arXiv subjects

P. Janardhan

Publications and source records attributed to P. Janardhan.

At least 19 recordsLinked to original sources

arXiv2025

Evidence for in situ particle energization during the May-2024 event based on ASPEX instrument on board Aditya-L1

The interaction between interplanetary Coronal Mass Ejection (ICME) structures can alter the geo-effectiveness of the ICME events in myriad ways. Many aspects of these interaction processes are not well-understood till date. Using the energy spectra measured in two mutually orthogonal top hat analyzers (THA 1 and 2), which are part of the Solar Wind Ion Spectrometer (SWIS) subsystem of the Aditya Solar Wind Particle EXperiment (ASPEX) on board India's Aditya L1 mission, we gain insights into intricate features of ICME ICME interactions during May 2024 solar event. We report here an unprecedented two-orthogonal-plane perspective of ICME ICME interactions for the first time from the L1 point. The investigation reveals a special interaction region formed by the propagation of the forward shock driven by complex ejecta in the preceding ICME. The interaction causes the formation of a downstream region spanning over 13 hours, which propagates in the interplanetary medium. The observations reveal that this region serves as a site for proton and alpha particle energization, and the particles within this region get distributed from one plane to the other. The presence of forward shock and particle energization is confirmed by the energetic particle flux measurements by the SupraThermal and Energetic Particle Spectrometer (STEPS) of ASPEX. These observations provide an unprecedented perspective on how solar wind ions become energized and distributed in an ICME-ICME interaction region.

astro-ph.SR↗

arXiv2025

One Year of ASPEX-STEPS Operation: Characteristic Features, Observations and Science Potential

The SupraThermal and Energetic Particle Spectrometer (STEPS), a subsystem of the Aditya Solar wind Particle EXperiment (ASPEX) onboard India's Aditya-L1 satellite, is designed to study different aspects of energetic particles in the interplanetary medium from the Sun-Earth L1 point using six detector units oriented in different directions. This article presents details of the one-year operation (08 January 2024 - 28 February 2025) of the AL1-ASPEX-STEPS after the insertion of the satellite into the final halo orbit around the L1 point with emphasis on performance, science observations, and scientific potentials. Four out of six AL1-ASPEX-STEPS units exhibit a stable detector response throughout the observation period, confirming operational robustness. This work also includes the temporal variation of particle fluxes, spectra of ions during selected quiet times and transient events, and cross-comparisons with existing instruments at the L1 point. A strong correlation (with coefficient of determination, R2 ~ 0.9) is observed in the cross-comparison study, establishing the reliability of the AL1- ASPEX-STEPS observations. AL1-ASPEX-STEPS also captures different forms of energetic ion spectra similar to those observed by previous missions. These results underscore the instrument's potential to contribute significantly to the study of energetic particle acceleration, transport, and long-term space weather monitoring from the Sun-Earth L1 vantage point.

physics.space-ph↗

arXiv2025

One year of ASPEX-SWIS operation -- Characteristic features, observations and science potential

The Aditya-L1 mission, India's first dedicated solar observatory positioned at the first Lagrange point (L1) of the Sun-Earth system, carries the Solar Wind Ion Spectrometer (SWIS) as part of the ASPEX payload suite. Even before settling into its Halo orbit, SWIS has been providing nearly continuous in-situ measurements of solar wind ion spectra. Moments of the velocity distribution functions (VDFs) have been calculated to derive key solar wind parameters such as density, bulk speed, and temperature. In this study, we assess the performance of SWIS (hereafter referred to as AL1-ASPEX-SWIS) by comparing its measurements with contemporaneous data from the Wind and DSCOVR missions. In this study, we assess the performance of SWIS (hereafter referred to as AL1-ASPEX-SWIS) by comparing its measurements with contemporaneous data from the Wind and DSCOVR missions. A detailed case study of the interplanetary coronal mass ejection (ICME) event on August 7, 2024, is presented, where sharp changes in bulk speed, thermal speed, and number density were found to be well-aligned with independent observations-confirming the instrument's ability to capture dynamic solar wind features. Spectral analysis of kinetic fluctuations revealed a well-defined inertial range with a spectral slope consistent with magnetohydrodynamic (MHD) turbulence. Furthermore, a 17-month statistical comparison (from January 2024 to May 2025) shows a strong correlation in bulk velocity (R2 = 0.94 with Wind), with expected variations in thermal speed and density arising from differences between instruments. These findings demonstrate the scientific value of AL1-ASPEX-SWIS for monitoring both transient solar events and long-term solar wind conditions.

astro-ph.SR↗

arXiv2025

Multi-directional investigations on quiet time suprathermal ions measured by ASPEX-STEPS on-board Aditya L1

The origin, acceleration and anisotropy of suprathermal ions in the interplanetary medium during quiet periods have remained poorly understood issues in solar wind physics. To address these aspects, we derive the spectral indices for the quiet time suprathermal ions based on the measurements by the four directionally separated sensors that are part of the Supra-Thermal and Energetic Particle Spectrometer (STEPS) of Aditya Solar Wind Particle EXperiment (ASPEX) on-board Aditya L1 spacecraft. Three out of four STEPS sensors Parker Spiral (PS), Inter-Mediate (IM), Earth Pointing (EP) are in one plane (nearly aligned with the ecliptic plane) while the fourth sensor North Pointing (NP) is in a mutually orthogonal plane. The energy ranges covered by the PS, IM, EP and NP sensors are 0.36-1.32 MeV, 0.14-1.22 MeV, 0.39-1.33 MeV and 0.12-1.23 MeV respectively. The quiet intervals are identified during January November, 2024 and the derived spectral indices (differential directional flux versus energy) are found to be in the range of 2.0 for all directions in the time scale of a few days revealing isotropic nature of their distribution. Further analysis of elemental abundance ratios (3He/4He, Fe/O, and C/O) during the same quiet intervals obtained from the Ultra-Low Energy Isotope Spectrometer (ULEIS) on board the Advanced Composition Explorer (ACE) spacecraft suggests possible contributions from the leftover ions from the previous impulsive (Solar flares) and gradual events (CMEs) in the quiet time suprathermal ion pool.

physics.space-ph↗

arXiv2025

Energetic ($<$ 2 MeV) ion fluxes measured by ASPEX-STEPS on board Aditya-L1 during its earth-bound phase

During its earth-bound phase of the Aditya-L1 spacecraft of India, the Supra-Thermal and Energetic Particle Spectrometer (STEPS) of the Aditya Solar wind Particle EXperiment (ASPEX) was operated whenever the orbit was above 52000 km during 11 - 19 September 2023. This phase of operation provided measurements of energetic ions (with energies 0.1--2 MeV) in the magnetosphere, magnetosheath, and interplanetary medium. Three interplanetary coronal mass ejections (ICME) hit the magnetosphere during this period. This provided opportunity to examine the relative roles of ICME-generated solar energetic particles (SEPs) and substorm generated energetic ions on the magnetosphere. We approach this objective by detailed spectral analyses of energetic ion fluxes measured by two units of ASPEX-STEPS. We identify three distinctly different conditions of the north-south component of the interplanetary magnetic field (IMF $B_z$ = 0, $>$ 0, and $<$ 0) and use the derived spectral indices to understand this relative role. By combining these with the simultaneous energetic ion flux variations from the Advanced Composition Explorer (ACE) around the Sun-Earth first Lagrangian (L1) point and the Geostationary Operational Environmental Satellite (GOES) in the Earth's magnetosphere, we show that the polarity of IMF $B_z$ influences the energetic ion spectra in the magnetosphere by modulating the interplay of the ICME-generated SEP with the energetic particles generated inside the magnetosphere by substorms. Interestingly, ASPEX-STEPS observations also indicate towards directional anisotropy based on spectral indices. This suggests spatially inhomogeneous mixing of energetic ions coming from different source processes.

physics.space-ph↗

arXiv2024

Prolonged and Extremely Non-radial Solar Wind Flows

We present a study of three highly non-radial solar wind events when the azimuthal solar wind flow angle exceeds > 6 degrees for one day or more. None of the events are associated with coronal mass ejections and co-rotating interaction regions observed at 1 AU. For all events, the solar wind outflows at 1 AU have low solar wind velocity and solar wind density. Based on the significant increase in the Oxygen charge state ratio of O7+/O6+ at 1 AU for all of the events, we have traced them back to the Sun and found that their source regions originated in an active region and coronal hole (AR-CH) pairs mainly located at the central meridian. Further, examining the dynamical evolutions in their source regions using both the Extreme ultra-violet Imaging Telescope and Michelson Doppler Imager, it is found that the changes taking place in AR-CH boundaries eventually disturbed the stable CH configurations, resulting in a reduction of the CH area and finally its disappearance, leaving only with the AR. Our study provides a possible explanation to discuss the origin of the prolonged and highly non-radial solar wind flows.

astro-ph.SR↗

arXiv2023

Evolution of elemental abundances in hot active region cores from Chandrayaan-2 XSM observations

The First Ionization Potential (FIP) bias, whereby elemental abundances for low FIP elements in different coronal structures vary from their photospheric values and may also vary with time, has been widely studied. In order to study the temporal variation, and to understand the physical mechanisms giving rise to the FIP bias, we have investigated the hot cores of three ARs using disk-integrated soft X-ray spectroscopic observation with the Solar X-ray Monitor (XSM) onboard Chandrayaan-2. Observations for periods when only one AR was present on the solar disk were used so as to ensure that the AR was the principal contributor to the total X-ray intensity. The average values of temperature and EM were ~3 MK and 3.0E46/cm3 respectively. Regardless of the age and activity of the AR, the elemental abundances of the low FIP elements, Al, Mg, and Si were consistently higher than their photospheric values. The average FIP bias for Mg and Si was ~3, whereas the FIP bias for the mid-FIP element, S, was ~1.5. However, the FIP bias for the lowest FIP element, Al, was observed to be higher than 3, which, if real, suggests a dependence of the FIP bias of low FIP elements on their FIP value. Another major result from our analysis is that the FIP bias of these elements is established in within ~10 hours of emergence of the AR and then remains almost constant throughout its lifetime.

astro-ph.SR↗

arXiv2022

Soft X-ray Spectral Diagnostics of Multi-thermal Plasma in Solar Flares with Chandrayaan-2 XSM

Spectroscopic observations in X-ray wavelengths provide excellent diagnostics of the temperature distribution in solar flare plasma. The Solar X-ray Monitor (XSM) onboard the Chandrayaan-2 mission provides broad-band disk integrated soft X-ray solar spectral measurements in the energy range of 1-15 keV with high spectral resolution and time cadence. In this study, we analyse X-ray spectra of three representative GOES C-class flares obtained with the XSM to investigate the evolution of various plasma parameters during the course of the flares. Using the soft X-ray spectra consisting of the continuum and well-resolved line complexes of major elements like Mg, Si, and Fe, we investigate the validity of the isothermal and multi-thermal assumptions on the high temperature components of the flaring plasma. We show that the soft X-ray spectra during the impulsive phase of the high intensity flares are inconsistent with isothermal models and are best fitted with double peaked differential emission measure distributions where the temperature of the hotter component rises faster than that of the cooler component. The two distinct temperature components observed in DEM models during the impulsive phase of the flares suggest the presence of the directly heated plasma in the corona and evaporated plasma from the chromospheric footpoints. We also find that the abundances of low FIP elements Mg, Si, and Fe reduces from near coronal to near photospheric values during the rising phase of the flare and recovers back to coronal values during decay phase, which is also consistent with the chromospheric evaporation scenario.

astro-ph.SR↗

arXiv2022

Multi-wavelength observations by XSM, Hinode and SDO of an active region. Chemical abundances and temperatures

We have reviewed the first year of observations of the Solar X-ray Monitor (XSM) onboard Chandrayaan-2, and the available multi-wavelength observations to complement the XSM data, focusing on Solar Dynamics Observatory AIA and Hinode XRT, EIS observations. XSM has provided disk-integrated solar spectra in the 1--15 keV energy range, observing a large number of microflares. We present an analysis of multi-wavelength observations of AR 12759 during its disk crossing. We use a new radiometric calibration of EIS to find that the quiescent AR core emission during its disk crossing has a distribution of temperatures and chemical abundances that does not change significantly over time. An analysis of the XSM spectra confirms the EIS results, and shows that the low First Ionization Potential (FIP) elements are enhanced, compared to their photospheric values. The frequent microflares produced by the AR did not affect the abundances of the quiescent AR core. We also present an analysis of one of the flares it produced, SOL2020-04-09T09:32. The XSM analysis indicates isothermal temperatures reaching 6 MK. The lack of very high-T emission is confirmed by AIA. We find excellent agreement between the observed XSM spectrum and the one predicted using an AIA DEM analysis. In contrast, the XRT Al-Poly / Be-thin filter ratio gives lower temperatures for the quiescent and flaring phases. We show that this is due to the sensitivity of this ratio to low temperatures, as the XRT filter ratios predicted with a DEM analysis based on EIS and AIA gives values in good agreement with the observed ones.

astro-ph.SR↗

arXiv2021

Evolution of Elemental Abundances During B-Class Solar Flares: Soft X-ray Spectral Measurements with Chandrayaan-2 XSM

The Solar X-ray Spectrometer (XSM) payload onboard Chandrayaan-2 provides disk-integrated solar spectra in the 1-15 keV energy range with an energy resolution of 180 eV (at 5.9 keV) and a cadence of 1~second. During the period from September 2019 to May 2020, covering the minimum of Solar Cycle 24, it observed nine B-class flares ranging from B1.3 to B4.5. Using time-resolved spectroscopic analysis during these flares, we examined the evolution of temperature, emission measure, and absolute elemental abundances of four elements -- Mg, Al, Si, and S. These are the first measurements of absolute abundances during such small flares and this study offers a unique insight into the evolution of absolute abundances as the flares evolve. Our results demonstrate that the abundances of these four elements decrease towards their photospheric values during the peak phase of the flares. During the decay phase, the abundances are observed to quickly return to their pre-flare coronal values. The depletion of elemental abundances during the flares is consistent with the standard flare model, suggesting the injection of fresh material into coronal loops as a result of chromospheric evaporation. To explain the quick recovery of the so-called coronal "First Ionization Potential (FIP) bias" we propose two scenarios based on the Ponderomotive force model.

astro-ph.SR↗

arXiv2021

Observations of the Quiet Sun During the Deepest Solar Minimum of the Past Century with Chandrayaan-2 XSM -- Sub-A Class Microflares Outside Active Regions

Solar flares, with energies ranging over several orders of magnitude, result from impulsive release of energy due to magnetic reconnection in the corona. Barring a handful, almost all microflares observed in X-rays are associated with the solar active regions. Here we present, for the first time, a comprehensive analysis of a large sample of quiet Sun microflares observed in soft X-rays by the Solar X-ray Monitor (XSM) on board the Chandrayaan-2 mission during the 2019-20 solar minimum. A total of 98 microflares having peak flux below GOES A-level were observed by the XSM during observations spanning 76 days. By using the derived plasma temperature and emission measure of these events obtained by fitting the XSM spectra along with volume estimates from concurrent imaging observations in EUV with the Solar Dynamics Observatory/Atmospheric Imaging Assembly (SDO/AIA), we estimated their thermal energies to be ranging from 3e26 to 6e27 erg. We present the frequency distribution of the quiet Sun microflares with energy and discuss the implications of these observations of small-scale magnetic reconnection events outside active regions on coronal heating.

astro-ph.SR↗

arXiv2021

Observations of the Quiet Sun During the Deepest Solar Minimum of the Past Century with Chandrayaan-2 XSM -- Elemental Abundances in the Quiescent Corona

Elements with low First Ionization Potential (FIP) are known to be three to four times more abundant in active region loops of the solar corona than in the photosphere. There have been observations suggesting that this observed "FIP bias" may be different in other parts of the solar corona and such observations are thus important in understanding the underlying mechanism. The Solar X-ray Monitor (XSM) on board the Chandrayaan-2 mission carried out spectroscopic observations of the Sun in soft X-rays during the 2019-20 solar minimum, considered to be the quietest solar minimum of the past century. These observations provided a unique opportunity to study soft X-ray spectra of the quiescent solar corona in the absence of any active regions. By modelling high resolution broadband X-ray spectra from XSM, we estimate the temperature and emission measure during periods of possibly the lowest solar X-ray intensity. We find that the derived parameters remain nearly constant over time with a temperature around 2 MK, suggesting the emission is dominated by X-ray Bright Points (XBPs). We also obtain the abundances of Mg, Al, and Si relative to H, and find that the FIP bias is ~2, lower than the values observed in active regions.

astro-ph.SR↗

arXiv2020

Global Solar Magnetic-field and Interplanetary Scintillations During the Past Four Solar Cycles

The extended minimum of Solar Cycle 23, the extremely quiet solar-wind conditions prevailing, and the mini-maximum of Solar Cycle 24 drew global attention and many authors have since attempted to predict the amplitude of the upcoming Solar Cycle 25, which is predicted to be the third successive weak cycle; it is a unique opportunity to probe the Sun during such quiet periods. Earlier work has established a steady decline, over two decades, in solar photospheric fields at latitudes above $45^{\circ}$ and a similar decline in solar-wind micro-turbulence levels as measured by interplanetary scintillation (IPS) observations. However, the relation between the photospheric magnetic fields and those in the low corona/solar-wind are not straightforward. Therefore, in the present article, we have used potential-field source-surface (PFSS) extrapolations to deduce global magnetic-fields using synoptic magnetograms observed with National Solar Observatory (NSO), Kitt Peak, USA (NSO/KP) and Solar Optical Long-term Investigation of the Sun (NSO/SOLIS) instruments during 1975-2018. Furthermore, we have measured the normalized scintillation index [m] using the IPS observations carried out at the Institute of Space Earth Environment Research (ISEE), Japan during 1983-2017. From these observations, we have found that, since the mid-1990s, the magnetic-field over different latitudes at 2.5 $\rm R_{\odot}$ and 10 $\rm R_{\odot}$(extrapolated using PFSS method) has decreased by $\approx 11.3-22.2 \%$. In phase with the declining magnetic-fields, the quantity m also declined by $\approx 23.6 \%$. These observations emphasize the inter-relationship between the global magnetic-field and various turbulence parameters in the solar corona and solar wind.

astro-ph.SR↗

arXiv2020

Ground Calibration of Solar X-ray Monitor On-board Chandrayaan-2 Orbiter

Chandrayaan-2, the second Indian mission to the Moon, carries a spectrometer called the Solar X-ray Monitor (XSM) to perform soft X-ray spectral measurements of the Sun while a companion payload measures the fluorescence emission from the Moon. Together these two payloads will provide quantitative estimates of elemental abundances on the lunar surface. XSM is also expected to provide significant contributions to the solar X-ray studies with its highest time cadence and energy resolution spectral measurements. For this purpose, the XSM employs a Silicon Drift Detector and carries out energy measurements of incident photons in the 1 -- 15 keV range with a resolution of less than 180 eV at 5.9 keV, over a wide range of solar X-ray intensities. Extensive ground calibration experiments have been carried out with the XSM using laboratory X-ray sources as well as X-ray beam-line facilities to determine the instrument response matrix parameters required for quantitative spectral analysis. This includes measurements of gain, spectral redistribution function, and effective area, under various observing conditions. The capability of the XSM to maintain its spectral performance at high incident flux as well as the dead-time and pile-up characteristics have also been investigated. The results of these ground calibration experiments of the XSM payload are presented in this article.

astro-ph.IM↗

arXiv2019

A new tool for predicting the solar cycle: Correlation between flux transport at the equator and the poles

The magnetic flux cancellation on the Sun plays a crucial role in determining the manner in which the net magnetic flux changes in every solar cycle, affecting the large scale evolution of the coronal magnetic fields and heliospheric environment. We investigate, in this paper, the correlation between the solar magnetic flux cancelled at the equator and the solar magnetic flux transported to the poles by comparing the net amount of magnetic flux in the latitude belt 0${^{\circ}}$-5${^{\circ}}$ and 45${^{\circ}}$-60${^{\circ}}$, estimated using synoptic magnetograms from National Solar Observatory at Kitt Peak, during Solar Cycles 21-24. We find a good correlation between the net flux in the latitude band 0${^{\circ}}$-5${^{\circ}}$ and 55${^{\circ}}$-60${^{\circ}}$ for the Northern as well as for the Southern hemispheres. However, we find a poor correlation if the net flux for the Northern and Southern hemispheres are considered together. In addition, we investigate the correlation between the net flux cancelled at the equator and the strength of solar polar field at cycle minimum, and find a good correlation between the two. We discuss the implication of the correlation of flux transported across the equator and to the poles that has an important bearing in the estimation of the residual polar cap field strength at the cycle minimum. This can be used a predictive tool for estimating the amplitude of subsequent cycles and we use this to estimate maximum smoothed sunspot numbers of 77$\pm$5 and 85$\pm$5 for the Northern and Southern hemispheres, respectively, for the upcoming Solar Cycle 25.

astro-ph.SR↗

arXiv2019

Another Mini Solar Maximum in the Offing: A Prediction for the Amplitude of Solar Cycle 25

We examine the temporal changes in both solar polar magnetic field (PMF) at latitudes $\ge$ $45^{\circ}$ and heliospheric magnetic field (HMF) at 1 AU during solar cycles 21--24 with emphasis on the recent activity changes after July 2015, the so called "mini solar maximum" of cycle 24. While unsigned PMF shows a solar-cycle-like variation in cycles 21 and 22, it shows an anti-solar-cycle-like variation in cycle 24. In addition, the floor level of the HMF (of 4.6 nT), i.e. the value that the HMF returns to at each solar minimum, is breached about two years prior to cycle 24 minimum, indicating a reduced HMF floor level in the upcoming cycle 24 minimum. In light of the change of unsigned PMF and the availability of a revised smoothed sunspot number (SSN) after July 2015, we have revisited the correlation of unsigned PMF and HMF at solar minimum. The correlation is used to estimate a new value of the HMF of 4.16$\pm$0.6 nT at the cycle 24 minimum and the amplitude of the upcoming cycle 25. The updated prediction is 82$\pm$8 and 133$\pm$11, on the original (V1.0) and revised (V2.0) SSN scales, respectively. These better and more reliable SSN values (due to the larger data set) imply that we will witness another mini solar maximum in the upcoming cycle 25 which will be relatively stronger than cycle 24 and a little weaker than cycle 23, even if the current solar cycle minimum occurs in 2021 instead of 2020.

astro-ph.SR↗

arXiv2019

Beyond the mini-solar maximum of solar cycle 24: Declining solar magnetic fields and the response of the terrestrial magnetosphere

The present study examines the response of the terrestrial magnetosphere to the long-term steady declining trends observed in solar magnetic fields and solar wind micro-turbulence levels since mid-1990's that has been continuing beyond the mini-solar maximum of cycle 24. A detailed analysis of the response of the terrestrial magnetosphere has been carried out by studying the extent and shape of the Earth's magnetopause and bow shock over the past four solar cycles. We estimate sub-solar stand-off distance of the magnetopause and bow shock, and the shape of the magnetopause using numerical as well as empirical models. The computed magnetopause and bow shock stand-off distances have been found to be increasing steadily since around mid-1990's, consistent with the steady declining trend seen in solar magnetic fields and solar wind micro-turbulence levels. Similarly, we find an expansion in the shape of the magnetopause since 1996. The implications of the increasing trend seen in the magnetopause and bow shock stand-off distances are discussed and a forecast of the shape of the magnetopause in 2020, the minimum of cycle 24, has been made. Importantly, we also find two instances between 1968 and 1991 when the magnetopause stand-off distance dropped to values close to 6.6 earth radii, the geostationary orbit, for duration ranging from 9$-$11 hours and one event in 2005, post 1995 when the decline in photospheric fields began. Though there have been no such events since 2005, it represents a clear and present danger to our satellite systems.

astro-ph.SR↗

arXiv2018

Clearing certain misconception in the common explanations of the aerodynamic lift

Air travel has become one of the most common means of transportation. The most common question which is generally asked is: How does an airplane gain lift? And the most common answer is via the Bernoulli principle. It turns out that it is wrongly applied in common explanations, and there are certain misconceptions. In an alternative explanation the push of air from below the wing is argued to be the lift generating force via Newton's law. There are problems with this explanation too. In this paper we try to clear these misconceptions, and the correct explanation, using the Lancaster-Prandtl circulation theory, is discussed. We argue that even the Lancaster-Prandtl theory at the zero angle of attack needs further insights. To this end, we put forward a theory which is applicable at zero angle of attack. A new length scale perpendicular to the lower surface of the wing is introduced and it turns out that the ratio of this length scale to the cord length of a wing is roughly $0.4930\pm0.09498$ for typical NACA airfoils that we analyzed. This invariance points to something fundamental. The idea of our theory is simple. The "squeezing" effect of the flow above the wing due to camber leads to an effective Venturi tube formation and leads to higher velocity over the upper surface of the wing and thereby reducing pressure according the Bernoulli theorem and generating lift. Thus at zero angle of attack there is no need to invoke vortex and anti-vortex pair generation. In fact vortex and anti-vortex pair generation cannot be justified. We come up with the equation for the lift coefficient at zero angle of attack: \[C_l = \frac{1}{c}\int_0^c dx \left(\frac{h_c^2}{(h_c - f(x))^2}-1\right).\] Here $C_l$ is the lift coefficient, $h_c$ is our new length scale perpendicular to the lower surface of the wing and $f(x)$ is the functional profile of the upper surface of the wing.

physics.pop-ph↗