Investigation of conditions necessary for inception of positive corona in air based on differential formulation of photoionization

Sharp point electrodes generate significant electric field enhancements where electron impact ionization leads to the formation of electron avalanches that are seeded by photoionization. Photoionization of molecular oxygen due to extreme ultraviolet emissions from molecular nitrogen is a fundamental process in the inception of a positive corona in air. In a positive corona system, the avalanche of electrons in the bulk of the discharge volume is initiated by a specific distribution of photoionization far away from the region of maximum electron density near the electrode where these photons are emitted. Here, we present a new approach to finding the inception conditions for a positive corona, which is based on a differential formulation of the photoionization problem. The proposed iterative solution considers the same inception problem that has been solved in the existing literature by using either an integral approach to photoionization or a differential formulation of photoionization and considering the inception problem as a boundary-value eigenvalue problem. The results are validated by comparisons with previous integral formulations and time dynamic plasma fluid solutions in planar and spherical geometries. The results illustrate ideas advanced in Kaptzov (1950 Elektricheskiye Yavleniya v Gazakh i Vacuume p 610) providing a physically transparent connection between an effective secondary electron emission coefficient due to volume photoionization in a positive corona system and the secondary electron emission in conventional Townsend discharge theory. The results also demonstrate the significance of boundary conditions for accurate corona solutions that are based on a differential formulation of photoionization.


Introduction
The photoionization of molecular oxygen, O 2 , due to extreme ultraviolet emissions from molecular nitrogen, N 2 , in the wavelength range 98-102.5 nm (Zheleznyak et al 1982) is a fundamental process in positive corona discharges in air where electrons avalanche toward a sharp point that focuses electric field lines leading to significant field enhancements and electron multiplication due to electron impact ionization (Naidis 1987). In a positive corona system, the avalanche of electrons in the bulk of the discharge volume is initiated by a specific distribution of photoionization far away from the electrode. Most of the photons responsible for this photoionization originate from the near electrode region, where the electron density and electric field are maximized (Naidis 2005). Interest in further research on positive corona discharges has increased in recent years due to the need to better understand the inception conditions of a cylindrical corona as a function of gas pressure and humidity (Mikropoulos and Zagkanas 2015) and investigations of the electro-hydro-dynamic effects of a cylindrical corona including coupled inner/glow and outer/drift regions (Monrolin et al 2018, Monrolin andPlouraboue 2021). There is a significant interest in corona discharges in geophysics, as these are directly linked to the physics of lightning, and in particular, the lightning initiation problem. The corona discharges initiated by hydrometeors play a significant role in the thundercloud electrical environment, with related research focusing on the effects of pressure and humidity on positive corona inception from thundercloud hydrometeors in spherical geometry (Liu et al 2012) and on the understanding of the ignition and propagation of streamers in a system of two approaching/colliding hydrometeors (e.g. Jansky and Pasko 2020, and the extensive list of references therein). Recent research on photoelectric feedback in relativistic runaway electron avalanches has benefitted from the existing knowledge on positive coronas, as the related processes are fully analogical to photoionization feedback in a positive corona (Pasko et al 2023).
A corona is typically observed in cylindrical and spherical geometries. However, the principal physical processes are realized in planar (one-dimensional) geometry as well. The mathematical formulation of the positive corona problem can be found in e.g. Naidis (2005), Liu et al (2012), Benilov et al (2021), and the references therein. In this work we report a new approach to finding the inception conditions for a positive corona based on the differential formulation of the photoionization problem (Bourdon et al 2007, Janalizadeh andPasko 2019). Our approach is based on the same principal set of stationary differential equations describing balance of volume ionization and losses in a positive corona that was recently used in planar, cylindrical and wire to plane geometries (Ferreira et al 2019, Almeida et al 2020. We emphasize that the idea of using stationary solvers and finding conditions for corona inception by adjusting model parameters is not new (Ferreira et al 2019, Almeida et al 2020. In (Ferreira et al 2019, Almeida et al 2020 the self-sustained condition of the corona discharge is characterized as a resonance phenomena that occurs in a non-self-sustained discharge in the same electrode configurations, and a system of stationary linear partial differential equations (similar to equations (2) and (3) described in this work below) is solved as a boundaryvalue eigenvalue problem. Specifically, in the (Ferreira et al 2019, Almeida et al 2020 solution, an artificial ionization source is introduced, that can be arbitrary in terms of magnitude and spatial distribution, and a voltage below the corona inception voltage is applied. The stationary problem is solved that gives a discharge current. This current is kept constant, and the artificial source is gradually reduced and the applied voltage is gradually increased until a steady state solution with zero artificial source is obtained that corresponds to the inception corona voltage. In mathematical terms, Ferreira et al (2019) and Almeida et al (2020) consider this problem as an eigenvalue problem and the voltage as the eigenparameter. In this work we solve the same problem in which the system is initiated by one seed electron and the produced electron density and the photoionization field are iterated in response to different applied voltages until a balanced steady state is obtained. From this perspective the iterative solution considers the same inception problem that has been solved in the existing literature by using either an integral approach to photoionization (Naidis 2005, Liu et al 2012, or a differential formulation of photoionization and considering the inception problem as a boundary-value eigenvalue problem (Ferreira et al 2019, Almeida et al 2020. All three approaches (i.e. this work, (Naidis 2005, Liu et al 2012 and (Ferreira et al 2019, Almeida et al 2020) are expected to lead to consistent results. Our proposed solution provides an additional physical insight on the spatial distributions of plasma species and the photoionization field in which the production of electrons by photoionization is balanced by their losses that has not been available in previous literature. We compare our results with previous integral formulations (Naidis 2005, Liu et al 2012 and with the time dynamic plasma fluid solutions obtained in planar and spherical geometries via a model developed for previous hydrometeor studies (Jansky and Pasko 2020). The results are compared to the Peek condition in spherical geometry, where the threshold electric field required for the inception of a positive corona is expressed as a function of the electrode radius (e.g. Lowke and D'Alessandro 2003, and references cited therein). In addition, we illustrate the dependence of solutions on the specifics of the photoionization model. The results also highlight corona regimes when an approximation that most of the photoionizing radiation originates next to the surface of the coronating electrode (Naidis 2005, Liu et al 2012 becomes inaccurate. This effect is especially important for large electrode dimensions ∼1 m and at air pressures <10% of atmospheric pressure. The present model results are used to illustrate ideas advanced in Kaptzov (1950), p 610 for providing a physically transparent connection of an effective secondary electron emission coefficient due to volume photoionization in the positive corona system to the secondary electron emission coefficient from the cathode surface due to the positive ion bombardment in conventional Townsend discharge theory (Townsend 1910, p 34). The results also demonstrate the significance of boundary conditions for accurate corona solutions based on the differential formulation of photoionization, and we provide additional interpretation of the results obtained with different boundary conditions in Ferreira et al (2019).

Model formulation
As part of this work we developed stationary and time dependent positive corona models in planar and spherical geometry and repeated previously published results on the inception of positive corona discharges in air (Naidis 2005). Following the convention in the corona literature we use a scaling factor δ = p/p 0 = n/n 0 , where p is the air pressure, n is the air number density, p 0 = 760 Torr, and n 0 = 2.688×10 25 m −3 are reference values corresponding to the standard (gas temperature 273 • K) atmospheric conditions at sea-level in the Earth's atmosphere. All calculations are performed in spherical coordinates with sole dependence on the spherical radial coordinate r. The one-dimensional planar cases were also considered and these were modeled by choosing a very large radius of inner electrode and considering small electrode gaps as giving a good approximation of planar gaps with a constant field (the specific parameters are listed below for the cases considered). We express our results in terms of the ionization integral K, that is defined as the integral of the net ionization coefficient α eff over the ionization producing region of the corona (where the ionization coefficient α eff >0, as in Naidis (2005)): versus the reduced radius of the spherical electrode r 0 δ. For some cases we also include the reduced applied electric field E 0 /δ required for the inception of a positive corona discharge in air as a function of r 0 δ, where E 0 refers to the electric field at the surface of the coronating electrode (i.e. at r = r 0 ) (referred to as spherical Peek condition). We model a spherical positive corona at three air pressures corresponding to δ = 0.1, 1 and 10, for a range of electrode reduced radii r 0 δ = 10 −2 -10 1 cm. Following Naidis (2005) the corona inception in air is studied without considering the space charge effects on the applied electric field, which is a very good assumption for low particle densities present in air before the discharge is ignited. Analytical expression of the space charge free electric field distribution around the anode is E(r) = E 0 r 2 0 /r 2 (Naidis 2005). Figure 1 illustrates three examples spanning a broad range of parameters, including a very small radius r 0 = 0.01 mm at high pressure (δ = 10), an intermediate radius r 0 = 1 mm at atmospheric pressure (δ = 1), and a very large radius r 0 = 1 m at low pressure (δ = 0.1). The panels also show the corresponding values for critical field E k defined by the equality of ionization and attachment coefficients in air (e.g. Raizer 1991, p 135). The observed deviation from simple E k /δ scaling in figure 1 is due to the three body attachment coefficient in air that, in contrast to ionization and two body attachment coefficients that scale ∝ δ, scales ∝ δ 2 and leads to higher reduced E k values at higher air pressures. In figure 1 the radius of the coronating electrode (solid line) and radius at which the electric field drops to E = E k (dashed line) are drawn, preserving their relative scales, to illustrate that at large electrode radii the corona region converts to a narrow near surface layer with a depth much less than the electrode radius. The left panel in figure 1 schematically shows electrons avalanching toward the positive electrode (anode). These electrons are responsible for the excitation of the UV producing states of N 2 that emit photons that subsequently produce electrons through the photoionization of O 2 . This is the photoionization feedback process, in which photons generated due to previous electron avalanches produce new seed electrons for new avalanches. This electron production happens in both the ionization region where E > E k such that the electrons multiply immediately as they drift toward the anode, and in the attachment region E < E k where electron density can be significantly depleted by the time the electrons enter the active ionization region. We include both groups in our modeling. As already mentioned, the problem has a spherical symmetry and is dependent on r only. The vertical lines in figure 1 illustrate the domain boundary that was used in Naidis (2005) and Liu et al (2012) for their integral solutions. In these solutions only the unshielded part of the spherical space to the right of these planar boundaries is considered in the calculations of the photoionization field produced by a photon source point at the electrode surface (Naidis 2005, Liu et al 2012. We repeated these integral solutions during the validation part of current work. In Naidis (2005) and Liu et al (2012) it is assumed that photons ionizing O 2 originate mostly from the narrow region near the electrode surface where the electric field and electron density are maximized. This is an accurate and justified approximation in most of the cases considered. In our differential solutions, all photons in the discharge volume (i.e. originating from both the attachment and the ionization regions) are included. We comment below on specific regimes when the near surface approximation becomes less accurate.

Differential formulation of photoionization
As explained in Naidis (2005), under inception conditions in a positive corona, each electron arriving at the anode creates on average just enough seed electrons in the discharge volume through photoionization to replicate itself. Under For the cases shown, the radius of the coronating electrode (solid line) and radius at which the electric field drops to E = E k value (dashed line) are drawn, preserving their relative scales, to illustrate that at large electrode radii the corona region converts to a narrow near surface layer with a depth much less than the electrode radius. The vertical lines are explained in the text. these self-sustained steady state conditions, photoionization feedback produces just enough secondary electrons upstream of the avalanche to maintain the system in a steady state. The stationary drift-diffusion-reaction equations for electron density n e [m −3 ] and the rate of production of electrons due to photoionization S ph [m −3 s −1 ] follow from Bourdon et al (2007), Janalizadeh and Pasko (2019) (n e ∝ S ph , all other notations are standard): where S ph = ∑ i S i ph , I = pq p+pq ν * n e , p q = 30 Torr, p O2 = 0.2091p, and n O2 = 0.2091n. Here, ν * is the total excitation frequency of all singlet states of molecular nitrogen responsible for photoionization in air, and l i , C i are fitting parameters provided in table 1 of Janalizadeh and Pasko (2019). The ion- where ⃗ v e is the electron drift velocity and ν i , ν a2 , and ν a3 [s −1 ] are ionization, two and three body attachment frequencies, respectively, is integrated over its positive values in region E > E k to obtain the ionization integral K defined in equation (1). The formulation does not include the effects of electron detachment from negative ions. The detachment effects may be important under conditions of large electrode dimensions r 0 δ when the corona inception field remains close to the critical field E k (Naidis 1987). For consistency, all transport and rate coefficients mentioned above and used in this work are taken from Morrow and Lowke (1997). The divergence term on the left hand side of equation (1) was discretized using a first order upwind numerical scheme. Equation (3) was solved using the Thomas algorithm (e.g. Hockney and Eastwood 1981, p 169) with boundary conditions for S i ph defined at the surface of the coronating electrode and at a remote location far away from the electrode surface. We note that the density of electrons n e and the photon emission rate I are maximized near the electrode and the results appeared not to be sensitive to the location of this remote boundary. In most examples shown below, that boundary was set at r = 5r 0 assuming S i ph = 0 at that location. The integral formulation of photoionization corresponding to equation (3) was used to accurately predict the boundary S i ph values at the coronating electrode surface at r = r 0 . The details of the numerical implementation are provided in the appendix. We emphasize that the integral formulation is used only for the definition of boundary conditions and does not represent a significant computational expense for model execution. In the corona system, the photon source region resides right next to the electrodes. As demonstrated in this work, the integral approach is necessary as the correctness of corona solutions is directly affected by accurate boundary conditions. Conceptually, the approach is similar to the one when integral solutions are used to predict electric potential boundary conditions in significantly more computationally expensive simulations of streamers, where more efficient differential techniques (i.e., mesh relaxation) are used to find solutions inside the simulation domain, e.g., Liu and Pasko (2004). For consistency, the time dynamic plasma fluid solutions based on the model of Jansky and Pasko (2020) employed identical transport and rate coefficients taken from Morrow and Lowke (1997) and the same integral approach for predicting the boundary value for the rate of photoionization S ph at the electrode surface at r = r 0 . The reported modeling assumes that the space charge effects are negligibly small. In practice, the inception of streamers, that is controlled by space charge effects, occurs at relatively high K values exceeding 14-18 (Naidis 2005, and references therein). We also emphasize in the next section 2.2 that due to the linear relationship between n e and S ph , the corona inception conditions can be mathematically obtained for arbitrarily small values of electron density. Physically, the studied inception conditions provide a threshold below which the electron density decays and above which it grows. The growth can be bounded by the outflow of plasma through the chamber walls or by space charge effects modifying the applied electric field.

Iterative solution of the positive corona onset problem
The solution of the photoionization feedback problem requires finding a specific value of the applied electric field E 0 and a related self-consistent distribution of S ph that provides full self-replication of the electrons n e in the positive corona discharge volume. For a given electric field E, differential equations (2) and (3) are self-consistently solved for n e and S ph , respectively. The threshold electric field E 0 required for the inception of positive corona discharges for different electrode radii r 0 is obtained by an iterative process. The following explanation is accompanied by an example of the resultant spatial distributions for the electric field E, the electron density n e and the photoionization rate S ph for the case r 0 = 1 mm and δ = 1 shown in figure 2. The solution is started by setting an arbitrary initial value of E 0 (see figure 2(a)), initiating a primary electron avalanche with an initial density n ep = 1 m −3 at a radial distance where E = E k , and assuming S ph = 0 everywhere in the simulation domain. The resultant primary electron avalanche distribution n ep is then obtained by solving equation (2) and an example of the result is shown in figure 2(b). The electrons n ep in the presence of a field produce source term I on the right hand side in equation (3) and therefore photoionization source S php is obtained as the solution of that equation, with an example shown in figure 2(c). The photoionization source distribution S php is then used to find the next iteration of the electron density n es by solving again equation (2) with initial seed n ep = 0 at E = E k , and only using the volumetric distribution of S php . The density n es produces S phs that gives the next iteration n ess . The final E 0 and related K value are obtained by simple iterations (by increasing or decreasing E 0 ) until the volume integrated n es and n ess agree. Due to the linear dependence of n e on S ph and vice versa, the results do not depend on the specific value n ep = 1 m −3 . The results generally indicate that these iterative solutions are not sensitive to the location of the initial seed n ep = 1 m −3 . As an illustration, figure S1 shows solutions where the seed n ep = 1 m −3 is placed at an arbitrary location r = 4r 0 where E < E k .
However, the results also indicate the importance of the inclusion of photoionization in the attachment region where E < E k for correct solution of the problem.

Results and discussion
Figure 3(a) shows the ionization integral K as a function of reduced electrode radius r 0 δ obtained using the Morrow and Lowke (1997) plasma fluid coefficients and employing the same integral model methodology as described in Naidis (2005) Naidis (2005). Some deviation trends at δ = 0.1 and high r 0 δ values appear to be consistent with those also seen in Liu et al (2012) figure 4, and are considered to be a result of details in the independent numerical implementation. These deviations are not considered to be significant for any conclusions of the present work. We also note that Liu et al (2012) used rate coefficients derived from the electron collision cross sections for N 2 , O 2 and Ar supplied as part of the BOLSIG+ package (Hagelaar and Pitchford 2005) that are different from the coefficients in Morrow and Lowke (1997) and in Naidis (2005). Figure 3(b) shows the results obtained using the differential model developed in the present work based on the Morrow and Lowke (1997) coefficients and with the SP3 photoionization model of Bourdon et al (2007). We note that the SP3 formulation is a differential approximation to the original photoionization model of Zheleznyak et al (1982). This formulation provides a fit over a broad range of reduced dimensions r 0 δ, including the small electrode dimensions that are studied in this work (please see figure 5 in Janalizadeh and Pasko (2019) and related discussion therein). We note that Naidis (2005) and Liu et al (2012) employed the integral formulation of the same original photoionization model of Zheleznyak et al (1982). Figure 3(b) results are consistent with those reported in Naidis (2005) and Liu et al (2012) for most of the δ and r 0 δ values. We note a more pronounced deviation from previous results at δ = 0.1 and high r 0 δ values. The reasons for this deviation are further discussed below. Figure 4(a) shows the ionization integral K as a function of reduced electrode radius r 0 δ obtained using Morrow and  (2005) and Liu et al (2012), based on the photoionization model of Zheleznyak et al (1982), but with a different dependence of the effective ionization coefficient on the reduced field defined by Morrow and Lowke (1997). (b) The differential model developed in the present work is used with Morrow and Lowke (1997) coefficients and with the SP3 photoionization model of Bourdon et al (2007).  Morrow and Lowke (1997) effective ionization coefficient and with the SP3 photoionization model of Bourdon et al (2007); (b) the present model is based on Morrow and Lowke (1997) and Janalizadeh and Pasko (2019) with and without electron three-body attachment effects. The open circles illustrate the results obtained with a time dependent plasma fluid model that was adopted from Jansky and Pasko (2020) and employed the Morrow and Lowke (1997) coefficients and the Janalizadeh and Pasko (2019) photoionization model. Lowke (1997) Figure 4(b) shows the present results based on Morrow and Lowke (1997), Janalizadeh and Pasko (2019) models with and without three-body electron attachment effects. The three-body electron attachment effects appear to be insignificant for the δ cases considered. They need to be accounted for at high air pressures when δ >10. Open circles in figure 4(b) show comparisons with fully timedependent plasma fluid model results (Jansky and Pasko 2020) for selected δ and r 0 δ values that were used for validation of the present results. The distributions of the electric field E, electron density n e , and photoionization rate S ph as a function of coordinate r for the cases shown by open circles in figure 4(b) are included in the Supplementary Information as figures S2 (for δ = 10 and r 0 = 0.01 mm), S3 (for δ = 1 and r 0 = 1 mm), and S4 (for δ = 0.1 and r 0 = 1 m). Specifically, figure S4 indicates that for large electrode radius the electric field that drives the corona remains relatively constant and close to the E k value (i.e., it does not exhibit the sharp enhancement observed for smaller electrode radii in figures S2 and S3). The right hand panel in figure 1 further illustrates this field geometry, which is expected to approach a planar one-dimensional distribution with increase in the electrode radius. This effect enhances the contribution to photon production from regions of the corona not in the immediate vicinity of the electrode surface and explains the differences observed in figure 3(b) at large r 0 δ values. These effects are especially pronounced at low air pressure (i.e. δ = 0.1) because the distance photons propagate before they produce photoionization (i.e., the photoabsorption length) also increases with the reduction of pressure. In summary, the observed differences in figure 3(b) are attributed to approximations in the previous integral formulation (Naidis 2005 and Liu et al 2012) not accounting for the geometry and the whole volume of the discharge in calculations of the photoionization field, in particular, to the assumption that most of the photoionizing radiation originates next to the surface of the coronating electrode.
As indicated in the Model Formulation section, all calculations presented in this work based on the differential formulation of photoionization account for photoionization in the attachment region where E < E k . Our calculations indicate that the contribution of the E < E k region is more pronounced for small electrode dimensions when the photoabsorption length of photons allows them to effectively generate seed electrons in the E < E k region and these seed electrons also have a sufficiently short residence time in that region to avoid significant depletion by attachment. For a large electrode dimension r 0 = 1 m and δ = 0.1 the photons effectively produce photoionization locally at the locations they are emitted from, and while it is important to account for the whole volume of the active corona region in calculations of the photoionization field as emphasized in the previous paragraph, practically, the relative contribution of the E < E k region is small. Specifically, omitting the photoionization in the E < E k region led to an increase in K value to 8.68 with respect to 8.66 (0.23%) for a large electrode r 0 = 1 m and δ = 0.1 shown in figure 4(b). Similar calculations performed for a small electrode r 0 = 1 mm and δ = 0.1 led to an increase in K value to 6.35 with respect to 5.93 (7%). Although these changes are relatively small, they are accounted for in all presented calculations.
The above discussion indicates that the present approach can be easily applied to model one-dimensional planar domains with a constant electric field. An example of this kind of solution is shown in figure 6 that shows the results for a 1 mm gap that is modeled using an inner electrode of radius 1 m. Figure 6 shows the (a) E, (b) n e , and (c) S ph spatial distributions obtained using the formulation described in the present work based on Morrow and Lowke (1997) and Janalizadeh and Pasko (2019) and compared with the corresponding time dynamic plasma fluid solution (Jansky and Pasko 2020). Figure 6 indicates excellent agreement between the two modeling approaches. The distribution of the photoionization field S ph in figure 6 emphasizes that neither S ph or dS ph dr take zero values at the electrode surface. In fact, as can be seen in figure 6(c) the dS ph dr = 0 condition is realized at a significant distance from the electrode surface. This behavior can be understood on physical grounds as follows. The source of the photoionizing radiation, i.e., the photon emission rate, I peaks at the electrode surface as the electron density is maximum near the electrode ( figure 6(b)). If I were symmetric with respect to the electrode surface (i.e., if the source effectively existed inside the electrode), the photoionization term S ph would also have a maximum and would be symmetric with respect to the electrode surface. Readers can find numerous examples illustrating these symmetric solutions in test cases reported in Bourdon et al (2007). Since I does not exist inside the electrode, the S ph peaks at some distance defined by the photoabsorption range of photons (i.e., the photons are not absorbed at the same location as they are produced). We emphasize that, as can be directly seen from other figures reported in this paper, these observations are applicable to all solutions based on the differential formulation of photoionization (i.e., not only to the one-dimensional case shown in figure 6) and indicate the importance of boundary conditions for the correct solution of corona problems.
To further illustrate these points and in agreement with the time dynamic plasma fluid solution we include figure 7, which provides a solution for the δ = 1 and r 0 = 1 mm case in the same format as figure 6 (figure 7 is the same as figure S3 that is included in the supplementary information to provide complete coverage of all considered cases there). As illustrated by a zoomed-in insert in figure 7(c), for a sharp electrode with radius 1 mm, the peak of S ph (where dS ph dr = 0) is realized at a closer distance from the electrode surface when compared to a 1-D case illustrated in figure 6(c). This effect is attributed to a more sharply peaked electron density combined with the absorption of photons in the same range as in  the 1-D case. This shifts the peak closer to the surface. As indicated in figure 5, the employment of S ph = 0 or dS ph dr = 0 boundary conditions at the surface of the coronating electrode leads to results that exhibit substantial quantitative differences with respect to the previous results of Naidis (2005) and Liu et al (2012).
The boundary condition S ph = 0 at r = r 0 was employed in Morrow (1997). Both of the boundary conditions S ph = 0 and dS ph dr = 0 were discussed in Benilov et al (2021) and Benilov (2022), and references therein. Specifically, Ferreira et al (2019) noted that the values of the inception field were sensitive to the choice of the boundary condition for photoionization (i.e., either S ph = 0 or dS ph dr = 0) at the anode for low pressures. Ferreira et al (2019) observed in their results that solutions for the inception field obtained with these two boundary conditions moved closer for higher values of pressure. The present work allows us to provide additional physical insight for these observations. Based on the above discussion of the S ph structure near the anode shown in the insert of figure 7(c), it is expected that setting S ph = 0 would underestimate the photoionization intensity near the coronating electrode and would require higher E 0 and K inception values. In the same vein, setting dS ph dr = 0 would overestimate the photoionization near the coronating electrode and reduce the E 0 and K inception values in comparison to the correct ones. This trend is illustrated by figure 5. The horizontal axis in figure 5 can be interpreted as an increase in pressure if r 0 is kept constant and the differences between the results obtained with S ph = 0 and dS ph dr = 0 boundary conditions are indeed reduced with increasing pressure. As further emphasized in the next paragraph, the results presented in the form of K values are more sensitive to changes in the boundary conditions than E 0 (i.e., graphically small variations in E 0 correspond to significant changes in K values). In figure S5, we included results corresponding to figure 5 showing the percentage change of E 0 /δ obtained with the S ph = 0 boundary condition with respect to E 0 /δ obtained, assuming dS ph dr = 0 at the boundary. This percentage change is reduced with increasing pressure. More specifically, for a fixed r 0 = 1 mm the percentage change is ∼15% at low pressure, corresponding to δ = 0.1 and is ∼4% at high pressure when δ = 10. Since the peak of S ph distribution, where dS ph dr = 0, is controlled by the range of photons producing photoionization, this peak moves closer to the electrode surface with increasing pressure and reduction in this photoionization range. Specifically, for r 0 = 1 mm these distances are ∼0.02 mm, 0.006 mm and 0.0019 mm for δ = 0.1, 1 and 10, respectively (these distributions are not included here for the sake of brevity). This provides a physically intuitive The results reported in figure 4(a) and discussion above indicate a pronounced sensitivity of positive corona onset conditions on the specifics of the photoionization model when expressed in terms of the ionization integral K. In spite of the differences, the resultant K values stay in the 5-11 range and exhibit similar trends as a function of the air pressure scaling factor δ. The results can also be expressed in terms of the threshold reduced electric field required for the inception of a positive corona E 0 /δ as a function of reduced electrode radius r 0 δ. By analogy to cylindrical corona, these are referred to as spherical Peek conditions, and we now discuss our modeling results with respect to the related data and analytical approximations in Peek (1920), p 93, Lowke and D'Alessandro (2003) and Bazelyan et al (2007Bazelyan et al ( , 2008, and references therein. The empirical expression for spherical geometry formulated by Peek (1920), p 93 is where E 0 is in units of kV cm −1 and r 0 is in units of cm. This distribution is illustrated by a dashed line in figure 8(a). We note that although the positive corona has a pronounced dependence on air pressure (i.e., the parameter δ seen in figure 4(a)), due to its non-similarity behavior introduced by the quenching of singlet states of molecular nitrogen responsible for photoionization in air (Naidis 2005), the existing formulations, including equation (4) above, often do not explicitly include this dependence (Lowke and D'Alessandro 2003, Bazelyan et al 2007, 2008. Figure 8 provides a summary of the results from this work when expressed in the above mentioned E 0 /δ versus r 0 δ form. Figure 8(a) results utilized the photoionization model of Janalizadeh and Pasko (2019). A detailed comparison of models based on the photoionization formulations of Janalizadeh and Pasko (2019) and Bourdon et al (2007) indicates that, when plotted in this format, the differences between these two modeling approaches appear to be minor (of the order of 2%-4%) and less than the scatter of related experimental data summarized in Lowke and D'Alessandro (2003). As an illustration, figure 8(b) shows the percentage increase of E 0 /δ obtained with the SP3 photoionization formulation (Bourdon et al 2007) with respect to E 0 /δ obtained with the photoionization formulation from Janalizadeh and Pasko (2019)  Comparisons of the present results with experimental data in spherical geometry summarized by Lowke and D'Alessandro (2003) are included in figure S7. The present modeling results can be used to illustrate the ideas advanced in Kaptzov (1950), p 610 to provide a physically transparent connection between an effective secondary electron emission coefficient due to volume photoionization in the positive corona system and the secondary electron emission from the cathode surface due to the ion bombardment in conventional Townsend discharge theory. In figure 9(a) the flux of electrons arriving at the anode is schematically shown as Γ ea . In conventional Townsend discharge theory, these electrons originate from the cathode with flux Γ es and avalanche in a constant electric field in the discharge volume with effective ionization coefficient α such that Γ ea = Γ es e αd . The electrons with flux Γ es are produced from the cathode as secondary electrons due to the flux Γ ic of positive ion bombardment. The electron flux is expressed in the standard form Γ es = γ se Γ ic , where γ se is the secondary electron emission coefficient from the cathode surface due to positive ion bombardment. Under steady state conditions of electrons just replicating themselves in the discharge volume, the Figure 9. Self-Sustained Townsend discharge driven by (a) secondary electron emission from the cathode (see e.g., Townsend (1910), p 34, Raizer (1991), p 72 andRoth (1995), p 277 and Lieberman and Lichtenberg (2005), p 544) and (b) by photoionization (Kaptzov (1950), p 610). charge conservation condition Γ ea − Γ es = Γ ic gives the selfsustaining condition for DC discharges (primitive reproduction of electrons) (e.g. Roth 1995, p 277): where e αd − 1 is the number of extra electrons-positive ions in the gap due to a single electron traversing the gap, and consequently, γ se (e αd − 1) denotes the number of electrons created due to these positive ions bombarding the cathode. In the positive corona system depicted schematically in figure 9(b), the remote cathode does not contribute as a source of secondary electrons and each electron arriving at the anode as part of flux Γ ea creates on average just enough seed electrons in the discharge volume through photoionization to replicate itself (Naidis 2005). Under these steady state conditions one can introduce an effective surface anywhere in the discharge volume (shown by the vertical dashed line in figure 9(b)) and express the relationship between ion flux at that location Γ ic and Γ es as Γ es = γ se Γ ic , where γ se is the effective secondary electron emission coefficient due to volume photoionization in the positive corona system (Kaptzov 1950, p 610). The relation Γ es = γ se Γ ic provides a physically transparent connection to secondary electron emission from the cathode surface due to the ion bombardment in conventional Townsend discharge theory, and after application of the charge conservation condition Γ ea − Γ es = Γ ic gives exactly the same condition of primitive reproduction of electrons seen in equation (5). The ion flux Γ ic and the effective coefficient γ se are generally not needed for a solution of the positive corona feedback problem as illustrated in Naidis (2005) and by the results of this work. Nevertheless, as described below, in figure 10 we use onedimensional planar geometry to provide a quantitative illustration of the physical quantities discussed above. Figure 10 illustrates the photoionization driven selfsustained Townsend discharge using a one-dimensional planar gap with d = 0.6 mm and δ = 1. The format and data shown in this figure are identical to figure 6, except the electron density n e and the photoionization rate S ph are not normalized and are shown in their original dimensional form. The location of the surface illustrating the effective secondary emission of electrons is arbitrarily set to create a d = 0.6 mm gap. Similar calculations can be performed anywhere in the discharge volume. The values of n ess electron density marked at two locations in figure 10(b) can be used to deduce e αd = 325.9 and α = 9.64 × 10 3 m −1 , and therefore γ se = 1/(e αd − 1) = 3.1 × 10 −3 . The electron flux Γ es = n e v e and ion flux Γ ic = n i v i can be readily evaluated using the electric field E = 4.98 × 10 6 V m −1 and electron and ion mobilities µ e = 0.041 m 2 V −1 s −1 and µ i = 2.3 × 10 −4 m 2 /V/s, respectively. At the specified location, the velocities of electrons and ions are v e = µ e E = 2.04 × 10 5 m s −1 and v i = µ i E = 1.14 × 10 3 m s −1 , respectively. The related densities are n e = 46.6 m −3 and n i = n e v e /γ se /v i = 2.67 × 10 6 m −3 . These numerical values indicate that a significant number of ions should be accumulated in the discharge volume to maintain the system in a steady state since n i ≫ n e .

Conclusions
The principal results and contributions that follow from the studies presented in this paper can be summarized as follows: 1. We repeated the previously published modeling results of Naidis (2005) and Liu et al (2012) on the inception conditions for a positive corona in spherical geometry based on an integral approach to photoionization. 2. We developed a new approach to the corona inception problem based on a differential formulation of photoionization due to gas discharges in air (Bourdon et al 2007, Janalizadeh andPasko 2019). The results have been validated using a time dependent plasma fluid model (Jansky and Pasko 2020). 3. The differential approach gives the ionization integral threshold results K that are broadly consistent with previous integral formulations (Naidis 2005, Liu et al 2012 when an identical photoionization model is used. 4. The differences between the present differential and the previous integral modeling are especially pronounced at large electrode dimensions when the extent of the corona region becomes much smaller than the electrode radius. These differences are observed when both classic (Zheleznyak et al 1982, Bourdon et al 2007, or more recent (Janalizadeh and Pasko 2019) photoionization models are used. 5. The observed differences between Naidis (2005), Liu et al (2012) are attributed to approximations not accounting for the geometry and the whole volume of the discharge in calculations of the photoionization field; in particular, to the assumption that most of the photoionizing radiation originates next to the surface of the coronating electrode. 6. The ideas advanced by Kaptzov (1950), p 610, providing a physically transparent connection of the effective secondary electron emission coefficient due to volume photoionization in the positive corona system to the secondary electron emission from the cathode surface due to the ion bombardment in conventional Townsend discharge theory, are quantitatively illustrated using the present model results. 7. The results demonstrate the significance of boundary conditions for accurate corona solutions based on a differential formulation of photoionization. In particular, the employment of S ph = 0 or dS ph dr = 0 boundary conditions at the surface of the coronating electrode leads to results that exhibit substantial quantitative differences with respect to the previous results of Naidis (2005) and Liu et al (2012). The boundary conditions for the photoionization source based on the integral formulation of photoionization provide accurate solutions. 8. For the air pressures considered (i.e. δ = 0.1, 1, 10) the three body electron attachment effects appear to be weak. Based on the reported results, these effects need to be accounted for only at higher pressures (i.e. δ >10).

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).

Acknowledgment
This research has been supported by the NSF grants AGS-1744099 and AGS-2010088 to Penn State University. J Jansky gratefully acknowledges financial support from the grant VAROPS (DZRO FVT 3) granted by the Ministry of Defense of the Czech Republic.

Appendix. Implementation of boundary conditions
The integral representation of the photoionization rates S i ph given by equation (3) is: (Bourdon et al 2007, Janalizadeh andPasko 2019), where R = |⃗ r −⃗ r ′ | is the distance between the point of source ⃗ r ′ and the point of observation ⃗ r. We use this integral representation to accurately calculate the boundary condition for S i ph (⃗ r) by choosing the point of observation to be on the surface of the coronating electrode at ⃗ r = r 0 ⃗ i r , where ⃗ i r is a unit vector corresponding to the radial coordinate in the spherical coordinate system, centered at the center of the coronating electrode. The modeling consistency was also tested by using the equation (A.1) for other points inside the simulation domain. For the sake of brevity we show details of the numerical implementation only for the boundary value S i ph (r 0 ) that was used in the practical implementation of the boundary conditions for all modeling results reported in this work. The integration over all points of source ⃗ r ′ = r ⃗ i r ′ yields: S i ph (r 0 ) =ˆr max r0 drˆθ max(r) 0 dθ 2π r 2 sin θ I(r) where θ is the angle coordinate in the spherical coordinate system (cos θ = ⃗ i r · ⃗ i r ′ ), R = √ r 2 0 + r 2 − 2r 0 r cos θ, and θ max (r) = arccos(r 0 /r). As most of the photons originate near the coronating electrode, where the electron density and electric field are maximized, r max was set to the location where E = E k (or at r = r 0 +d for planar electrode configurations discussed in the main paper text). With an increase in the electrode radius, in particular for cases used for the modeling of 1-D planar gaps, the maximum values of θ max (r) angles are reduced. A sufficiently fine grid resolution in θ angle was chosen and tested to ensure convergent and accurate results in both the iterative and fully time dependent solutions.