝑅 𝜈 𝜈 𝑒 ) 2 (1)
where:

(𝑅 𝜈 𝜈 𝑒 ) 2 = ∫ ∞ 0 𝜓 𝜈 (𝑟) 𝑅𝑒(𝑟) 𝜓 𝜈 (𝑟)𝑑𝑟 2 (2)
𝑅 𝜈 𝜈 𝑒 is the electronic-vibrational transition moment, 𝜓 𝜈 and 𝜓 𝜈 represent the vibrational wavefunctions of the upper and lower states respectively, Λ and Λ represent the projection of the orbital angular momentum of the upper and lower states respectively and 𝛿 𝑖, 𝑗 is the Kronecker delta function. The above expressions are taken from Schadee [START_REF] Schadee | Unique definitions for the band strength and the electronic-vibrational dipole moment of diatomic molecular radiative transitions[END_REF] and CGS units are used. These lead to a set of equations linking the square of the electronic-vibrational transition moment (𝑅 𝜈 𝜈 𝑒 ) 2 to other conventional values as follows:

𝐴 𝜈 𝜈 = 64𝜋 4 𝜈 3 2 -𝛿 0,Λ +Λ 3ℎ𝑐 3 2 -𝛿 0,Λ (𝑅 𝜈 𝜈 𝑒 ) 2 = 2.026 × 10 -6 (𝜈 𝜈 𝜈 ) 3 2 -𝛿 0,Λ +Λ 2 -𝛿 0,Λ 𝑅 𝜈 𝜈 𝑒 𝑒𝑎 𝑜 2 𝑓 𝑎𝑏𝑠 𝜈 𝜈 = 8𝜋 2 𝑚 𝑒 𝜈 2 -𝛿 0,Λ +Λ 3ℎ𝑒 2 2 -𝛿 0,Λ (𝑅 𝜈 𝜈 𝑒 ) 2 = 3.0376 × 10 -6 𝜈 𝜈 𝜈 2 -𝛿 0,Λ +Λ 2 -𝛿 0,Λ 𝑅 𝜈 𝜈 𝑒 𝑒𝑎 𝑜 2 𝑓 𝑒𝑚𝑖𝑠 𝜈 𝜈 = 2 -𝛿 0,Λ 2 -𝛿 0,Λ 𝑓 𝑎𝑏𝑠 𝜈 𝜈 ∑ (𝑅 𝜈 𝜈 𝑒 ) 2 = 2 -𝛿 0,Λ +Λ (2𝑆 + 1) (𝑅 𝜈 𝜈 𝑒 ) 2 (3)
where (𝑅 𝜈 𝜈 𝑒 ) 2 represents the sum of the electronic-vibrational transition moment across all electronic subtransitions.

The above equations hold as long as the electronic-vibrational transition moment 𝑅 𝑒 (𝑟) has been calculated according to the convention of Whiting et al [START_REF] Whiting | Recommended conventions for defining transition moments and intensity factors in diatomic molecular spectra[END_REF]. In the above expressions, ℎ = 6.626 × 10 -27 erg•s, 𝑐 = 3.00 × 10 10 cm/s, 𝑒 = 4.803 × 10 -10 statC, 𝑚 𝑒 = 9.11 × 10 -28 g, 𝑎 𝑜 = 5.291 × 10 -9 𝑐𝑚 (Bohr radius), 𝜈 is the frequency in 𝐻𝑧, 𝜈 𝜈 𝜈 is the wavenumber of the transition in cm -1 and 𝑅 𝜈 𝜈 𝑒 is in atomic units (multiple of 𝑒𝑎 0 ). Λ and Λ are the projected orbital angular momentum quantum numbers of the upper and lower states respectively. Meanwhile, the Hönl-London factors are calculated using the method of Whiting et al [START_REF] Whiting | Recommended conventions for defining transition moments and intensity factors in diatomic molecular spectra[END_REF]. These are then used along with the Einstein coefficients to determine the line strength of a specific rovibrational transition.

For calculations of the Einstein coefficients for molecular band systems that are not impacted by perturbations, the Rydberg-Klein-Rees (RKR) method is first used to determine the potential energy curves for the relevant electronic states. The vibrational wavefunctions are then calculated and used, along with the electronic-vibrational transition moment function taken from the literature, to calculate a set of Einstein coefficients for a given band system. This is the approach used for the 𝛾, 𝜖 and 𝛽 systems of 𝑁𝑂. Table 1 shows the sources and values used for these calculations. This methodology does not take into account perturbations between quantum states. The wavefunctions of states perturbed by Huber and Herzberg [START_REF] Huber | Molecular spectra and Molecular Structure. IV. Constants of diatomic molecules[END_REF] Lavrov [START_REF] Lavrov | Calculation of Thermodynamic Functions for N 2 , O 2 and NO[END_REF] avoided crossings assume a hybrid form, characterized by a mixture of the quantum states involved in the perturbation. This is the case for emission from excited 2 Π states of 𝑁𝑂 and the methodology for these systems will be detailed below.

Emission from NO 2 Σ + states: The 𝛾( 𝐴 2 Σ + -𝑋 2 Π) and 𝜖 (𝐷 2 Σ + -𝑋 2 Π) are modeled using the approach outlined above -no perturbations were taken into account. The ETMF of Langhoff, Bauschlicher and Partridge was used for the NO 𝛾 system emission [START_REF] Langhoff | Theoretical study of the NO 𝛾 system[END_REF]. The ETMF for the NO 𝜖 system was also taken from calculations done by Langhoff and co-workers [START_REF] Sheehy | Theoretical study of the nitric oxide 𝜖 and 11000 Å bands[END_REF]. Emission from vibrational levels up to 𝑣 = 8 in the A-state and 𝑣 = 5 in the D-state are accounted for.

In this paper, the maximum vibrational level of the D-state was increased from 𝑣 = 5 to 𝑣 = 6. Figure 6a shows the effect of this modification on the SPECAIR calculation, which is minor and limited to wavelengths below 175 nm.

The NO 𝛾 (E 2 Σ + -X 2 Π) system is included in SPECAIR as discussed in Ref. [START_REF] Laux | Optical Diagnostics and Radiative Emission of Air Plasmas[END_REF]. The electronic transition dipole moment used was that of De Vivie and Peyerimhoff [START_REF] De Vivie | Theoretical spectroscopy of the NO radical. I. Potential curves and lifetimes of excited states[END_REF]. The spectroscopic constants for the E 2 Σ + state were taken from Huber and Herzberg [START_REF] Huber | Molecular spectra and Molecular Structure. IV. Constants of diatomic molecules[END_REF]. The resulting calculations largely overpredict the measured photoabsorption cross-sections reported by Chan, Cooper and Brion [START_REF] Chan | Absolute optical oscillator strengths for the photoabsorption of nitric oxide (5-30 eV) at high resolution[END_REF] discussed in Section V. Furthermore, the impact of this system on the high temperature air emission spectrum was minimal. Appendix A contains figures that illustrate these points. For these reasons, this system is not included in the SPECAIR calculations shown below.

Emission from NO 2 Δ states: Emission from the 𝛽 (𝐵 2 Δ -𝑋 2 Π) was modeled using the standard approach and no perturbations were taken into account. The 𝛽 system primarily affects the spectrum at wavelengths below approximately 180 nm, and even then only weakly with an average magnitude less than 1 𝑚𝑊/𝑐𝑚 2 /𝑛𝑚/𝑠𝑟. The ETMF for this band was taken from de Vivie and Peyerimhoff [START_REF] De Vivie | Theoretical spectroscopy of the NO radical. I. Potential curves and lifetimes of excited states[END_REF]. Emission from vibrational levels up to 𝑣 = 7 in the B'-state are accounted for.

For this work, the maximum vibrational level of the B'-state was increased from 𝑣 = 7 to 𝑣 = 9. Figure 6b shows the effect of this modification on the SPECAIR calculation, which is negligible and limited to wavelengths below 160 nm.

The NO F 2 Δ -X 2 Π system was also added to SPECAIR. As with the 𝛾 system of NO, the resulting calculations largely overpredict the measured photoabsorption cross-sections reported by Chan, Cooper and Brion [START_REF] Chan | Absolute optical oscillator strengths for the photoabsorption of nitric oxide (5-30 eV) at high resolution[END_REF]. The corresponding impact on the high temperature air emission spectrum is also minimal. Appendix B contains figures that illustrate these points. For these reasons, this system is not included in the SPECAIR calculations shown below. c) The curve "w/perturbations" is the same as the curve "additional B' states" but with the perturbation calculation for the oscillator strengths of the 2 Π states accounted for as described in the text. d) The curve "additional C states" is the same as the curve "w/perturbations" but with the maximum vibrational level of the 𝑁𝑂 C-state increased from 𝑣 = 4 to 𝑣 = 9. e) The curve "with 𝑁 2 (𝑎 -𝑋) and 𝑁𝑂 + ( 𝐴 -𝑋)" is the same as the curve "additional C states" but with the 𝑁 2 (𝑎 -𝑋) and 𝑁𝑂+ ( 𝐴 -𝑋) systems added.

Emission from NO 2 Π states: Perturbations amongst the various 2 Π states of NO have been noted in the literature [START_REF] Gallusser | Multistate vibronic coupling between the excited 2 Π states of the NO molecule[END_REF][START_REF] Qu | A spectroscopic model for the low-lying electronic states of NO[END_REF]. These perturbations affect both the line strength and position of individual lines within these systems. This paper uses the notation of Gallusser and Dressler [START_REF] Gallusser | Multistate vibronic coupling between the excited 2 Π states of the NO molecule[END_REF] and SPECAIR accounts for emission from the 𝐵 2 Π -𝑋 2 Π,

𝐶 2 Π -𝑋 2 Π, 𝐿 2 Π -𝑋 2 Π, 𝐾 2 Π -𝑋 2 Π and 𝑄 2 Π -𝑋 2 Π
systems. The primary 2 Π -2 Π transitions of 𝑁𝑂 contributing to the observed emission in the UV/VUV are the 𝐵 -𝑋 and 𝐶 -𝑋 systems, also known as the 𝛽 and 𝛿 systems. These 2 Π -2 Π transitions require a special treatment because of the strong perturbations among the various 2 Π states of 𝑁𝑂.

These perturbations affect both the line positions and the line strengths. The original SPECAIR calculates the perturbed energies and line positions using the perturbation method outlined by Gallusser and Dressler. However, perturbations in the line strengths were not taken into account. * In this paper, SPECAIR was modified as follows:

• The maximum emitting vibrational level for the 𝐶 2 Π state was increased from 𝑣 𝑚𝑎𝑥 = 4 → 9. This impacts emission from the 𝑁𝑂 𝛿 system.

• Emission from the 𝐿 2 Π -𝑋 2 Π, 𝐾 2 Π -𝑋 2 Π and 𝑄 2 Π -𝑋 2 Π systems was accounted for.

• The effect of perturbations on the oscillator strength was accounted for the

𝐵 2 Π -𝑋 2 Π, 𝐶 2 Π -𝑋 2 Π, 𝐿 2 Π -𝑋 2 Π, 𝐾 2 Π -𝑋 2 Π and 𝑄 2 Π -𝑋 2 Π systems.
Note that the 2 Π states have a multiplicity of two. Each branch of the multiplet is impacted separately by the perturbation calculation of Gallusser and Dressler. The calculations in SPECAIR fully account for this separate dependence on the perturbed energy. However, as a first approximation, this separate dependence on the line strength is ignored. Rather, the oscillator strength for each branch of the multiplet is taken to be the average of the two branches. As a check on this assumption, we replaced the oscillator strength by the maximum or minimum of the two branches (rather than the average) to see what changes this resulted in and found the difference to be minimal.

As a result of these modifications, the SPECAIR calculations in this paper accounts for emission from vibrational levels up to 𝑣 = 9 in the 𝛿 system, 𝑣 = 25 in the 𝛽 system, 𝑣 = 11 in the L-X system, 𝑣 = 3 in the Q-X system and 𝑣 = 4 in the K-X system. Additionally, the effect of perturbations on the energy levels and on the oscillator strengths is accounted for. Figures 6c-6d show the effect of this modification on the SPECAIR calculation. In Fig. 6c, the perturbation on the energy levels and oscillator strengths is accounted for. In addition, the 𝑁𝑂 𝐿 -𝑋, 𝐾 -𝑋 and 𝑄 -𝑋 systems are added. However, the maximum vibrational quantum number of the C-state is limited to 𝑣 = 4 as done in the original SPECAIR calculation, even though the perturbation calculations permit calculations up to 𝑣 = 9. In Fig. 6d, the maximum vibrational quantum number is extended to 𝑣 = 9. In comparing Figures 6c and6d, it is apparent that increasing the maximum vibrational quantum number of the C-state from 𝑣 = 4 to 𝑣 = 9 has a very minimal impact.

However, the impact of the perturbations on the oscillator strengths has a substantial impact. * For a few vibrational levels, the perturbed line positions computed with the Gallusser and Dressler method were then replaced by the more accurate experimental line positions of Amiot and Verges [START_REF] Amiot | Fine Structure of the C 2 Π-A 2 Σ + and D 2 Σ + -A 2 Σ + Band Systems of the NO Molecule: Homogeneous and Heterogeneous Perturbations[END_REF] for the 𝑣 = 0 level of the C-state and the 𝑣 = 7 -10 levels of the B-state, and of Imajo et al [START_REF] Imajo | The application of a VUV Fourier transform spectrometer and synchrotron radiation source to measurements of: II. The 𝛿(1,0) band of NO[END_REF] for the 𝑣 = 1 level of the C-state.

Emission from NO + (A -X) and N 2 (a-X): Two new electronic systems were added to SPECAIR: the NO + A 1 Π -X 1 Σ + and N 2 𝑎 1 Π 𝑔 -𝑋 1 Σ + 𝑔 (Lyman-Birge-Hopfield) transitions. For the NO + A 1 Π -X 1 Σ + system, the electronic transition dipole moment function of Partridge, Langhoff and Bauschlicher [START_REF] Partridge | Theoretical study of the spectroscopy of NO+[END_REF] was used (See Fig. 7 and Table 2 of their article).

For the Lyman-Birge-Hopfield system of 𝑁 2 , the magnetic dipole transition was modeled using the oscillator strengths reported by Pilling, Bass and Braun [START_REF] Pilling | A curve of growth determination of the ƒ-values for the fourth positive system of CO and the Lyman-Birge-Hopfield system of N2[END_REF]. Figure 6e shows that the addition of these two systems has a negligible impact on the predicted emission.

Relative contribution from various systems: Figure 7 shows the contributions from various NO bands discussed above as well as other residual sources of the predicted emission calculated based on the temperature profile shown in Fig. 2a. The calculations in Fig. 7 were performed after updating SPECAIR as described in this section.

𝑁 2 a 1 Π 𝑔 -X 1 Σ + 𝑔 , NO+ 𝐴 1 Π -X 1 Σ + , NO 𝐿 2 Π -𝑋 2 Π, NO 𝐾 2 Π -𝑋 2 Π and NO 𝑄 2 Π -𝑋 2 Π systems. 14

B. Atomic Emission -174 nm feature

The atomic nitrogen feature at 174 nm is optically thick for the conditions studied (see Fig. 8). To account for the extreme self-absorption along the line-of-sight and correctly predict the measured intensity, it is necessary to correctly predict the lineshape. Line broadening mechanisms include Doppler, Stark, resonance, van der Waals and natural line broadening mechanisms. The dominant mechanism here is Doppler broadening, which is described by a Gaussian lineshape of full width at half maximum (FWHM) given by:

Δ𝜆 𝐷𝑜 𝑝 𝑝𝑙𝑒𝑟 = 𝜆 𝑢𝑙 √ 8𝑘𝑇 ln 2 𝑚𝑐 2 (4)
Calculations indicate the Van der Waals broadening is also important. Though the FWHM of the van der Waals broadening is smaller than that of the Doppler broadening, the lineshape falls off more slowly from line center due to the Lorentzian profile. Van der Waals broadening is estimated based upon the approach detailed by Griem [START_REF] Griem | Principles of Plasma Spectroscopy[END_REF][START_REF] Griem | Spectral Line Broadening by Plasmas[END_REF]. The following formula, adapted from Griem [START_REF] Griem | Principles of Plasma Spectroscopy[END_REF], was used for the Lorentzian FWHM of the nitrogen line:

Δ𝜆 𝑣 𝑑𝑤 = ∑ 𝑝 𝜆 2 𝑐 √ 8𝑘𝑇 𝜋 3/5 ℏ 5 𝑅 2 𝑚 3 𝑒 2/5 𝑝 𝑘𝑇 𝑋 𝑝 𝐸 4/5 𝑝 𝑚 3/10 𝑟 𝑝 (5)
where all values are in SI units. 𝑚 𝑒 is the mass of the electron, 𝑋 𝑝 the mole fraction of the perturber, 𝐸 𝑝 the energy of the first excited state of the perturber connected to its ground state by an allowed transition and 𝑚 𝑟 𝑝 the reduced mass of the radiator and the perturber. The formula for the parameter 𝑅 is taken from Griem:

𝑅 2 = 1 2 𝐸 𝐻 𝐸 ∞ -𝐸 𝑖 5 𝐸 𝐻 𝐸 ∞ -𝐸 𝑖 + 1 -3𝑙 𝑖 (𝑙 𝑖 + 1) (6)
where 𝐸 𝐻 is the ionization energy of the hydrogen atom, 𝐸 ∞ is the ionization energy of the radiating atom and 𝐸 𝑖 is the excited state energy of the radiator. 𝑙 𝑖 is the orbital angular momentum quantum number of the radiating state. The FWHM calculated by Eqn. 5 for each perturber was increased by a multiplicative factor of 𝑛 2/5 𝑒 , where 𝑛 𝑒 is the number of valence electrons of the perturber. This is done based upon a remark by Griem in section 4.8 of his text [START_REF] Griem | Principles of Plasma Spectroscopy[END_REF]. The calculations above apply only to atom-atom collisions as discussed by Griem. However, they are nonetheless applied to the molecular 𝑁 2 perturber. While the results give good agreement, the value for the Van der Waals broadening should therefore be considered as approximate.

Tables 2 and3 show the parameters used in Eqn. 6 for the van der Waals calculation. For conditions near the center of the plasma jet (T = 6700 K, p = 1 atm), Doppler broadening is calculated to have a FWHM (full-width at half maximum) of 2.3 𝑝𝑚 whereas van der Waals broadening is calculated to have a FWHM of 0.3 𝑝𝑚.

IV. Results

Figure 8 shows a SPECAIR calculation of equilibrium air emission corresponding to the measured temperature profile in Fig. 2a. The radiative transfer equation is solved along the line-of-sight to yield the calculated spectrum. The uncertainty in the measured temperature profile (Fig. 2a) leads to a corresponding uncertainty in the calculated intensity profile. This uncertainty is less than 5% across the entire wavelength range and, for wavelengths above 160 nm, is less than 3%. Prominent atomic features include the 174 nm atomic lines of nitrogen as well as three atomic carbon lines at 156 nm, 166 nm and 193 nm. The molecular structure is primarily composed of nitric oxide molecular bands: the

𝛾( 𝐴 2 Σ + -𝑋 2 Π), 𝛽(𝐵 2 Π -𝑋 2 Π), 𝛿(𝐶 2 Π -𝑋 2 Π), 𝜖 (𝐷 2 Σ + -𝑋 2 Π), 𝛽 (𝐵 2 Δ -𝑋 2 Π) systems.
The molecular bands of NO are not heavily optically thick in Fig. 8. The difference in intensity between the calculations of 𝑁𝑂 emission with and without self-absorption is very small over a large portion of the spectrum and is maximal around 180 nm where the difference grows to a factor of 2. The primary effect of self-absorption is on the atomic lines such as the nitrogen feature at 174 nm, which is optically thick. Figure 9a shows a comparison between the experimental results and the original SPECAIR calculation, after convolving with the measured slit function. Figure 9b shows a comparison between experimental results and the updated SPECAIR calculation after convolving the SPECAIR calculation in Fig. 8 with the measured slit function. The updates to the SPECAIR calculation significantly improve the agreement with the experiment at wavelengths below 200 nm.

The agreement between the updated SPECAIR calculation and the experiment is quite good down to approximately 180 nm. The amplitude of the nitrogen feature at 174 nm -which is very sensitive to line broadening estimates because of strong self-absorption -appears to be well predicted. Below 180 nm, SPECAIR is found to underpredict the observed molecular structure of the spectrum. This suggests that sources of emission are missing in the SPECAIR calculation.

V. Discussion

Despite the good agreement between the SPECAIR calculation and experiment, small discrepancies remain. The goal of this section is to discuss potential sources of this disagreement.

A. SPECAIR comparison with low temperature photoabsorption data

Chan et al [START_REF] Chan | Absolute optical oscillator strengths for the photoabsorption of nitric oxide (5-30 eV) at high resolution[END_REF] used electron energy loss spectroscopy to measure photoabsorption cross-sections from nitric oxide.

In their experiments, NO was taken from a compressed cylinder and would have been essentially at or close to room temperature. In 2007, Kato et al [START_REF] Kato | Excitation of the A 2 Σ + , C 2 Π and D 2 Σ + Rydberg-electronic states in NO by 100eV electrons[END_REF] performed another set of electron energy loss spectroscopy measurements targeting NO and noted good agreement between their measurements and those of Chan et al.

Figures 10a and 10b

suggest that there is a continuum source of emission/absorption from nitric oxide in the VUV.

To our knowledge, the source of this continuum has not been identified. In Fig. 10a, the underlying broadband structure appears to be from a true continuum, as opposed to a large broadening of spectral features such that they are merged together. If this is indeed the case, then inverse photodissociation would appear to be the culprit for the missing source -as opposed to pre-dissociation which would lead to a broadening and weakening of spectral features, but not to a broad continuum. The data reported by Chan et al was taken from nitric oxide at room temperature. Therefore, the reported cross-sections would have been measured from the ground electronic and ground vibrational level and would not allow us to account for vibrational excitation of the nitric oxide within the plasma jet studied in this work. However, to arrive at an order-of-magnitude estimation of the impact of the observed continuum on the spectrum from high temperature air as measured in the torch, these cross-sections were assumed to be independent of vibrational quantum number. In other words, the cross-sections are simply shifted by the energy of the particular excited vibrational level with respect to the ground vibrational level. This enabled an estimation of the emission that might be expected from the recombination 𝑁 + 𝑂 → 𝑁𝑂, if this is indeed the source of the observed continuum. Figure 11 shows the resulting comparison. The fairly good agreement suggests that such a continuum may indeed explain part or all of the observed discrepancy. Note that the overprediction of SPECAIR below 150 nm when accounting for this continuum is not necessarily problematic because a portion of the actual emitted signal could be absorbed by cold 𝑂 2 in the boundary of the plasma jet. Vibrationally specific photoabsorption cross-sections for NO, such as those presented for the 𝑂 2 molecule by Allison et al [START_REF] Allison | Absorption by vibrationally excited molecular oxygen in the Schumann-Runge continuum[END_REF], would help in assessing this hypothesis.

B. Perturbations between the 2 Σ + states of NO

A second potential candidate for the observed discrepancy would be additional excited state perturbations in the NO molecule not accounted for. De Vivie and Peyerimhoff [START_REF] De Vivie | Theoretical spectroscopy of the NO radical. I. Potential curves and lifetimes of excited states[END_REF] performed a comprehensive set of calculations with the goal of identifying and characterizing all electronic states of NO -including perturbations among the various states -up to the dissociation limit. They used a basis set composed of molecular and atomic orbitals in order to fully describe the molecular eigenstates and to calculate electronic transition dipole moments. They noted that their calculations are slightly less accurate than studies focused on a more limited subset of electronic states because of computational limitations. Indeed, the electronic transition dipole moment functions provided by De Vivie and Peyerimhoff are not used in SPECAIR calculations as other models were found to provide a better comparison with experimental data [START_REF] Laux | Optical Diagnostics and Radiative Emission of Air Plasmas[END_REF].

However, a big advantage of their calculation is the relatively complete picture that it provides of the various electronic states. They address the aforementioned perturbations among the various 2 Π states that were modeled by Gallusser and Dressler [START_REF] Gallusser | Multistate vibronic coupling between the excited 2 Π states of the NO molecule[END_REF] and that are accounted for in SPECAIR. In addition, they also discuss perturbations among certain 2 Σ + states. They note that the 𝐴 2 Σ + and 𝐷 2 Σ + (associated with the 𝛾 and 𝜖 molecular systems) interact with the 𝐼 2 Σ + state and a second repulsive state that they label the 2 2 Σ + 𝑣 state (referred to as the 𝐴 2 Σ + state by Miescher [START_REF] Miescher | The 𝐼 2 Σ + state of the NO molecule[END_REF]). These interactions lead to perturbations in the 𝑣 = 8 vibrational level of 𝐴 2 Σ + and 𝑣 = 3 -4 of 𝐷 2 Σ + . Figure 2 of the article by De Vivie and Peyerimhoff [START_REF] De Vivie | Theoretical spectroscopy of the NO radical. I. Potential curves and lifetimes of excited states[END_REF] shows the result of these perturbations on the electronic structure. A SPECAIR calculation of high temperature air emission (using the temperature profile in Fig. 2a) accounting only for the emission coming from these vibrational levels is shown in Fig. 12. These levels are responsible for a large portion of the emission at wavelengths between 160 and 180 nm where the discrepancy between calculation and experiment is largest (see Fig. 9b). Perturbations of these levels could therefore potentially account for the discrepancy seen with the measurements.

VI. Conclusion

Measurements of high temperature air emission spectra between 150 -250 nm (VUV/UV) are reported. This data is included with the article on the journal website -files containing the measured temperature profile, the slit function of the VUV spectrometer and the measured VUV spectrum are included. Subsequent comparisons with calculations led to several improvements in the SPECAIR radiation code, resulting in overall improved agreement. Perturbations amongst the various 2 Π states of NO were taken into account to better predict emission from the 𝛽 and 𝛿 systems. Several other new molecular systems were also added. Finally, atomic line broadening was also taken into account as this was found to have a significant impact on the predicted intensity of the atomic nitrogen 174 nm line, namely because the emission from this atomic line is impacted by self-absorption. The overall agreement is considered to be good. Below 180 nm, the amplitude is underpredicted by SPECAIR, and the finer structure of the spectrum is not fully captured. With the improvements that were made, SPECAIR was also found to provide accurate predictions of photoabsorption data reported by Chan et al down to approximately 145 nm, with the exception of a continuum that becomes significant at wavelengths below 160 nm.

One possible explanation for the observed discrepancies between SPECAIR and the emission measurements is an unaccounted for continuum emission source. Though its source is unknown, order-of-magnitude estimates suggest that the continuum observed by Chan et al (red curve, Fig. 10a) could potentially explain the discrepancy between SPECAIR predictions and measurements below 180 nm. A second possible explanation could be linked to perturbations among the various 2 Σ + states that are unaccounted for.

A. NO 𝛾 (E 2 Σ + -X 2 Π) band system Figure 13 shows how the inclusion of the 𝛾 system impacts the cross-section calculation for comparison with the data of Chan et al as well as the calculation of the high temperature emission. This can be compared with Fig. 10b which does not include this system. Including this system adversely impacts the comparison with the data of Chan et al while having a negligible impact on the predicted emission from the plasma torch. Two 𝑅𝑒(𝑟) curves were tested for this system. Both were scaled to give the measured lifetime of the 𝜈 = 0 level of the F-state reported by Brzozowski, Erman and Lyyra [START_REF] Brzozowski | Predissociation Rates and Perturbations of the A, B, B , C, D and F States in NO Studied Using Time Resolved Spectroscopy[END_REF]. The first curve was simply a constant. The second was a scaled 𝑅𝑒(𝑟) derived from the 𝛾 system. This second option was motivated by an observation that the 𝑅𝑒(𝑟) curves used in SPECAIR for the 𝛾( 𝐴 2 Σ + -𝑋 2 Π), 𝜖 (𝐷 2 Σ + -𝑋 2 Π) and 𝛿(𝐶 2 Π -𝑋 2 Π) systems could be scaled by constants to produce a single curve -they appeared self-similar. Meanwhile, the potential energy curves for these states and the F-state have a similar form as well. Figure 14a shows how the different 𝑅𝑒(𝑟) curves used affect the modeling of the F-X system cross-section calculation for NO at room temperature. Figure 14b shows how the full SPECAIR calculation compares with the measurements of Chan et al when accounting for the F-X system. Figure 15 shows how the inclusion of the F-X system impacts the SPECAIR prediction of the emission from the plasma torch. The inclusion of the F-X system adversely impacts the comparison with the data of Chan et al while having a negligible impact on the predicted emission from the plasma torch.

  𝜈 𝜈 𝑒 = electronic-vibrational transition moment between vibrational states 𝜈 and 𝜈 𝑆 = total spin of electrons 𝑋 𝑝 = mole fraction of species 𝑝 𝑎 𝑜 = Bohr radius 𝑐 = speed of light 𝑒 = absolute value of electron charge 𝑓 𝑎𝑏𝑠 𝜈 𝜈 = absorption oscillator strength between vibrational states 𝜈 and 𝜈 𝑓 𝑒𝑚𝑖𝑠 𝜈 𝜈 = emission oscillator strength between vibrational states 𝜈 and 𝜈 ℎ = Planck constant 𝑙 𝑖 = orbital angular momentum quantum number (individual electron) 𝑘 = boltzmann constant 𝑚 𝑒 = mass of electron 𝑝 = pressure 𝑇 = temperature Δ𝜆 𝐷𝑜 𝑝 𝑝𝑙𝑒𝑟 = doppler broadening linewidth Δ𝜆 𝑣 𝑑𝑤 = Van der Waals linewidth Λ = projection of the total orbital angular momentum of a molecular electronic state 𝛿 𝑖 𝑗 = Kronecker delta function 𝜆 = wavelength 𝜈 = frequency 𝜈 = vibrational quantum number of upper state 𝜈 = vibrational quantum number of lower state 𝜈 𝜈 𝜈 = wavenumber of vibrational transition calculated according to r-centroid approximation 𝜓 𝜈 = vibrational wavefunction associated with vibrational state 𝜈 ℏ = ℎ/2𝜋 superscripts = upper state involved in an optical transition = lower state involved in an optical transition

Fig. 1

 1 Fig. 1 Plasma torch head and nozzle assembly. The gas injectors include radial, swirl and axial injectors.

Fig. 2 a

 2 Fig. 2 a) Temperature profile obtained by measuring the 777 nm oxygen triplet emission. The error bars are small, indicating that the temperature is accurately known. b) Composition of the plasma jet as a function of radius based upon the measured temperature profile. 𝐴𝑟, 𝐶𝑂 and 𝐶 are present in mole fractions on the order of 6 × 10 -3 , 2 × 10 -4 and 6 × 10 -5 , respectively.

( a )

 a Spectroscopic system used for making spectrally resolved VUV measurements of air emission. (b) Detail of the tube that protrudes from the imaging box and extends the optical line to the plasma. The final portion in orange touches the boundary of the plasma jet -it is manufactured out of copper and is water cooled. A 2 mm hole provides optical access. The portion in red is manufactured out of Teflon. Both the red and orange sections -which together makeup the final 10 cm of the optical path -are put under an argon purge. The portion in green is manufactured out of PEEK plastic and is put under vacuum. A MgF 2 window separates the vacuum section from the argon purge section.

Fig. 4 Fig. 5

 45 Fig. 4 Measured slit function of the VUV system.

Fig. 6

 6 Fig. 6 Evolution in the SPECAIR calculation after several modifications. The modifications are made sequentially from subplots a) -e). Once a modification is made, it is preserved for the remaining subplots that follow. a) The original SPECAIR and an initial update of the SPECAIR calculation. The curve "additional D states" is the same as the original calculation but with the maximum vibrational level of the D-state increased from 𝑣 = 5 to 𝑣 = 6. b) The curve "additional B' states" is the same as the curve "additional D states" but with the maximum vibrational level of the B'-state increased from 𝑣 = 7 to 𝑣 = 9.c) The curve "w/perturbations" is the same as the curve "additional B' states" but with the perturbation calculation for the oscillator strengths of the 2 Π states accounted for as described in the text. d) The curve "additional C states" is the same as the curve "w/perturbations" but with the maximum vibrational level of the 𝑁𝑂 C-state increased from 𝑣 = 4 to 𝑣 = 9. e) The curve "with 𝑁 2 (𝑎 -𝑋) and 𝑁𝑂 + ( 𝐴 -𝑋)" is the same as the curve "additional C states" but with the 𝑁 2 (𝑎 -𝑋) and 𝑁𝑂+ ( 𝐴 -𝑋) systems added.

Fig. 7

 7 Fig.7Contributions to predicted SPECAIR emission using the temperature profile in Fig.2a. All calculations were made after SPECAIR was updated as described in Section III. Contributions from a) NO 𝛾 system, b) NO 𝛽 system, c) NO 𝛿 system, d) NO 𝜖 system, e) NO 𝛽 system and f) the curve labeled CO(4+) contains emission from the CO 4+ system while the curve labeled 'minor systems' contains the combined emission from 𝑁 2 a 1 Π 𝑔 -X1 Σ + 𝑔 , NO+ 𝐴 1 Π -X 1 Σ + , NO 𝐿 2 Π -𝑋 2 Π, NO 𝐾 2 Π -𝑋 2 Π and NO 𝑄 2 Π -𝑋 2 Π systems.14

Fig. 8 Fig. 9 a

 89 Fig. 8 SPECAIR calculation based upon the temperature profile in Fig. 2a. Two blackbody curves are included for reference.

Fig. 10 a

 10 Fig. 10 a) Photoabsorption cross-sections measured by Chan et al (Figs. 2 -3 of their paper) [24]. The red curve denotes the underlying continuum that is observed. b) Comparison between measurements of Chan et al and SPECAIR. The continuum shown in Fig. 10a has been manually added to the SPECAIR calculation.

Fig. 11

 11 Fig. 11 Experiment and SPECAIR comparison -SPECAIR is red and experiment is blue. SPECAIR calculation includes estimate of continuum based upon Chan et al data [24].

2 /Fig. 12

 212 Fig. 12 Calculated emission from specific vibrational levels of the 𝜖 and 𝛾 systems of NO using the temperature profile in Fig. 2a.

Fig. 13

 13 Fig. 13 a) SPECAIR calculations of high temperature air emission with (black) and without (red) the 𝛾 system included. The red curve corresponds to the SPECAIR calculation in Fig. 9b. b) NO cross-section calculations including the 𝛾 system in red are compared with the data of Chan et al in blue. The 𝛾 transition results in an over-prediction of several features.

Fig. 14 a

 14 Fig.14 a) Comparison between SPECAIR calculations of the F-X absorption cross-sections using a constant 𝑅𝑒(𝑟) or an 𝑅𝑒(𝑟) self-similar to that of the NO 𝛾 system. In both cases, the 𝑅𝑒(𝑟) curve has been scaled so that the lifetime of the 𝑣 = 0 level matches the lifetime reported by Brzozowski et al[START_REF] Brzozowski | Predissociation Rates and Perturbations of the A, B, B , C, D and F States in NO Studied Using Time Resolved Spectroscopy[END_REF]. In the calculations/figures to follow, the scaled 𝑅𝑒(𝑟) curve was used for F-X calculations. b) NO cross-section calculations including the 𝐹 -𝑋 system in red are compared with the data of Chan et al in blue.

2 /

 2 Fig. 15 SPECAIR calculation of line-of-sight emission with the temperature profile in Fig. 2a. The red curve corresponds to the SPECAIR calculation in Fig. 9b. The black curve includes the NO F-X system.

Table 1 Sources for the parameters used in the RKR portion of the 𝑁𝑂 molecular emission calculations from the A, D and B' upper states to the X lower state.

 1

	State	Molecular Constants	Diss. Energy (cm -1 )
	X 2 Π 𝑟	Amiot [18]	Lavrov [19]
	A 2 Σ +	Huber and Herzberg [20]	Lavrov [19]
	D 2 Σ +	Huber and Herzberg [20]	Lavrov [19]
	B' 2 Δ

Table 2 Parameters used in Eqn. 6 for the line broadening calculation for the 174 nm line.

 2 𝐸 𝐻 (𝑒𝑉) 𝐸 ∞ (𝑒𝑉) 𝐸 𝑖 (𝑒𝑉) 𝑙 𝑖

	13.6	14.5	10.7	0

Table 3 Species-specific parameters used in Eqn. 6 for the line broadening calculation for the 174 nm line. 𝑛 𝑒 is the number of valence electrons taken for the particular perturber. The value of

 3 𝑛 𝑒 = 10 for 𝑁 2 is twice the value for an individual 𝑁 atom. 𝐸 𝑝 for 𝑁 2 corresponds to the energy of the 𝑏 1 Π 𝑢 state

which is the first electronic state connected to the ground state by a dipole allowed transition.

	𝑋 𝑝	𝐸 𝑝 (𝑒𝑉) 𝑛 𝑒
	𝑁 0.31	10.3	5
	𝑂 0.29	9.1	6
	𝑁 2 0.39	12.6	10

Acknowledgments

We would like to acknowledge the contributions of Umar A Sheikh, Richard Morgan and Victor Gondret who participated in the development of this VUV system. The VUV spectrometer was loaned to the authors by NASA Ames as part of an International NASA Space Act Agreement for equilibrium radiation measurements in collaboration with Brett Cruden of NASA Ames.


Fernando Bussiman

Ching-Yi Chen

Justin Holl

Matias Bermann

|| Andres Legarra

Ignacy Misztal

Daniela Lourenco

Boundaries for genotype, phenotype, and pedigree truncation in genomic evaluations in pigs

Keywords: data truncation, genomic selection, old genotypes, pedigree depth, single-step

Historical data collection for genetic evaluation purposes is a common practice in animal populations; however, the larger the dataset, the higher the computing power needed to perform the analyses. Also, fitting the same model to historical and recent data may be inappropriate. Data truncation can reduce the number of equations to solve, consequently decreasing computing costs; however, the large volume of genotypes is responsible for most of the increase in computations. This study aimed to assess the impact of removing genotypes along with phenotypes and pedigree on the computing performance, reliability, and inflation of genomic predicted breeding value (GEBV) from single-step genomic best linear unbiased predictor for selection candidates. Data from two pig lines, a terminal sire (L1) and a maternal line (L2), were analyzed in this study. Four analyses were implemented: growth and "weaning to finish" mortality on L1, pre-weaning and reproductive traits on L2. Four genotype removal scenarios were proposed: removing genotyped animals without phenotypes and progeny (noInfo), removing genotyped animals based on birth year (Age), the combination of noInfo and Age scenarios (noInfo + Age), and no genotype removal (AllGen). In all scenarios, phenotypes were removed, based on birth year, and three pedigree depths were tested: two and three generations traced back and using the entire pedigree. The full dataset contained 1,452,257 phenotypes for growth traits, 324,397 for weaning to finish mortality, 517,446 for pre-weaning traits, and 7,853,629 for reproductive traits in pure and crossbred pigs. Pedigree files for lines L1 and L2 comprised 3,601,369 and 11,240,865 animals, of which 168,734 and 170,121 were genotyped, respectively. In each truncation scenario, the linear regression method was used to assess the reliability and dispersion of GEBV for genotyped parents (born after 2019). The number of years of data that could be removed without harming reliability depended on the number of records, type of analyses (multitrait vs. single trait), the heritability of the trait, and data structure. All scenarios had similar reliabilities, except for noInfo, which performed better in the growth analysis. Based on the data used in this study, considering the last ten years of phenotypes, tracing three generations back in the pedigree, and removing genotyped animals not contributing own or progeny phenotypes, increases computing efficiency with no change in the ability to predict breeding values.

Lay Summary

Recording data for long years is common in animal breeding and genetics. However, the larger the data, the higher the computing cost of the analysis, especially with genomic information. This study aimed to investigate the impact of removing data, namely, genotypes, phenotypes, and pedigree, on the computing performance and prediction ability of genomic breeding values. We tested four scenarios to remove genotyped individuals in pig populations. For each scenario, phenotypes were removed according to birth year, and the pedigree was either kept complete or traced back from two to three generations. Reliabilities for young, genotyped animals did not differ after removing genotypes for older or less important animals. However, using only two generations of data slightly reduces the reliability for young, genotyped animals. The dispersion did not change across the studied scenarios, and its worst value was observed when using only one generation in the pedigree. Using the last ten years of phenotypes, a pedigree depth of three generations, and removing genotyped animals not contributing own or progeny phenotypes reduces computing cost with no change in the ability to predict breeding values.

Introduction

Over the past decades, breeding companies and breed associations have accumulated large amounts of historical data, including pedigree, phenotypes, and genotypes. Adding all ancestors to the relationship matrix increases the accuracy of breeding values if the model of analysis is the true model [START_REF] Henderson | Applications of linear models in animal breeding models[END_REF]. However, our models are usually approximations [START_REF] Lourenco | Are evaluations on young genotyped animals benefiting from the past generations?[END_REF], and this discrepancy between fitted and true models can result in a steadily cumulative bias [START_REF] Macedo | Removing data and using metafounders alleviates biases for all traits in Lacaune dairy sheep predictions[END_REF]. Moreover, the genetic contributions from previous generations decay over time. Still, using those large historical data requires high computing power and could result in biased predictions for young animals [START_REF] Macedo | Removing data and using metafounders alleviates biases for all traits in Lacaune dairy sheep predictions[END_REF]. This is especially the case in single-step genomic best linear unbiased predictor (ssGBLUP), where genomic and pedigree-based relationships are combined, but the number of generations in the pedigree is often much larger than the generations of genotyped animals [START_REF] Lourenco | Are evaluations on young genotyped animals benefiting from the past generations?[END_REF].

New algorithms to compute the inverses of the genomic (G) and pedigree-based (A) relationship matrices have been proposed, making large ssGBLUP evaluations feasible. Among them is the algorithm for proven and young (APY), which generates a sparse representation of the inverse of G (Misztal et al., 2014a). For the computation of the inverse of A 22 , the pedigree relationship matrix for genotyped animals, partitions of the inverse of A can be used [START_REF] Strandén | Comparison of some equivalent equations to solve single-step GBLUP[END_REF][START_REF] Masuda | Technical note: avoiding the direct inversion of the numerator relationship matrix for genotyped animals in single-step genomic best linear unbiased prediction solved with the preconditioned conjugate gradient[END_REF]. Together with implementing the preconditioned conjugate gradient and parallel multi-processing programming, these two algorithms allowed the implementation of ssGBLUP evaluations for almost 30 million pedigreed dairy cattle, of which million were genotyped [START_REF] Cesarani | Multibreed genomic evaluation for production traits of dairy cattle in the United States using single-step genomic best linear unbiased predictor[END_REF]. However, whether all genotyped animals should be used in ssGBLUP is still a subject under investigation.

Under ssGBLUP, G must be scaled to be compatible with A 22 [START_REF] Chen | Effect of different genomic relationship matrices on accuracy and scale[END_REF][START_REF] Vitezica | Bias in genomic predictions for populations under selection[END_REF], and the size of A 22 will depend on the length of the pedigree for genotyped animals. Thus, one can expect that a too-large G results in upward bias for genotyped animals, whereas a too-small G does the opposite. In addition, as this scaling is for an average of A 22 , genomic predicted breeding value (GEBV) for genotyped animals can be biased depending on the length of the pedigree [START_REF] Lourenco | Are evaluations on young genotyped animals benefiting from the past generations?[END_REF]. Data truncation was suggested as a potential solution to this issue because 1) by removing old generations in the pedigree, the base population in A 22 becomes closer to the base population in G; 2) by deleting historical data, selection bias accumulated over time can be alleviated.

Data truncation has been successfully applied in dairy cattle, pigs, and dairy sheep [START_REF] Lourenco | Are evaluations on young genotyped animals benefiting from the past generations?[END_REF][START_REF] Howard | The impact of truncating data on the predictive ability for single-step genomic best linear unbiased prediction[END_REF][START_REF] Cesarani | Genomic predictions for yield traits in US Holsteins with unknown parent groups[END_REF][START_REF] Hollifield | Determining the stability of accuracy of genomic estimated breeding values in future generations in commercial pig populations[END_REF][START_REF] Macedo | Removing data and using metafounders alleviates biases for all traits in Lacaune dairy sheep predictions[END_REF]. However, depending on the species, non-informative genotyped animals are not always removed from the analyses, and when the truncation is also applied to the genotyped animals, it is based on birth year [START_REF] Howard | The impact of truncating data on the predictive ability for single-step genomic best linear unbiased prediction[END_REF]. Genotyped animals without own or progeny phenotypes are not expected to contribute to predictions of young animals, and in principle could safely be removed. Therefore, this study aimed to investigate how different criteria to remove genotyped animals along with pedigree and phenotype truncation would impact the prediction of GEBV of genotyped selection candidates.

Material and Methods

Data

Data for two pig lines, a terminal sire (L1) and a maternal (L2) were provided by PIC (a Genus Company, Hendersonville, TN). Four data sets were analyzed: growth on L1; "weaning to finish" mortality on L1; preweaning traits on L2; and reproductive traits on L2. Variance components were estimated considering the entire population and no genomic information. Genetic parameters and variance components are shown in Table 1. Because data were obtained from existing databases, the approval of the Animal Care and Use Committee was not needed for this study.

Phenotypes, pedigree, and genotypes

Purebred (L1 and L2) and crossbred (L1) animals were recorded for nine different traits: average daily gain and backfat on purebred and crossbred pigs (average daily gain on purebreds [ADG P ], backfat thickness on purebreds [BF P ], average daily gain on crossbreds [ADG X ], and backfat thickness on crossbreds [BF X ], respectively), "weaning to finish" mortality (WFM), birth weight and pre-weaning mortality (BW and PWM, respectively), litter size and number of stillborn (LS and NS, respectively). ADG P , ADG X , BF P , BF X , ). For all analyses, pedigree, genotypes, and phenotypes were analyzed together using ssGBLUP. This method uses the inverse of a combined pedigree-genomic relationship matrix (H -1 ), which requires the computation of the inverse of the genomic relationship matrix (G -1 ). Because of the number of genotyped animals, APY (Misztal et al., 2014a) was used to compute G -1 (G -1 APY - [START_REF] Misztal | Core-dependent changes in genomic predictions using the algorithm for proven and young in single-step genomic best linear unbiased prediction[END_REF]. Core animals were set to 11,000 for M1 and M2, and 7,500 for M3 and M4, according to the dimensionality of the genomic information (Pocrnic et al., 2016a;Pocrnic et al., 2016b) within each line, calculated as the number of eigenvalues explaining at least 99% of the variability in G. The core animals were chosen to be consistent with the truncation scenarios (defined later). Only animals with phenotypes for at least one of the considered traits within the analyses, born between 2017 and 2019, were randomly sampled to compose the core, and the core was kept constant throughout the analyses.

Finally, for M4, unknown parent groups (UPG) were assigned according to account for population admixture and modeled as a random effect. Therefore, H -1 (for M4) was given by [START_REF] Misztal | Unknown-parent groups in single-step genomic evaluation[END_REF]:

H -1 UPG, Σ = A * Σ +     00 0 0G -1 APY -A -1 22 - Ä G -1 APY -A -1 22 ä Q 2 0-Q 2 Ä G -1 APY -A -1 22 ä Q 2 Ä G -1 APY -A -1 22 ä Q 2     ,
Where A * Σ is the inverse of additive relationship matrix based on Quaas-Pollak (QP) transformation [START_REF] Quaas | Modified equations for sire models with groups[END_REF][START_REF] Quaas | Additive genetic model with groups and relationships[END_REF][START_REF] Westell | Genetic groups in an animal model[END_REF]; G -1 APY was defined above, A -1 22 is the inverse of the pedigree relationship matrix among genotyped animals, and Q 2 is a matrix relating genotyped animals to UPG.

All the analyses were performed using iteration on data with preconditioned conjugate gradient and parallel processing by OpenMP (OpenMP Architecture Review Board, 2015) using the BLUPF90IOD2OMP1 (Misztal et al., 2014b) software, which is an optimized version of BLUPF90IOD2 [START_REF] Tsuruta | Use of the preconditioned conjugate gradient algorithm as a generic solver for mixed-model equations in animal breeding applications[END_REF][START_REF] Tsuruta | Technical note: computing options for genetic evaluation with a large number of genetic markers[END_REF]. Computations were carried out on a Linux server (x86_64) with 1 TB of RAM and an Intel Xeon E7-8857 v2 (3.00 GHz) processor (24 computing cores), and the convergence criteria [START_REF] Tsuruta | Use of the preconditioned conjugate gradient algorithm as a generic solver for mixed-model equations in animal breeding applications[END_REF] was set to 1 -15 (for M1, M2, andM4) or 1 -12 (for M3).

Truncation of genotypes, phenotypes, and pedigree

Four scenarios were tested to assess the possibility of cutting old genotypes. For the first scenario (noInfo), we removed all genotyped animals not contributing phenotypes to the evaluation, i.e., genotyped animals without own or progeny phenotypes. The second scenario (Age) was created by removing genotyped animals according to their birth year (regardless of if phenotyped). The third scenario combined the first two approaches (noInfo + Age), where all non-informative genotyped animals were removed first, and then removal was based on birth year. In the fourth scenario, named AllGen, no genotyped animals were removed. For all scenarios, phenotypes were removed consecutively, according to the birth year, assuming a generation interval of three years for this population [START_REF] Lourenco | Are evaluations on young genotyped animals benefiting from the past generations?[END_REF], that is, every truncated dataset had the three oldest years of phenotypic information removed. For the pedigree truncation, we either kept all the pedigree information or traced back two or three generations in each phenotype truncation year and across the four genotype truncation scenarios. Since each dataset presented different ranges of birth year, the truncation points were different for each analysis (Table 2). The number of genotyped animals kept after removing non-informative animals (i.e., non-phenotyped and with no progeny) was 141,242 for M1 (16.29% removed), 26,894 for M2 (84.06% removed), 86,054 for M3 (49.42% removed), and 42,282 for M4 (75.15% removed). For the AllGen scenario, all genotyped animals were used. The number of genotyped animals kept for Age and noInfo + Age scenarios is in Figure 1.

Validation

The LR validation [START_REF] Legarra | Semi-parametric estimates of population accuracy and bias of predictions of breeding values and future phenotypes using the LR method[END_REF] was used to evaluate the truncation schemes. Focal individuals (males and females with phenotyped progeny) were chosen from the group of genotyped animals born in 2019 and subsequent years. For validation purposes, focal animals had their phenotypes masked, along with their contemporaries and progeny. The number of focal animals was different for each trait: 2,165 for ADG P , 1,897 for BF P , 2,221 for ADG X , 1,944 for BF X , 415 for WFM, 2,096 for BW, 3,949 for PWM, 910 for LS, and 933 for NS. Those numbers varied according to the availability of phenotyped progeny for validation candidates in each dataset. Let the masked phenotypes file be represented by the subscript p (partial), whereas the whole dataset is represented by the subscript w (whole). Under the LR method, the reliability (rel), and the dispersion (b) of GEBV can be calculated as:

rel = cov ûp , ûw σ 2 * u b = cov ûp , ûw var ûp
Where û is the vector of GEBV for focal animals, and σ 2 * u is the trait additive genetic variance in the focal individuals [START_REF] Macedo | Selection and drift reduce genetic variation for milk yield in Manech Tete Rousse dairy sheep[END_REF], calculated as in [START_REF] Sorensen | Inferring the trajectory of genetic variance in the course of artificial selection[END_REF].

Changes in ranking and predictions were evaluated on the scale of GEBV, using the analysis with no data truncation as a benchmark, differences were then assessed by the Spearman (ranking) and Pearson (GEBV) correlation coefficients across the truncation years within each depth of the pedigree and genotype truncation scenario. The function "ggplot" from the "ggplot2" [START_REF] Wickham | ggplot2: elegant graphics for data analysis[END_REF]) R package (R Core [START_REF] Core | R: a language and environment for statistical computings[END_REF] was used for all graphs in this study.

Results and Discussion

Validation

Correlations between GEBV and the ranking of validation animals did not change significantly after removing any quantity of historical data, except for ADG X (Table 3). The weakest correlations were observed for the truncation on the last tested year for all analyzed data. This may suggest the need for at least two generations of data to predict validation individuals because the amount of data kept in the last tested year corresponds to one-generation interval. We also observed no benefits by keeping all generations in the pedigree when truncation was applied to either phenotypes or genotypes and phenotypes. Increasing the number of generations traced back in the pedigree, without the corresponding phenotypes reduces rank correlations among focal individuals when a few years of data are analyzed.

Figure 2 presents reliability and dispersion coefficients for M1. The initial level of dispersion for the focal indi-viduals did not change for any truncation year across all tested scenarios. In contrast, the reliability decreased after 2014 for BF P and BF X , no matter how many generations were included in the pedigree. For ADG X , the reliability was slightly underestimated in all scenarios, but a more significant drop happened when keeping data from 2017. No dispersion problem was observed for ADG P , ADG X , BF P , and BF X in any scenario and truncation point. Conversely, GEBV for focal individuals were overdispersed in WFM in all truncation scenarios, whereas their reliability did not change significantly up to 2013 (Figure 3). Growth traits (ADG P , ADG X , BF P , and BF X ) are widely recorded, and the number of phenotyped animals within each year is higher than for mortality. The data structure and the type of analysis could also explain those results.

While growth traits are recorded on pure and crossed animals, WFM is recorded only on the commercial progeny of purebred animals. Therefore, there was a one-generation gap between the phenotypic measurement and the focal individuals for WFM. Furthermore, growth traits are analyzed in a four-traits model, which usually increases the reliability due to genetic correlations, whereas WFM is in a single-trait model. This can also explain why the original level of dispersion and reliability did not change for pre-weaning When all the animals were kept in the pedigree; the number of animals was 3,601,369 (for M1 and M2) and 11,240,865 (for M3 and M4).

The number of records is equal to the number of animals for M1, M2, and M3, because the traits in these models were not repeated. The "-" implies that this analysis was not performed (i.e., pedigree truncation was not applied to the full dataset). mortality (Figure 4). Even though the heritability (h 2 ) for WFM is 0.05, and for PWM d is 0.01, the genetic correlation between PWM d and BW m (0.39) helps improve predictions because h 2 for BW m is 0.20. Finally, the number of records for pre-weaning traits is larger than for WFM.

For BW, neither the reliability nor the dispersion was affected by the phenotype and genotype truncation (Figure 4); however, tracing back only two generation in the pedigree slightly increased dispersion of GEBV for young animals. The maternal effect of PWM was slightly overdispersed when truncating data from 2009, whereas the reliability did not change with truncation up to 2012. The truncation did not affect the reproductive traits, and in some cases, the reliability of LS improved after removing old data (Figure 5). In most cases, the different scenarios for removing genotyped animals resulted in the same reliability and dispersion, and discrepancies among scenarios were observed only when keeping less than 10 years of data.

Pedigree and phenotypic truncation

Our results mostly agree with previous studies. Increasing the number of generations traced back in the pedigree from 2 to 3 did not increase the predictive ability of US Holstein bulls [START_REF] Cesarani | Genomic predictions for yield traits in US Holsteins with unknown parent groups[END_REF]. Similarly, keeping all the gener-ations in the pedigree did not improve the predictive ability of young, genotyped pigs as compared to tracing back two generations [START_REF] Lourenco | Are evaluations on young genotyped animals benefiting from the past generations?[END_REF]. In our analyses, reliabilities within the truncated dataset did not increase when we changed from tracing back two to three generations to including all known ancestors in the pedigree. However, using only one generation of data and tracing back one generation in the pedigree reduced GEBV accuracies in a simulated population [START_REF] Howard | The impact of truncating data on the predictive ability for single-step genomic best linear unbiased prediction[END_REF].

The number of useful generations with associated phenotype depends on the trait heritability: the lower the heritability, the more generations are needed [START_REF] Mrode | Linear models for the prediction of animal breeding values[END_REF][START_REF] Hollifield | Determining the stability of accuracy of genomic estimated breeding values in future generations in commercial pig populations[END_REF]. The reliability depends on the heritability and the relationship between training and validation sets (Habier , 2010;[START_REF] Pszczola | Reliability of direct genomic values for animals with different relationships within and to the reference population[END_REF]. [START_REF] Weng | Effects of number of training generations on genomic prediction for various traits in a layer chicken population[END_REF] found that for lowly heritable traits, having genotyped animals in the training population more related to the validation animals is better. This may be related to changes in the relationships across other animals because genomic relationships are implicitly imputed for non-genotyped animals through A 22 in H, the joint relationship matrix in ssGBLUP [START_REF] Legarra | A relationship matrix including full pedigree and genomic information[END_REF]. If that is the case, truncating genotypes may affect those traits more when genotyped animals have no phenotypes.

In ssGBLUP, GEBV are predicted as a weighted sum of information from different sources: parent average (PA), yield deviation (YD), progeny contribution (PC), and genomic information (GI) [START_REF] Aguilar | Hot topic: a unified approach to utilize phenotypic, full pedigree, and genomic information for genetic evaluation of Holstein final score[END_REF]Lourenco et al., 2015a;[START_REF] Vanraden | Measuring genomic pre-selection in theory and in practice[END_REF]. GI can be further split into two parts [START_REF] Aguilar | Hot topic: a unified approach to utilize phenotypic, full pedigree, and genomic information for genetic evaluation of Holstein final score[END_REF]: direct genomic value (DGV) and pedigree prediction (PP) that comes from A 22 . No genotyped animals had phenotypes for WFM; thus, YD = 0. As the information from PA and PP is similar, GEBV will be reduced to PC and DGV. If the progeny number is large, PC will dominate the GEBV; if not, DGV will explain a more significant fraction of GEBV. In such a case, truncating genotypes and progeny phenotypes may impact predictions more.

The GEBV accuracy is less affected by the genotyping structure in ssGBLUP than in multi-step methods because of the additional pedigree information, which accounts for the relationships between a given genotyped animal and the rest of the population (Lourenco et al., 2015a). The composition of the reference population still influences GEBV predictivity; however, the inclusion of genotyped animals without progeny seems not to increase the predictability (Lourenco et al., 2015b). Therefore, removing those genotyped animals with no progeny and no phenotype should not impact the reliability, which was observed in our analyses (noInfo scenario). In fact, the noInfo scenario slightly improved the reliability of BF X . [START_REF] Howard | The impact of truncating data on the predictive ability for single-step genomic best linear unbiased prediction[END_REF], argued that a slight numerical increase in the predictive ability from truncated data is related to changes in trait definition over time. This definition change can be because of different measurement techniques or selection criteria and environmental changes [START_REF] Tsuruta | Changing definition of productive life in US Holsteins: effect on genetic correlations[END_REF]. Considering a trait under strong selection and a short generation interval, the population average will rapidly increase or decrease depending on the direction of selection. If that is the case, removing old information from the dataset moves the average of the solutions closer to the desired direction once those animals with more distant breeding values are removed.

If the trait is selected for increasing the average, such as growth and some reproductive traits, the GEBV average from the partial, truncated dataset will be higher than the whole, non-truncated dataset; therefore, the bias will be positive. Conversely, when animals are selected for a reduction in the average, like in mortality, the GEBV average from the partial, truncated dataset will be smaller than the whole, non-truncated, and the bias will be negative. This was observed for all studied traits (data not shown), even though the truncation is supposed to reduce the level of bias [START_REF] Macedo | Removing data and using metafounders alleviates biases for all traits in Lacaune dairy sheep predictions[END_REF]. The selection effect changes when the partial and whole data are truncated or not. A bias reduction from truncation was observed for noInfo and noInfo + Age scenarios (data not shown).

The depth of the pedigree has a minor influence on the reliability of genomic prediction [START_REF] Lourenco | Are evaluations on young genotyped animals benefiting from the past generations?[END_REF][START_REF] Howard | The impact of truncating data on the predictive ability for single-step genomic best linear unbiased prediction[END_REF][START_REF] Cesarani | Genomic predictions for yield traits in US Holsteins with unknown parent groups[END_REF], which was observed in our study. Additionally, no major difference was found after tracing back two to three generations in the pedigree for all traits. No additional benefit was found by using all pedigree information; in some cases, the impact on predictions was the same as using two generations (PWM d ). According to [START_REF] Macedo | Removing data and using metafounders alleviates biases for all traits in Lacaune dairy sheep predictions[END_REF], if models used to estimate breeding values are imperfect, a small percentage of bias will affect PA each year, accumulating bias over the years. The same effect can appear if the genetic correlation across distant generations but in the same trait is not one [START_REF] Tsuruta | Genetic correlations among production, body size, udder, and productive life traits over time in Holsteins[END_REF], which is the case of genotype by environment interaction. Therefore, some analyses can be biased per se, and correcting for this bias would involve improving the model, which is not always easy as linear models have limitations if important factors are not accounted for [START_REF] Macedo | Removing data and using metafounders alleviates biases for all traits in Lacaune dairy sheep predictions[END_REF]. In that case, removing old information can avoid this bias accumulation. In our study, data truncation helped release the level of bias in many cases (results not shown) but did not affect the dispersion. Thus, data truncation can be an effective alternative to alleviate the bias level without harming the reliability and dispersion of genotyped validation animals. Interestingly, the last phenotypic truncation point showed reduced reliability and increased overdispersion for some traits in M1, M3, and M4. For M4, this could be explained by the extensive removal of phenotypes causing poor estimates of UPG, which have been previously regarded as a source of bias if not enough phenotypes are used to estimate their effects [START_REF] Tsuruta | Controlling bias in genomic breeding values for young genotyped bulls[END_REF][START_REF] Masuda | Invited review: unknown-parent groups and metafounders in single-step genomic BLUP[END_REF]. As M1 included phenotypes from pure and crossbred animals, the last truncation point removed phenotypes from nearly all purebred ancestors of crossbred individuals. The validation was performed only on purebred animals, parents of crossbreds; therefore, more crossbred phenotypes are needed to estimate the GEBV on purebreds for crossbred performance (as in M1). In M3, the prediction of the maternal effects was possibly affected in the last truncation point. As the maternal effects had greater h 2 (Table 1), they caused indirect changes in the direct genetic effect through genetic correlations.

Computing efficiency

Figure 6 shows the number of rounds to convergence for all truncation points across all tested pedigree depths. Mainly, with fewer data, fewer iterations were needed to reach convergence. There was a big difference between using the entire pedigree and tracing back three or two generations from each phenotype truncation year across the four genotype truncation scenarios. This difference can be attributed to the inclusion of many animals with no records and no progeny records when using the complete pedigree, and to the reduced number of equations when analyzing truncated data. Similar results were reported by [START_REF] Pocrnic | Technical note: impact of pedigree depth on convergence of single-step genomic BLUP in a purebred swine population[END_REF]. The number of iterations is related to the convergence rate: the better the convergence rate, the lesser iterations are needed. The convergence rate on the preconditioned conjugate gradient solver is related to the condition number of the coefficient matrix and the distribution of its eigenvalues (van der Sluis and van der [START_REF] Van Der Sluis | The rate of convergence of conjugate gradients[END_REF][START_REF] Strakoš | On the real convergence rate of the conjugate gradient method[END_REF]. The eigenvalues distribution will change depending on the number of non-zero elements of the coefficient matrix, which depends on the number and structure of equations.

With truncated data, increasing the number of generations in the pedigree would decrease the convergence rate, with a corresponding increase in the number of iterations [START_REF] Pocrnic | Technical note: impact of pedigree depth on convergence of single-step genomic BLUP in a purebred swine population[END_REF] due to the increase in the number of zeros. For the same system of equations, i.e., for the same truncation point and pedigree depth, differences in the number of iterations could be attributed to numerical accuracy in the computations [START_REF] Strandén | Solving efficiently large single-step genomic best linear unbiased prediction models[END_REF]. On the other hand, keeping all genotyped animals but truncating phenotypes and pedigree generally required more iterations to converge. This could be related to the number of non-zero elements of the coefficient matrix since the genomic information usually increases it [START_REF] Misztal | Emerging issues in genomic selection[END_REF], affecting the distribution of eigenvalues and, consequently, the condition number.

Practical remarks

Overall, data truncation may be affected by generation interval, which has decreased over time due to genomic selection. It can also be subject to changes in the trait definition because the genetic gain accumulated over the past years could result in a trait measured in the current animals being biologically different (and having different distributions) than in older animals. The ability to remove old data can also be affected by changes in variance components over time, which modifies breeding values for the same individuals in the same trait but in different years. Lastly, data accumulate fast, especially genotypes. The more genotyped animals, the higher computing costs. Therefore, constantly reevaluating how many years of data is needed for predicting GEBV on selection candidates is increasingly relevant. For new datasets, one can expect the more widely recorded and the more heritable the trait, the more data can be removed, depending on the generation interval. The number of phenotyped progeny and/or progeny per genotyped animal should also be considered when removing genotyped animals, along with the old information. For a precise assessment of pedigree inbreeding and genetic trends, one should remember that using the whole pedigree is recommended, as these two measures rely on the complete data.

Conclusions

Data truncation is a feasible alternative to reducing the computing costs of genomic evaluations without compromising the GEBV of selection candidates. The generation interval, number of records, and trait heritability influence the decision regarding how many years of data to keep. About ten years of data on widely recorded traits are enough for predictions, whereas more years are needed for traits with fewer records or evaluated only on the progeny. The heritability and the number of traits in the model should also be considered. For lowly heritable traits, more years of data are required to maintain the same level of reliability as in the non-truncated dataset. Fewer years of data can be used if a lowly heritable trait is analyzed with a moderately heritable trait because they are genetically correlated. Using more than three generations of pedigree for animals with phenotypes and/or genotypes does not improve predictions. Additionally, using up to three generations reduces computing costs and helps improve the compatibility between genomic and pedigree relationship matrices. Old, genotyped animals can be eliminated from the analyses without harming predictions for the selection candidates, and the best strategy is to remove genotyped animals without own or progeny phenotypes.

Figure 1 .

 1 Figure 1. Number of genotyped animals across truncation years for each model in Age (cutting off old, genotyped animals) and noInfo + Age (cutting off old, genotyped animals along with the non-informative) scenarios. The numbers in parenthesis are the percentage of removed genotyped animals.

Figure 2 .

 2 Figure 2. Reliability (rel) and dispersion (b) coefficients on focal individuals for growth traits in the terminal-sire line. The dashed line is the original level of reliability or dispersion for the nontruncated dataset, with all genotyped animals and the entire pedigree. The numbers in parenthesis are the percentage of phenotypes analyzed according to the trait. Abbreviations: noInfo, cutting off non-informative genotyped animals; Age, cutting off old genotyped animals; noInfo + Age, cutting off old genotyped animals, along with non-informative; and AllGen, no genotype removal.

Figure 3 .

 3 Figure 3. Reliability (rel) and dispersion (b) coefficients on focal individuals for mortality in the terminal-sire line. The dashed line is the original level of reliability or dispersion for the non-truncated dataset, with all genotyped animals and the entire pedigree. The numbers in parenthesis are the percentage of phenotypes analyzed according to the trait. Abbreviations: noInfo, cutting off non-informative genotyped animals; Age, cutting off old, genotyped animals; noInfo + Age, cutting off old, genotyped animals, along with non-informative; and AllGen, no genotype removal.

Figure 4 .

 4 Figure 4. Reliability (rel) and dispersion (b) coefficients for focal individuals for preweaning traits in the maternal line. The dashed line is the original level of reliability or dispersion for the non-truncated dataset, with all genotyped animals and the entire pedigree. The numbers in parenthesis are the percentage of phenotypes analyzed according to the trait. Abbreviations: noInfo, cutting off non-informative genotyped animals; Age, cutting off old, genotyped animals; noInfo + Age, cutting off old, genotyped animals, along with non-informative; and AllGen, no genotype removal.

Figure 5 .

 5 Figure 5. Reliability (rel) and dispersion (b) coefficients for focal individuals for reproductive traits in the maternal line. The dashed line is the original level of reliability or dispersion for the non-truncated dataset, with all genotyped animals and the entire pedigree. The numbers between parenthesis are the percentage of phenotypes analyzed according to the trait. Abbreviations: noInfo, cutting off non-informative genotyped animals; Age, cutting off old, genotyped animals; noInfo + Age, cutting off old, genotyped animals, along with non-informative; and AllGen, no genotype removal.

Figure 6 .

 6 Figure 6. Number of iterations to converge for each phenotypic truncation point across different pedigree depths for all genotype truncation scenarios. The dashed line represents the number of iterations in the non-truncated dataset, with all phenotypes, genotypes, and pedigree. Abbreviations: noInfo, cutting off non-informative genotyped animals; Age, cutting off old, genotyped animals; noInfo + Age, cutting off old, genotyped animals, along with non-informative; and AllGen, no genotype removal.

Table 1 .

 1 Heritability (diagonal, bold), genetic (above diagonal), and residual (below diagonal) correlations estimate for all evaluated traits ADG P , average daily gain on purebreds; BF P , backfat on purebreds; ADG X , average daily gain on crossbreds; BF X , backfat on crossbreds; WFM, weaning to finish mortality; BW

	Traits	Traits
		ADG P	BF P	ADG X	BF X	WFM	BW d	BW m	PWM d	PWM m	LS	NS
	ADG P	0.26	0.40	0.66	0.38
	BF P	0.26	0.41	0.14	0.90
	ADG X			0.24	0.27
	BF X WFM BW d BW m PWM d PWM m LS NS			0.32	0.29	0.05	0.06 -0.33	-0.38 0.20	-0.05 0.39 0.01	0.22 -0.47 -0.03 0.02	0.14 0.27	0.52 0.11	Downloaded from https://academic.oup.com/jas/article/doi/10.1093/jas/skad273/7243206 by INRA AVIGNON user on 29 September 2023

d , birth weight (direct additive effect); BW m , birth weight (maternal additive effect); PWM d , pre-weaning mortality (direct additive effect); PWM m , pre-weaning mortality (maternal additive effect); LS, litter size; and NS, snumber of stillborn.

Table 2 .

 2 Pedigree and phenotype count for each model across truncation years over the tested pedigree depths

	Model	Year 1	NR 2	NA 3	Number of animals kept in the pedigree 4
					Two generations	Three generations
	M1	Full	1,452,257	1,452,257	-	-
		2005	1,333,802	1,333,802	1,419,508	1,420,973
		2008	1,211,597	1,211,597	1,294,452	1,296,461
		2011	1,049,955	1,049,955	1,129,402	1,131,870
		2014	844,955	844,955	925,305	927,839
		2017	582,469	582,469	686,190	688,677
	M2	Full	324,397	324,397	-	-
		2007	302,200	302,200	502,704	503,961
		2010	252,300	252,300	448,823	450,069
		2013	178,317	178,317	369,443	370,686
		2016	125,504	125,504	312,745	313,988
	M3	Full	517,446	517,446	-	-
		2003	489,076	489,076	594,060	596,666
		2006	459,273	459,273	564,931	567,526
		2009	432,505	432,505	538,931	541,540
		2012	375,692	375,692	484,296	486,928
		2015	307,425	307,425	427,155	429,793
	M4	Full	7,853,629	2,322,474	-	-
		2002	7,753,676	2,296,858	2,480,557	2,482,212
		2005	7,557,900	2,248,913	2,436,270	2,439,407
		2008	7,108,436	2,136,521	2,336,026	2,341,316
		2011	6,319,428	1,946,867	2,158,129	2,166,354
		2014	5,033,113	1,641,339	1,871,378	1,882,611
		2017	3,107,674	1,135,171	1,398,058	1,412,626
	1 Phenotype truncation year.

2 Number of records. 3 Number of animals with records.

4

  Downloaded from https://academic.oup.com/jas/article/doi/10.1093/jas/skad273/7243206 by INRA AVIGNON user on 29 September 2023

Table 3 .

 3 Pearson (GEBV) and spearman (Rank) correlations (minimum value) between GEBV from the full dataset and each genotype truncation scenario for validation individuals

Model Trait GEBV Rank noInfo Age noInfo + Age AllGen noInfo Age noInfo + Age AllGen

	M1	ADG P	0.98	0.97	0.98	0.97	0.97	0.97	0.97	0.97
		BF P	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.99
		ADG X	0.90	0.87	0.87	0.90	0.89	0.86	0.86	0.89
		BF X	0.98	0.97	0.97	0.98	0.97	0.97	0.97	0.97
	M2	WFM	0.98	0.97	0.97	0.97	0.96	0.96	0.96	0.96
	M3	BW d	0.98	0.98	0.98	0.98	0.98	0.98	0.98	0.98
		BW m	0.98	0.98	0.98	0.98	0.98	0.98	0.98	0.98
		PWM d	0.94	0.93	0.93	0.94	0.93	0.92	0.93	0.93
		PWM m	0.96	0.95	0.96	0.95	0.95	0.95	0.95	0.95
	M4	LS	0.97	0.96	0.96	0.97	0.96	0.96	0.96	0.96
		NS	0.97	0.95	0.95	0.97	0.96	0.95	0.94	0.97
	ADG P , average daily gain on purebreds; BF P , backfat on purebreds; ADG X , average daily gain on crossbreds; BF X , backfat on crossbreds; WFM, weaning to finish mortality; BW d , birth weight (direct additive effect); BW m , birth weight (maternal additive effect); PWM d , pre-weaning mortality (direct additive effect); PWM

m , pre-weaning mortality (maternal additive effect); LS, litter size; and NS, number of stillborn.

Acknowledgments

Discussions and edits from Jorge Hidalgo, Jennifer Richter, and Joe-Menwer Tabet are highly appreciated. We gratefully acknowledge the valuable comments of the anonymous reviewers.This study was supported by the Pig Improvement Company.

Conflict of interest statement

Ching-Yi Chen and Justin Holl are employees of the Pig Improvement Company. The authors declare no real or perceived conflicts of interest.