it has been known for several decades that energy and particle confinement in tokamak plasmas are mainly degraded by turbulence driven by steep temperature and density gradients . for this reason ,the characterization and understanding of these turbulent processes is a very important task in order to improve the performance of present experiments as well as future fusion reactors .  due to the strong background magnetic field and the low collisionality in tokamak plasmas ,gyrokinetic theory has been established as the most appropriate theoretical framework for the study of turbulent transport in the plasma core . in order to improve confidence in the numerical results obtained with gyrokinetics and to establish a solid understanding of turbulent transport across the whole range of plasma parameters , it is very important to perform direct comparisons between simulations and experimental measurements . in this respect , with the recent development and improvements in fluctuations diagnostic, it is now possible to measure turbulence features with high precision , allowing for quantitative comparisons between experimental data and results of nonlinear gyrokinetic simulations .these validation studies are crucial in developing confidence in the models and improving the predictive capabilities of the numerical simulations .  in a recent paper  ,we have compared density fluctuation levels measured with a new doppler reflectometer installed in asdex upgrade and simulation results obtained with the gyrokinetic gene code .we extend the previous work by additionally presenting simulation results of density wavenumber spectra , electron temperature fluctuation levels as well as cross - phases between different quantities .one of the reasons for analyzing electron temperature fluctuation levels and cross - phases is that a new correlation electron cyclotron emission ( cece ) system is expected to be installed and to be in operation in 2015 in asdex upgrade .therefore , the gyrokinetic results presented in this paper can provide guidance for the on - going development of the diagnostic . from a more fundamental point of view, we will also investigate the variation of these quantities with respect to various physical input parameters .this information can be used to characterize core turbulence features in asdex upgrade plasmas .the paper is organized as follows . in section [ discharge ] , an overview of the chosen plasma discharge analyzed is given .a description of the gyrokinetic simulation method used is described in section [ overview ] .micro - instability studies from linear gyrokinetic simulations are outlined in section [ linear ] .the main results of the paper are shown in section [ nonlinear ] .core turbulence features such as heat fluxes , density fluctuation amplitudes and spectra , temperature fluctuation amplitudes as well as cross - phases between these quantities will be presented in detail , followed by a discussion in section [ discussion ] . finally , conclusions and future work will be discussed in section [ conclusions ] .the asdex upgrade discharge analyzed in this paper was operated in the high - confinement regime ( h - mode ) .it was planned to study the turbulence characteristics in both the ion temperature gradient ( itg ) and trapped electron mode ( tem ) regimes , through a transition from one regime to another .this transition can be achieved by modifying locally the electron temperature gradient , which affects the tem instability .this is obtained by changing the electron cyclotron resonance heating power ( ecrh ) by steps during the discharge .an overview of several relevant time traces is given in fig .[ fig_exp ] .the ecrh power is deposited at from seconds . here, is the normalized poloidal flux radius . at this heating location , is varied subsequently between and ( a ) .the influence of the stepped heating power can be clearly observed in the electron temperature ( b ) . finally , the strongest increase of the logarithmic temperature gradient is observed at when an ecrh power of mw is applied ( c ) . for this reason, this case and the one without ecrh will be analyzed in detail , as examples of the two extreme cases .they correspond to the time windows shaded in grey in fig .[ fig_exp ] . within these time windows, we have simulated three different radial locations : , making a total of six different scenarios to be studied with gyrokinetic simulations .the physical parameters for each of these six cases are given in table [ table_exp ] .there , the reference length is defined as , where is the toroidal flux at the separatrix and is the magnetic field on axis . typically , this reference length is comparable but not identical to the tokamak minor radius .the logarithmic gradients are defined as in ref .: with , and the normalized toroidal flux radius .the magnetic shear is given as , where is the safety factor , and the electron beta is defined as .here , and are the ion and electron temperature , respectively , and is the electron density .the turbulence data obtained in this paper are produced with the gyrokinetic code gene , which solves self - consistently the gyrokinetic - maxwell system of equations on a fixed grid in five dimensional phase space ( plus time ) : two velocity coordinates and three field - aligned coordinates . here , is the coordinate along the magnetic field line , while the radial coordinate and the binormal coordinate are orthogonal to the equilibrium magnetic field .the velocity coordinates are , respectively , the velocity parallel to the magnetic field and the magnetic moment .gene has the possibility to simulate either a flux - tube ( local simulations ) or a full torus ( global simulations ) . in the former option, it is assumed that the relevant turbulent structures are small with respect to the radial variation of the background profiles and gradients .this allows to use periodic boundary conditions and thus the coordinates perpendicular to the magnetic field are fourier transformed . for this work ,only the local version of the code has been employed .the gene code is physically quite comprehensive and includes many features ( see ref .  for more details ) . for the asdex upgrade scenario studied in this paper ,the following features of the gene code were used : two particle species ( deuterons and electrons ) , electromagnetic effects by solving the parallel component of ampre s law , external shear , parallel flow shear and a linearized landau - boltzmann collision operator with energy and momentum conserving terms  .unless stated otherwise , the magnetic equilibrium geometry is taken from the tracer - efit interface  .additionally , gyroles techniques have been used to reduce the accumulation of energy at the smallest scales ( see refs . [ ] ) .further simulation details , such as resolution grid , box sizes , etc . , are given in the following sections .in order to calculate turbulent transport fluxes , density and temperature fluctuation amplitudes , etc . , nonlinear gyrokinetic simulations are necessary .nevertheless , linear gyrokinetic simulations can provide useful insights .for instance , they may allow us to identify the underlying micro - instabilities which drive the turbulence present in the experiments .they can also be used for convergence studies and , since they are usually computationally cheap , they can also be used to do scans in different physical parameters .   in linear simulationswe calculate the growth rate and frequency of the most unstable mode present in the system for a given binormal wave vector and . in this paperwe choose to present in instead of the more common units .this has been done in order to compare to experimental results . for reference , valuesare given in table [ table_exp ] .  in fig [ fig_nom_par ], we display the growth rates ( ) and frequencies for the cases at , because similar conclusions are obtained for the other cases . in the figure ,the negative frequencies are represented by dashed lines .the grid resolution was .convergence tests were performed at higher resolutions and confirm the validity of the results .several observations can be made .first , for low wavenumbers , itg is the dominant instability .this is indicated by a positive frequency , which with the present normalization represents a frequency in the ion diamagnetic direction .second , for all the cases analyzed , the tem mode is stable ( studied with an eigenvalue solver ) .in fact , as was already shown in ref . , only with the combination of a much higher electron temperature gradient and a lower ion temperature gradient than the ones measured experimentally , do tem modes become unstable  .third , for higher wavenumbers , etg is the dominant instability ( indicated by a negative frequency ) .finally , there is practically no effect of the ecrh on the itg growth rates .the main effect of ecrh is to increase the growth rates of etg modes , possibly leading to a subsequent increase of the electron heat flux for these cases .however , in both cases , we expect their contribution to be small with respect to the itg contribution .moreover , etg modes are not expected to influence density and temperature fluctuations at low wavenumbers , which are dominated by itg . since these are the scales measured by the diagnostics we are considering here , in this work we will limit ourselves to wavenumbers up to , thus excluding etg modes , but allowing for a significant reduction in computational resources .physical parameters such as temperature , density , magnetic equilibrium profiles , etc . , are measured with experimental uncertainties .since these values are used as input in the gyrokinetic codes , the uncertainty in these quantities could affect the simulation results .therefore , sensitivity studies are carried out , with the aim of studying the effect of the different uncertainties on the simulation results .ideally , these studies should be done in nonlinear gyrokinetic simulations . however , due to the expensive computational effort associated , this is in practice unfeasible .for this reason , this sensitivity study is done within linear gyrokinetic simulations .we have studied the sensitivity of the linear growth rate with respect to a variation of in the nominal value for different physical parameters , such as : logarithmic ion temperature gradient ( ) , logarithmic electron temperature gradient ( ) , logarithmic electron density gradient ( ) , electron to ion temperature ratio ( ) , collisionality ( ) , safety factor ( ) and magnetic shear ( ) .the main results are summarized in table [ tab_lin_var ] . for simplicity, we show only cases at , although similar conclusions were obtained to the other radial positions .the sensitivity studies for and for have been done using a miller - type magnetic equilibrium  . as is shown in table [ tab_lin_var ] ,the peak of the growth rates are practically insensitive with respect to changes of in , , , and .changes in the peak of the growth rate up to are found for variations .the most critical parameter is , whose variation modifies the growth rate by up to ( see fig .[ fig_lin_var ] ) . based on these results, one could expect that the uncertainties in will have the largest influence in nonlinear gyrokinetic simulations . in the following ,we will mainly focus on the influence of this parameter on nonlinear simulations .the influence of the shear is in general also expected to have a relevant impact on the transport in nonlinear simulations .however , the low value of the shear for this particular discharge is such that its influence can safely be neglected ..percentage difference in maximum growth rate with respect to the nominal values for cases at for various parameters ( , , , , , , and ) .[ cols="^,^,^,^",options="header " , ]in order to predict and compare with experimental results , nonlinear simulations are required . for the selected discharge ,the grid resolution needed is points in coordinates .a convergence test of the results with that resolution has been performed by comparison with nonlinear simulations with a double resolution in the perpendicular directions for a few cases . moreover , perpendicular box sizes have been chosen in such a way that several correlation lengths fit in the box and convergence checks have also been done on this respect to ensure the validity of this choice .results of the simulations are time - averaged over a range well exceeding the correlation time of the underlying turbulence .   in this section ,we compare the experimental ion and electron heat fluxes obtained through power balance analysis with the astra code and the results from nonlinear gene simulations .as was done previously , we have grouped in the same plot the cases taken at the same radial position .the results are shown in fig .[ fig_stiff ] .ion ( electron ) heat fluxes are shown in the left ( right ) columns in fig .[ fig_stiff ] .the rows represent radial positions .the dashed lines indicate the astra results and their shaded regions are used to indicate the uncertainty of the astra values , where a uncertainty is assumed  . based on the linear sensitivity studies , for each case ,several simulations were performed varying the logarithmic ion temperature gradient in steps of with respect to the nominal values , up to a maximum of variation .gene simulation results are represented by the markers in the figure .the cases without ecrh are colored in blue and with ecrh in red and for each set , the simulations with the nominal parameters are colored differently ( in green for the cases without ecrh and in yellow for the cases with ecrh ) .the statistical error bar is an estimation of the standard deviation of the set of means of consecutive temporal sub - domains of the saturated state .several conclusions can be obtained from fig .[ fig_stiff ] . for the nominal parameters ,the cases with ecrh produce more ion and electron heat flux than the ones without ecrh for all positions .moreover , at , the ion heat fluxes match the experimental values without having to vary the gradient with respect to the nominal value . at this position, only the electron temperature must be increased by . however , for the other positions , the values of the heat fluxes obtained with the nominal parameters clearly overestimate the heat fluxes obtained with astra by a factor of .we need to decrease by in order to match the experimental results for the cases without ecrh .whilst , for the cases with ecrh heating , it has to be decreased by a maximum of . in fig .[ fig_heat_comp ] , only the flux - matched results are compared with the experimental heat fluxes . we can conclude that agreement of the transport levels within the errors bars can be achieved , since even small uncertainties in the temperature profile itself may translate to relatively large ones ( up to ) in the logarithmic gradients  .since turbulent fluxes are caused by plasma fluctuations on microscopic scales , it is necessary to validate gyrokinetic codes on a microscopic level . in this regard ,a new doppler reflectometer has been recently installed in asdex upgrade ( see ref  for more details on the diagnostic ) , which is able to measure electron density fluctuation amplitudes ( ) . in order to compare experimental and numerical results , a synthetic diagnosticmust be implemented in gene to reproduce the measurement process of the reflectometer .two kinds of synthetic diagnostics can be employed .a first approach consists in simply filtering the data in order to taking into account only the location and the wavenumbers that the diagnostic measures .this translates to take into account only fluctuations at the outboard mid - plane ( ) , averaged over a finite radial length and then to select the range of perpendicular wavenumber that are measured in each case ( in gene ) .a more sophisticated method uses a full - wave code to simulate also the incidence and reflection of the wave into the gyrokinetic turbulent data  .this work is in progress and only preliminary results are available with this synthetic diagnostic ( see ref . ) .for this reason , we have only used the filtering method for this work .comparison of the experimental and simulated are shown in fig .[ fig_spec_dens ] .the simulations that match the experimental ion heat fluxes are shown with a different marker to distinguish them from the rest .the fluctuation data is analyzed considering only perpendicular wavenumbers between $ ] .the experimental values are scaled by a common factor since the measurements are in arbitrary units . for this reason, only the shape of radial turbulence level profiles and the effect of ecrh can be used for comparison .we decided to scale the experimental values to try to match the case without ecrh . as is shown in the left plot of fig .[ fig_spec_dens ] , we obtain a remarkable agreement between experimental and simulations results in the radial trend . for the case of mw ecrh , there is also a good agreement in the turbulence level profile . however , with the scale used , the fluctuation levels are clearly underestimated with respect to the ones measured experimentally . in particular , the flux - matched results present the biggest discrepancy with respect to the experimental measurements .finally , from this figure we can also observe how sensitive the density fluctuations are with respect to variations in the ion logarithmic temperature gradient .for instance , a reduction in the logarithmic gradient can reduce density fluctuation levels by more than a factor of .the knowledge of the power - law spectra of a physical quantity is important for the understanding of the underlying physics and useful for providing constraints for simple physical models .based on kolmogorov - type arguments  , turbulence is generally associated with universal power - law spectra . however , as shown in refs , this is generally not the case in plasma turbulence , and different power - law spectra indices can be found depending on the type of mechanism which drives or dissipates energy in the system .furthermore , the knowledge of wavenumber spectra could be important for a clear identification of the turbulent regimes driven by different microinstabilities and can be used to further validation of the gyrokinetic model .the results of the electron density fluctuation spectra at for different logarithmic ion temperature gradients are shown in fig .[ fig_spec_dens_1 ] .the solid lines represent the wavenumber range where a fit to a power law is shown : , where is the spectral index , and represents an average over all the coordinates except .spectral indices for the flattest and steepest spectra are indicated in the figure .the calculated spectral indices are also shown in figs .[ fig_spec_dens_2 ] and [ fig_spec_dens_3 ] with respect to the ion temperature gradient and to the radial position , respectively .these figures present a clear qualitative behavior : the spectral index decreases with the increase of turbulence drive .additionally , the spectral indices also decrease when going from the inner to the outer core position .moreover , the magnitude of the exponents cover a wide range of values , approximately from to .although density fluctuation spectra were not measured with the doppler reflectometer for this discharge , a similar trend has been also reported in ref .  for various asdex upgrade discharges .these results indicate that the turbulence driven by itg modes exhibits non - universal power laws , whose spectral indices could depend on several physical parameters .future work in this respect will be to study also if tem modes exhibit similar properties .if this was the case , then it would become very difficult to distinguish a type of instability by measuring only its characteristic spectral index .   at the time the discharge was performed ,no temperature fluctuation measurements were available .however , a correlation electron cyclotron emission ( cece ) diagnostic is currently installed on asdex upgrade , and electron temperature fluctuation profiles will be available in the future campaigns .the cece diagnostic measures perpendicular electron temperature fluctuations ( ) in the long wavelength range ( relevant for itg and tem modes ) and is not sensitive short wavelengths ( etg modes ) .this diagnostic presents an inherent limitation in the lowest fluctuation level that can be detected .this noise level depends on the physical parameters of the discharge , but typical values are between  .a detailed description of cece modeling in diii - d is given in ref . .this synthetic diagnostic has already been implemented in gene for diii - d discharges  and a similar synthetic diagnostic will be implemented for asdex upgrade discharges .however , this diagnostic could be not used in this work since it requires the knowledge of the cece configuration during the discharge .for this reason , we have considered a simpler synthetic diagnostic , which consists in filtering the gyrokinetic data to the positions and wavenumbers that are expected to be measured in asdex upgrade .consequently , the gyrokinetic data analysis results are restricted to the outboard mid - plane position ( ) , averaged over the finite radial length and summing all perpendicular wavenumbers ( since short wavelengths are not simulated ) . in order to better model the actual diagnostic, one should also apply a filter in the frequency space .however , this filter will also depend on the specific range of frequencies measured .since we do not have access to this information , we have considered all the frequencies in the analysis .therefore , the following results should be only used as an approximated indication of the fluctuation amplitudes that could be detected with cece for this discharge .nevertheless , we do not expect radial trends to change with respect to a more sophisticated synthetic diagnostic approach and only the amplitudes are likely to be rescaled  .the main results are shown in fig .[ fig_fluc_temp ] .the perpendicular electron temperature fluctuation amplitudes go from a minimum of at the inner position to a maximum of at the outer core position . therefore , assuming a noise - level in the order of for the cece diagnostic , this could imply that only the fluctuations at the outer core positions ( starting from ) could be detected .as for the case of the density amplitudes , we also observe a large variation of the fluctuation amplitudes with respect to the changes in the logarithmic ion temperature gradient .doppler reflectometers can be coupled to the cece diagnostics to calculate cross - phases between electron density and temperature fluctuations  .this measurement is important for gyrokinetic validation studies since it represents a relationship between different fluctuating quantities ( density and temperature in this case ) .in addition , this cross - phase could be also related to the cross - phase that determines the turbulent heat fluxes ( electrostatic potential and temperature fluctuations ) .for this reason , in addition to the cross - phase that can be measured experimentally , we will also show the cross - phase between electrostatic potential and electron density fluctuations , so we can relate the cross - phases measured experimentally to the turbulent heat fluxes .the cross - phases are here defined as , and , .  fig .[ fig_cross_2 ] shows the variation of the cross - phase versus the ion temperature gradient integrated over binormal wavenumbers in the range of . for most of the cases ,the phases remain rather invariant with respect to the variation of this parameter .therefore , this result seems to indicate that the cross - phase is a better observable to identify the type of instability which drives the turbulence in experiments , since tem instability is expected to exhibit different cross - phases  .  in fig .[ fig_cross_3 ] , the cross - phase are displayed versus the radial positions . regarding the cross - phase between density and electrostatic potential fluctuations ( a ) and ( b ), we see that they are practically in phase , i.e. close to .in addition , an increase of the phase with the radial position is also observed , going from practically degrees at to approximate by degrees at the outer core position . on the contrary , for the cross - phase between electron density and temperature fluctuations ( c ) and ( d ), we see a decrease with the radial position , going from around degrees in the inner position to a value of degrees , which result in an increase of the electron heat flux .similar values of this cross - phase have also been measured in diii - d and calculated with gyro in ref . .furthermore , these values have also been found by gene for these discharges , see ref . .these observations could be explained in the following way . in the inner position ,the population of the trapped particles that contributes to the itg instability is small , so the electrons behave almost adiabatically . because of this , density and potential fluctuations are in phase and density and perpendicular temperature fluctuations are almost out of phase ( i.e. , close to degrees ) , thus producing negligible electron heat flux . with increasing the radial position ,the population of trapped particles increase and a deviation of the electron adiabaticity is observed . for this reasons , both cross - phases approach to degrees , with the subsequent increase of electron heat flux .  finally , in fig .[ fig_cross ] , the colored contours display the cross - phases obtained from the nonlinear simulations , while the red squares are used to display the cross - phases of the linear simulations for the case without ecrh at .the agreement between linear and nonlinear cross - phases is remarkably good .this is also observed for the other cases  .this result implies that linear simulations could be enough to compare with experimental results .focusing on the case at with mw ecrh , the key results obtained in this paper can be illustrated in fig .[ fig_variation ] . here, we show the impact of the variation of the logarithmic ion temperature gradient around the nominal value on various observables .ion and electron heat fluxes ( a ) , electron and temperature fluctuation amplitudes ( b ) and electron density spectral indices ( c ) are all very sensitive with respect to small changes in the ion logarithmic temperature gradient .for instance , by decreasing by , the amplitudes can be reduced by a factor of for the density and temperature fluctuations and by a factor of for the heat fluxes .this result implies that the comparison of gyrokinetic simulation and experimental measurements for these observables are very sensitive to uncertainties in the experimental input profiles . on the contrary ,cross - phases between density and temperature fluctuations and between density and electrostatic potential ( d ) are rather insensitive with respect to .this , together with the fact that linear and nonlinear cross - phases agreed also remarkably well , indicates that cross - phases could be a good observable to compare ( fast ) linear gyrokinetic simulations with experimental measurements .we have analyzed , by means of gyrokinetic simulations with gene , core turbulence features of an h - mode discharge in asdex upgrade .the main results of this paper can be summarized as follows .flux - matched simulations were achieved by varying the nominal ion temperature gradient by a factor of , which is within the uncertainty range of the experimental profiles .in addition , density fluctuation levels show an agreement in the shape of the radial turbulence level profiles , although the effect of the ecrh on the fluctuation levels was not reproduced .non - universal power - law spectra were found for turbulence driven by itg modes .in particular , itg instability exhibits spectral indices for the density fluctuation spectra which cover a broad range of values .these values depend on the radial position and also on the specific ion temperature gradient .gyrokinetic simulations predict a decrease of the exponents with respect to both the increase of the ion temperature gradient and the increase of the radial position .these results could help validate future analytical theories and are useful for comparisons with other gyrokinetic codes and future measurements .regarding the electron temperature fluctuations , we observe for the inner position ( ) fluctuation amplitudes which are close to the sensitivity of the cece diagnostic .therefore , it is possible that only measurements at positions larger than could be detected with this diagnostic in such discharges .we hope that these results can provide guidance for the development of the cece diagnostic that is currently being installed in asdex upgrade .finally , by analyzing cross - phases between density and temperature fluctuations and between density and electrostatic potential , we observed that linear and nonlinear cross - phases agree remarkably well , and that they are rather insensitive with respect to the variation of the ion temperature gradient , indicating that cross - phases could be a good observable with experimental measurements for comparisons .  for future work , gene and asdex upgrade comparisons will continue with the study of similar h - modes plasmas but with higher ecrh power ( up to mw ) .these discharges are expected to have peaked electron temperature profiles and allow tem modes to be dominant .this will allow us to study fundamental differences between itg and tem modes from a microscopic level .furthermore , additionally dedicated discharges have already been conducted in which detailed wavenumber spectra have been measured with the doppler reflectometer .these comparisons , along with the inclusion of future cece measurements , will help in further validating gyrokinetic codes and the development of synthetic diagnostics .the authors would like to thank a. e. white , g. d.  conway and u. stroth for fruitful discussions .the simulations presented in this work were carried out using the helios supercomputer system at the computational simulation centre of international fusion energy research centre ( iferc - csc ) , aomori , japan , and the hydra supercomputer at the rechenzentrum garching ( rzg ) , germany .this work has been carried out within the framework of the eurofusion consortium and has received funding from the euratom research and training programme 2014 - 2018 under grant agreement no 633053 .the views and opinions expressed herein do not necessarily reflect those of the european commission .the research leading to these results has also received funding from the european research council under the european unions seventh framework programme ( fp7/2007v2013)/erc grant agreement no .277870 .t. happel , a. ban navarro , g. d. conway , c. angioni , m. bernert , m. dunne , e. fable , b. geiger , t. grler , f. jenko , r. m. mcdermott , f. ryter , u. stroth and the asdex upgrade team , submitted to _ phys .plasmas_.                    a. e. white , l. schmitz , g. r. mckee , c. holland , w. a. peebles , t. a. carter , m. w. shafer , m. e. austin , k. h. burrell , j. candy , j. c. deboo , e. j. doyle , m. a. makowski , r. prater , t. l. rhodes , g. m. staebler , g. r. tynan , r. e. waltz , and g. wang , _ phys . plasmas _ * 17 * , 056116 ( 2008 ) .    j. c. hillesheim , j. c. deboo , w. a. peebles , t. a. carter , g. wang , t. l. rhodes , l. schmitz , g. r. mckee , z. yan , g. m. staebler , k. h. burrell , e. j. doyle , c. holland , c. c. petty , s. p. smith , a. e. white , and l. zeng , _ phys . rev . lett ._ * 110 * , 045003 ( 2013 ) .

Abstract: gyrokinetic validation studies are crucial in developing confidence in the model incorporated in numerical simulations and thus improving their predictive capabilities . as one step in this direction , we simulate an asdex upgrade discharge with the gene code , and analyze various fluctuating quantities and compare them to experimental measurements . the approach taken is the following . first , linear simulations are performed in order to determine the turbulence regime . second , the heat fluxes in nonlinear simulations are matched to experimental fluxes by varying the logarithmic ion temperature gradient within the expected experimental error bars . finally , the dependence of various quantities with respect to the ion temperature gradient is analyzed in detail . it is found that density and temperature fluctuations can vary significantly with small changes in this parameter , thus making comparisons with experiments very sensitive to uncertainties in the experimental profiles . however , cross - phases are more robust , indicating that they are better observables for comparisons between gyrokinetic simulations and experimental measurements .


with the upcoming launches of the transiting exoplanet survey satellite ( tess ) and the james webb space telescope ( jwst ) , the amount and quality of exoplanetary transmission spectra available for interpretation are expected to increase dramatically . to analyze the anticipated wealth of data in a timely and statistically meaningful fashion ,many members of the exoplanet community combine sophisticated statistical algorithms ( ; ; ; ; ; ; ) with instrument models and radiative transfer codes to retrieve atmospheric properties of transiting exoplanets from their lightcurves ( see e.g. analyses by ; ; ; ; ; , ; ; ; ; ) and to plan observational strategies for jwst ( ; ) .as many of these algorithms must compute numerous exoplanetary spectra as they sample across the allowed multi - dimensional space of atmospheric parameters , the forward models which compute these spectra must strike a proper balance between speed and accuracy .  in light of this ,analytical formalisms to compute exoplanetary spectra are extremely useful .they allow one to understand how key atmospheric parameters shape exoplanetary spectra , can constitute the backbone of forward models , and increase computational speed tremendously if they are general and accurate enough .they can also test , under the same conditions for which they are valid , the accuracy of more general numerical methods .one important formalism for transmission spectroscopy , against which radiative transfer codes are sometimes compared ( see e. g. ; ) and which a few analyses use ( see e.g. ; , 2016 ) is the semi - analytical derivation of for a homogeneous and isothermal atmosphere .  from numerical simulations , show that for a broad range of planetary radius to scale height ratios ( ) , the effective radius ( ) of a transiting exoplanet is located at an altitude where the integrated optical depth ( or slant optical depth , ) along a ray grazing the planetary limb is about 0.56 .they build upon this numerical result to show analytically that the variation with wavelength of the effective radius of an exoplanet is proportional to the atmospheric scale height .later , confirm in a purely analytical fashion s results and also show that the relevant scale height of the atmosphere changes with the type of the dominant source of opacity , with collision - induced absorption ( cia ) having half the effective scale height of rayleigh scattering . also demonstrate that in the presence of an opacity source which obeys a simple known power law with wavelength , such as rayleigh scattering by gases or well - mixed hazes , one can deduce the scale height of the atmosphere by measuring the spectral slope of the rayleigh signature ( rayleigh slope ) , provided it is the only source of opacity in the observed spectral region .both formalisms are very useful for clear atmospheres , or atmospheres where hazes are well - mixed .however , a planetary surface or an optically thick cloud deck creates a sharp boundary ( or surface " ) below which stellar radiation is cut - off , which neither s nor s analytical expressions capture . to illustrate the impact of a surface " , fig .[ fig1 ] shows the limb absorption ( ) profile , given by of a constant scale height atmosphere as a function of distance in scale height ( ) above this surface " for different values of the slant optical depth of the rays grazing the surface " ( ) .as decreases , the limb absorption profile shifts downward . when , the surface " has no impact on the effective radius of an exoplanet because the atmosphere is completely opaque at the surface " .a slant optical depth of 0.561 is thus representative of the location of the effective radius of the exoplanet for this purely atmospheric profile .when , the limb absorption profile deviates from the purely atmospheric one because it increases sharply to one at the surface " , violating the assumption behind s derivation .as decreases , the difference in absorption between the surface " and the atmosphere increases , and the limb absorption profile deviates increasingly from the pure atmospheric profile . when , the absorption profile is sufficiently distorted from the atmospheric one that s formalism completely breaks down and predicts that the effective radius of the exoplanet is below the surface " .however , common sense dictates that it should be above , as there is still some atmospheric absorption .indeed , for an airless rocky exoplanet where atmospheric absorption is non - existent , its effective radius is simply located at its surface .  ) . according to and , the effective radius of an exoplanetis located at a distance above the surface " where the optical depth of a ray traversing the atmosphere is about 0.561 ( i.e. where the triple - dot - dashed line intersects the various limb absorption profiles ) .[ fig1 ] ]  so not only are these formalisms ( ; ) not appropriate for terrestrial exoplanets with optically thin atmospheres , but neither are they for atmospheres where a cloud deck mimics a surface . unfortunately , there is a growing body of evidence that many exoplanets may have clouds or hazes .gj1214b is the most compelling example , as its transmission spectrum remains featureless to the best photometric precision , a fact which is more likely explained by a high altitude cloud deck than a high mean molecular weight atmosphere .indeed , cloud decks decrease the contrast of spectral features ( e. g. ; ; ; ) and are the likely cause for the reduced contrast , compared with that expected from a clear atmosphere , of the 1.4  m water band ( ; ; ) , sodium and potassium lines ( ; ) observed in a few hot jupiters .even thick cloudless atmospheres , or one with well - mixed hazes , require an analysis incorporating a surface " as recent results ( ; ; ) show that thick atmospheres have a sharp change in flux due to refraction .indeed , a refracted distorted image of the stellar surface is superimposed on the limb absorption profile , and the refractive boundary can be predominantly due to a change in the back - illumination of the atmosphere at the imaged stellar limb , unlike clouds and surfaces which only change the limb absorption profile .the spectroscopic signature of this first - order refractive effect and an optically thick cloud deck are similar in nature because they both cause lower atmospheric regions to appear dark in transmission spectroscopy ( see fig . 2 in ) .the major difference is the altitude where they occur , where for example the refractive boundary of an earth - sun analog is located above tropospheric clouds , significantly reducing the contrast of water features in the visible and near infrared ( ; , 2014 ; ) .moreover , the location of a refractive boundary is easier to compute than that of a cloud top because the chemical species which contribute to an atmosphere s refractive properties are the same that contribute to its mean molecular weight , whereas the chemical nature of clouds are currently difficult to retrieve from the lightcurve of a transiting exoplanet . shows that refraction modifies the rayleigh slope , and that it can break the degeneracy between retrieved abundances of chemical species and the planet s radius .nevertheless , a few recent papers use s formalism as a starting point for their own analytical derivations . interpret the strength of sodium and potassium lines of hd 189733b in term of vertical temperature gradients . explore the effects of a non - homogeneous limb distribution of vertically well - mixed hazes on an exoplanet s transmission spectrum . derives an index quantifying the haziness of an atmosphere from the contrast between the core and wings of sodium and potassium lines . apply s formalism to an atmosphere s sudden change in transmissive properties , due to the presence of a cloud base , to derive its purported signature .clearly , a new formalism which can also account for surfaces " ( surfaces , optically - thick cloud decks , or refractive boundaries ) is badly needed by the community .  in this paper , we derive the effective radius and the effective atmospheric thickness of a transiting exoplanet by proceeding along the line of s derivation .we highlight where our results differ from theirs , and show that we find an extra term which accounts for the presence of a surface " .we explore the various ramifications in terms of the location of the exoplanet s effective radius , its change with wavelength , and its effects on the rayleigh slope .we also discuss how surfaces " decrease the contrast of spectral features , and devise a figure of merit for characterizing the location of a surface " and a formalism for quickly estimating its impact on the spectral modulation of an atmosphere .finally , we develop a numerical recipe to generalize our formalism to atmospheres with a non - constant scale height and arbitrary sources of opacity .  ) between and .the above picture is for a refractive atmosphere which bends rays , where is the grazing radius ( i.e. point of closest approach ) of a ray . in the non - refractive approximation, rays follow straight lines parallel to the dashed line and .[ newfig2 ] ]our derivation for an exoplanet s effective radius is similar to the one in the main article and the supplementary materials of .it is accurate for an atmosphere with a constant scale height , ignoring stellar limb darkening and second - order refractive effects .we highlight differences in results , as well as differences in notation when necessary .two quantities appear regularly throughout this paper : the impact parameter of a ray ( or projected distance to the center of the exoplanet with respect to the observer , ) , and the grazing radius of a ray ( or closest approach to the center of the exoplanet , ) .both quantities are illustrated in fig .[ newfig2 ] which shows the geometry of observation of a transiting exoplanet .even though we do not include the subtle secondary effects of refraction which creates differences between and , we use the appropriate quantities in the various equations as if refraction was fully included. however , throughout the discussions in this paper , and are interchangeable because they are identical in our essentially non - refractive treatment .nevertheless , one can include first - order surface " effects of refraction by locating the surface " at the impact parameter of the refractive critical boundary , whose location depends both on the properties of the atmosphere and the planet - star geometry .a transiting exoplanet causes a momentary drop ( ) in the observed stellar flux ( ) , which is expressed in term of the effective radius ( ) of the exoplanet which occults the stellar surface by where and are the observed fluxes outside of and in the middle of the transit , respectively , and is the stellar radius .strictly speaking , is the sum of the stellar flux and scattered radiation from a thin crescent - shaped region of the observable dayside of the exoplanet , as well as thermal emission from the nightside of the exoplanet .however , the exoplanet s contribution to the observed out - of - transit flux is negligible compared to the stellar flux .the effective radius of an exoplanet is the radius that a completely opaque spherical body occulting a uniformly bright star must have in order to duplicate the observed flux drop . ignoring stellar limb darkening , the effective radius of an exoplanet occulting the center of its host star is theoretically computed by  where is the slant optical depth of a ray through the atmosphere , and and are the impact parameters of the ray at the surface " and the top of the atmosphere , respectively ( shown in fig . [ newfig2 ] ) .the contribution to the effective radius ( i.e. to the flux drop ) is split between the completely opaque body " which lies below the surface " ( first term ) , and that of the semi - transparent atmosphere above it ( second term ) .we represent the atmospheric term by just like in , but replace their first term ( i.e. planetary radius at the surface , ) with .the effective atmospheric thickness ( ) is the difference between the effective radius of the exoplanet and the surface " , so that using equations [ effradius ] and [ effradius2 ] , we can relate to the effective atmospheric thickness by which has only one physical solution : we note the difference inside the square root between our expression and equation 6 in .their expression is erroneous because their units are not consistent .since is the ratio of the effective projected surfaces of the thin atmospheric annulus to that of the planetary body , we can usually expect that . in that case, we can rewrite equation [ exacth ] as a taylor series and obtain  \notag \\   & \approx \frac{c}{2 b_s } \left [ 1 - \frac{1}{4 } ( c / b_s^2 ) + ... \right ] \label{effheight } .\end{aligned}\ ] ] the series inside the square bracket is a correction factor which is close to and always less than one .the slant optical depth along the path of a ray depends on the nature of the opacity through the relationship of the absorption coefficient with number density , as well as the vertical distribution of the chemical species responsible for the stellar radiation s extinction . assuming a homogeneous atmosphere , such that the mole fraction ( ) of each species is constant with altitude , and that the absorption coefficient is proportional to number density ( e.g. rayleigh scattering ) , the slant optical depth ( ) is given by where is the extinction cross section of each species, is the mean atmospheric cross section , and ( ) is the column - integrated density along the path of the ray ( called horizontal integrated density by ) , or simply column abundance . in the case of collision - induced absorption ( cia ) , where the absorption coefficient is proportional to the square of the number density , the slant optical depth ( )is given by where is the collision - induced cross - section of the and species , is the mean atmospheric collision - induced cross section , and is the square of the density integrated along the path of the ray .we adopt the same units as hitran for the collision - induced cross section , namely molecule .  in the case of an isothermal atmosphere , and assuming that the gravitational acceleration does not change much over the altitude regions which contribute significantly to an exoplanet s transmission spectrum , the neutral density ( ) through the atmosphere is related to its value ( ) at a reference radius ( ) by where the density scale height of the atmosphere ( ) is constant .since the absorption coefficient of various sources of opacity can have different dependences with number density , let us consider the general case where the absorption coefficient is proportional to .the integrated density to the power along the path of the ray ( ) is given by where is a distance along the ray with respect to its grazing point .as already pointed out , for rayleigh scattering , and for cia .the situation for line transitions is more complicated as is equal to one near the core of the line and the far lorentzian wings , and goes up to two in the voigt - to - doppler transition region of the line ( see the comprehensive discussion in s supplementary materials ) .however , molecular bands are the sum of many overlapping individual transition lines , and we postulate that only the sides of bands , where the cross section is rapidly changing with wavelength , are dominated by this transition region . hence , most spectral regions of a molecular cross section should fall outside this region , and into the domain where is about one .provided no extra exponential term is hidden in , we can see by inspection that the exponential in equation [ colabun2 ] is identical to that of a constant scale height atmosphere with a slant optical depth scale height ( ) , which we define by that is indeed the case in a non - refractive treatment , as the exponential dependence with then only comes from the density dependence , in opposition to the general case which is fairly complex ( see equations 32 and 17 , respectively , in ) .thus , the solution to equation [ colabun2 ] in our non - refractive treatment is simply the leading term in equation 43 of with replacing .it is then where we neglect the dependence ( ) as the variation of with altitude is dominated by the exponential factor . is simply where the grazing radius is the reference radius .  in general, the slant optical depth comes from a mixture of opacities with different , and the slant optical depth scale height will be between those of rayleigh scattering and cia ( i.e. is between one and two ) .moreover , the relative importance of cia opacities versus other opacities changes with altitude , so that might also change significantly with large changes in altitude . however , much of the discussions in the remainder of the paper assume that one form of opacity dominates in a given spectral region ( i.e. is constant with altitude ) .we treat the general case in section [ integration ] .when is constant with altitude , it follows that where is given by either equation [ opacity1 ] or [ opacity2 ] depending on the opacity source , and is the slant optical depth at the reference radius . when selecting the surface " as the reference radius , the effective slant optical depth ( ) is by definition simply where is the slant optical depth at the surface " .to determine the effective atmospheric thickness , we must evaluate to solve this integral , we first use equation [ slantdepth ] with the surface " as the reference radius , then replace with because of our non - refractive treatment ( see comments in section [ derivation ] ) .we thus derive the relationships and which we use to change the integral variable from to .this leads to where as it is the slant optical depth at the top of the atmosphere .the second term is on the order of larger than the first term . since is usually much smaller than one , we neglect the second term .this term probably contributes to the departure of the effective slant optical depth from a constant when is small ( see fig . 1 of ) . however , we focus here on the domain which is applicable to the largest ensemble of exoplanets .   solving the integral in the first term of equation [ intform ]is a multi - step process .the first step consists in splitting the integral by changing the limits of integration equation [ split ] is true whether is larger or smaller than one . we then add and subtract identical integral terms to obtain where ( ) is the euler - mascheroni constant , whose definition can be found in ( by inspection from equations 9 and 10 in appendix 1 ) and is we then rewrite the integral terms in equation [ inter ] as the second term in equation [ complex ] is trivial and simply .the last term is the exponential integral ( see equation 7 in appendix 1 of ) .hence , we get the very useful result that   using this result in equation [ intform ] , and combining it with equations [ effradius ] or [ effheight ] , we can see that to first - order the effective radius of an exoplanet is given by and the effective atmospheric thickness by we can compare our result for the effective atmospheric thickness to that in equation 7 of , noting that we use the symbol instead of , and instead of .we find our solutions are identical except that we have an extra term .what is the physical significance or interpretation of this term ?to understand the physical significance of our solution , we first rearrange the terms in equation [ effatmthick ] into we then take the exponential on both sides and use equation [ effslantdepth ] to get the very simple relation it is interesting to note that ( ) is the effective slant optical depth associated with s solution which matches the simulations of for a clear atmosphere . however , our solution has an extra multiplicative factor which depends on the optical properties of the atmosphere at the surface " .does our solution matches their solution for a clear atmosphere ? a clear atmosphere can be thought of as an atmosphere with a surface " buried so deeply that it can not be observed ( i.e. ) . since ,our extra multiplicative factor is indeed one for a clear atmosphere ., at the effective exoplanet radius ) as a function of the slant optical depth of the atmosphere at the surface " ( ) .also shown are the clear atmosphere solution of ( triple - dot - dashed line ) and when the effective radius is located at the surface " ( dashed line ) . [ fig2 ] ]  fig .[ fig2 ] shows the behaviour of with .as , asymptotically approaches the clear atmosphere solution of .below , diverges from this solution as decreases , and asymptotically approaches the surface " described by the  " line .this makes sense because the effective radius is located at a slant optical depth within the bounds of the observable atmospheric region , i.e. above the surface " .thus , our extra integral exponential term from equation [ effatmthick ] accounts for the effect of a surface " , and decreases the effective slant optical depth associated with the effective radius of the exoplanet .   how does this new result change the variation of the effective exoplanet radius with wavelength , originally discussed by ?assuming does not change much with wavelength , differentiation of the exponential integral gives so that as , we find that the size of spectral features depends on the nature of the opacity through , with cia spectral features proportional to half of the scale height ( see equation [ sldepthscale ] ) .when the absorption coefficient is proportional to density , such as for rayleigh scattering , this reduces to because the cross section is the only quantity which is wavelength - dependent in the expression for the slant optical depth . for cia , we simply replace with ( see equation [ opacity2 ] ) , and with ( see equation [ sldepthscale ] ) .we find an extra multiplicative factor compared to .this holds true also for differentiation with respect to . for a clear atmosphere where tends to infinity, this extra factor tends to one .as tends to zero , this extra factor tends to and becomes vanishingly small .hence , the size of spectral features is not strictly proportional to the scale height , but also depends on the slant optical thickness of the atmosphere at the surface " , which can become vanishingly small when the surface " number density is low and the observable atmosphere is optically thin .similarly , this factor also modifies the rayleigh slope in spectral regions where rayleigh scattering dominates .as the rayleigh scattering cross section decreases from the ultraviolet to the infrared , so does its slant optical thickness . in spectral regions where the rayleigh scattering slant optical thickness is less than about four , the rayleigh slope decreases with wavelength until it becomes flat .the decrease of the contrast of spectral features has been demonstrated several times for both clouds and refraction through radiative transfer numerical simulations of exoplanetary transmission spectra ( see e. g. references in section [ intro ] ) .however , the gradual change of the rayleigh slope in the visible and its flattening in the infrared was only recently shown numerically for jovian - type exoplanets .although and point out that the effects of refraction on transmission spectra are similar to clouds , we have now demonstrated analytically in sections [ solution ] that all these surfaces " cause these same first - order effects .since the effective atmospheric thickness is proportional to the density scale height of the atmosphere , one quantity is particularly helpful in thinking about exoplanetary transmission spectra : the spectral modulation of an exoplanet .the spectral modulation is the difference in effective atmospheric thickness expressed in units of scale height where the subscripts 1 and 2 refers to two different wavelengths which are chosen such that .it is thus useful to also express the effective atmospheric thickness in units of scale height where is defined in section [ opticaldepth ] ( see equations [ colabun2 ] and [ sldepthscale ] ) .  ) for ( see equations [ colabun2 ] and [ sldepthscale ] ) for different solutions : our solution , given by equation [ hinscale ] ( solid line ) , s solution ( triple - dot - dashed line ) , and the approximation ( red line ) outlined in section [ index ] . [ fig3 ] ]  fig .[ fig3 ] shows how varies with for , both for our expression ( equation [ hinscale ] ) and s ( or s ) . above , there is no meaningful difference between the two solutions .as decreases , our solution for approaches the surface " without ever going below , unlike s solution which reaches the surface " when , and continues below at smaller  values .since rayleigh scattering cross sections continuously decrease with wavelength , these two solutions illustrate how the signature of rayleigh scattering changes with the presence of a surface " ( see also figs . 1 - 3 in ) .the significance of the difference between the two solutions is also evident when considering its impact on the spectral modulation of molecular absorption , which we can infer from fig .let us consider a single line transition where at the core of the line .in the presence of a surface " , the spectral modulation between the core and the wing of the line is never greater than about , irrespective of the value of in the wings of the line .however , it can be much larger with s solution ( e.g. about 6.9 with the part of the wing ) .hence , a surface " decreases the atmosphere s spectral modulation .we can derive this analytically by combining equations [ specmod ] and [ hinscale ] .it requires an assumption about the nature of the dominant form of opacity ( i.e. about the value of ) at each of the two wavelengths that we are comparing .when is the same at both wavelengths , the spectral modulation simplifies to \label{sizefeatures}\ ] ] where .hence , cia ( ) create half the spectral modulation as rayleigh scattering or molecular absorption ( ) .this simplifies further to \label{smrayl}\ ] ] for rayleigh scattering and molecular absorption , and to \label{smcia}\ ] ] for cia .  in a clear atmosphere , , and the spectral modulationis then given by the first term in equations [ smrayl ] or [ smcia ] , i.e. a simple natural logarithm of the ratio of mean cross sections . for rayleigh scattering or molecular absorption , is an average of the cross sections of various species weighted by their respective mole fractions ( ) .if only one chemical species contribute significantly to the slant optical depth in each spectral region , then , and , where subscripts and refer to different chemical species , and the ratio of the mean cross section at two different wavelengths is then determined not only by the ratio of the cross section of each chemical species but also by their relative abundance . in each spectral region , the dominant species is the one which has the largest product .when several species have comparable products , these should be added together . when the dominant species is the same in both spectral regions , is independent of abundances . in atmospheres with many chemical species , molecular bands of different specieswill partially fill - in each others minima , potentially resulting in a lower spectral modulation than when the transmission spectrum is dominated by a single species .  in the presence of a surface " , is negative , and the spectral modulation is smaller than that of a clear atmosphere . comparing the observed spectral modulation to that of the clear case may yield clues to the presence of a surface " .recent work have indeed tried to infer the cloudiness of hot jupiters ( ; ; ; ) through studies of the spectral modulations of water features , as well as that of potassium and sodium lines .their results hinge upon getting the proper estimate of the density scale height to translate an observed spectral flux difference into a spectral modulation , and comparing it to that of the corresponding species in a clear isothermal atmosphere .  the theoretical contrast ratio between the surface " case and the clear case , or surface "- to - clear spectral modulation ratio ( ) , is given by and . in the more general casewhen one compares two spectral regions each with different , it is necessary to go back to the more fundamental expressions ( equations [ specmod ] and [ hinscale ] ) to determine and .as different spectral regions probe different atmospheric depths , one spectral region may be affected by the surface " while another is not , depending on the location of the surface " .equation [ rsc ] shows this because is dependent on which combines the opacity of the gas with the location of the surface " via the number density of the atmosphere a the surface " .( see equations [ colabun2 ] and [ sldepthscale ] ) with the approximation outlined in section [ index ] as a function of the surface " slant optical depth .[ fig4 ] ]   although useful for computing transmission spectra of isothermal well - mixed atmospheres , equation [ hinscale ] does not provide a quick understanding of which spectral features rise above the continuum created by a surface " , because exponential integrals fall outside the realm of our intuition .we can get around this problem with a simple approximation : we first characterize the location of the surface " in term of an equivalent mean atmospheric cross section , or surface " cross section ( for ) within the context of a clear atmosphere , and then define this surface " cross section to be the minimum mean cross section which can be observed in the atmosphere .  to determine this surface " cross section , we ask the question : what mean atmospheric cross section in a clear atmosphere produces an effective atmospheric thickness at the surface"?. it is mathematically equivalent to setting , which in a clear atmosphere becomes and gives in our non - refractive treatment . the number density at the surface " is converted into a mean cross section which is constant at all wavelengths , and which depends on the properties of the exoplanet through its radius ( ) and its scale height .we then set to any part of the mean cross section which is less than this surface " cross section thereby hiding spectral features located below this continuum .we can then do a quick back - of - the - envelope " calculation of the effective atmospheric thickness , using this modified mean cross section ( ) , by using equation [ sizefeatures ] for a clear atmosphere the red curve in fig .[ fig3 ] depicts our approximation , which follows the solution of when , and is zero otherwise .[ fig4 ] shows the amount by which our approximation underestimates the effective atmospheric thickness .it is at most when .we can apply the same line of reasoning to quickly determine which cia features rise above the surface " continuum .we use instead of equation [ approxs ] , where is the modified mean collision - induced cross section with its minimum set at the surface " collision - induced cross section ( ) , given by we then determine the spectral modulation between any two spectral regions with equation [ specmod ] .the surface " decreases the spectral modulation only if or in one of the two spectral regions .can we use this approximation to quickly determine the location of a surface " from observations ?if the observed spectral modulation is due to a single species and is less than that expected in a clear atmosphere , we can relate the observed spectral modulation , , with respect to a local maximum in its cross section ( ) by unfortunately , the surface " location can not be unambiguously determined because the mole fraction ( ) of the species is unknown .indeed , equation [ abuneffect ] shows that the spectral modulation of a species increases with abundance in the presence of a surface " , as demonstrated numerically by for a refractive boundary .the only way to break this degeneracy are two - fold : either the spectral signature is that of a major species ( i.e. ) and we can determine the location of the surface " , or we can predict the location of the surface " and then determine the abundance of the minor species . given that refractive boundaries are dependent on the bulk properties of the exoplanetary atmosphere as well as the planet - star geometry , quantities which are retrieved during the analysis of transit data , location of refractive boundaries can be predicted in principle and used to place a lower limit on the abundance of detected minor species .although the analytical approximation outlined in this section is intended to build physical intuition about the effects of surfaces " , it does not produce large differences in the computed transmission spectrum ( see fig .[ fig4 ] ) compared with the exact solution embodied by equation [ hinscale ] .its most obvious artifacts are sharp changes in the slope of the effective atmospheric thickness with ( see red line in fig . [ fig3 ] ) , i.e. with wavelength , similar to those which are produced by the analytical model of ( see their figs . 11 and 12 ) in the transition from rayleigh scattering to cloud - dominated spectral regions .however , to compute transmission spectra with the proper gradual change of slope with wavelength , characteristic of the presence of a surface " , it is necessary to use our exact analytical solution .although applicable only to an homogeneous isothermal atmosphere , our analytical formalism can be generalized to any atmosphere .indeed , equation [ results ] suggests a way to compute the effective radius from a set of numerically - computed slant optical depth ( ) at various impact parameters ( ) . to achieve this, we first divide the atmosphere into layers , so that where and .we can choose the impact parameter bounding the various layers ( ) so that the slant optical depth scale height is roughly constant in each layer .in this case , it is given by where the indices and refer to the upper and the lower boundaries of each layer , respectively .we need make no assumption about the nature of the opacity as we derive the slant optical depth scale height directly from the results of numerical integration , which can combine arbitrary sources of opacities .instead of using as our reference slant optical depth for the change of variable ( see equation [ varchange ] ) , we use the average of the slant optical depth at the boundaries of each layer ( ) , or simply the change of variable then becomes where is the impact parameter whose slant optical depth equals  .the effective radius then becomes   one advantage to choosing as a reference , is that is positive over half of the range of slant optical depth within a layer , and negative over the other half , so that the integral in the third term of equation [ genint ] is close to zero .combined with the fact that the factor in front of the third term is a factor of larger than the second term , and that this factor is generally much smaller than one , the third term is completely negligible .we can then use equation [ results ] to solve the integral in the second term , and obtain \notag \\  & \approx b_s^2 + 2 \sum_{l = 0}^{m-1 } ( b_{l+1 } - b_l ) b_{0l } \left ( 1 - \left <  e^{-\tau } \right>_l \right ) \end{aligned}\ ] ] where is the atmospheric transmission averaged over the atmospheric layer . the effective atmospheric thicknessis then given by \notag \\  & \approx \sum_{l = 0}^{m-1 } ( b_{l+1 } - b_l ) \left ( \frac{b_{0l}}{b_s } \right ) \left ( 1 - \left <  e^{-\tau } \right>_l\right )  \end{aligned}\ ] ] where is generally close to but larger than one .our generalized vertical integration scheme is applicable to arbitrary changes of the number density with impact parameter , whether it is due to a change in the mixture of types of opacity or change in density scale height . since it only requires a series of slant optical depths computed at various impact parameters , the complexity of the physics which is included in the transmission spectrum ( e.g. changing gravity with altitude , arbitrary temperature profile , collision - induced absorption , etc . )is only limited by the complexity of the radiative transfer code which computes the slant optical depths .we have derived analytical expressions to compute the effective radius , the effective atmospheric thickness , and the spectral modulation of transiting exoplanets for any source of opacity ( rayleigh scattering , molecular absorption , and collision - induced absorption ) .our analytical expressions assume that the atmosphere has a constant density scale height , and ignore stellar limb darkening and second - order refractive effects , just as the analytical expressions by and do .however , unlike those previous expressions , our analytical expressions include the effects of a sharp change in flux ( surface " ) which exists on all planets and is due to an actual surface , an optically thick cloud deck , or a refractive boundary .we find that the spectral modulation of an exoplanet depends on the nature of the opacity source via its effect on the slant optical depth scale height of the atmosphere .we thus confirm that collision - induced absorption have half the effective scale height as rayleigh scattering .more importantly , we also find that the effective radius of an exoplanet is located at a slant optical depth which varies with the location of the surface " .for a clear atmosphere , our solution matches that of and .however , as the number density , or the slant optical depth , at the surface " decreases , the effective radius of the exoplanet asymptotically approaches the surface " without going below , unlike and s solutions .our analytical formalism explains many of the effects of clouds and refraction on transmission spectra that have been previously discussed through numerical simulations .it explains the changing slope of the rayleigh scattering feature with wavelength first noted by for refraction. it also explains why clouds and refraction decrease the contrast of spectral features ( e. g. ; ; ; ) and the fact that the spectral modulation of the atmosphere increases with the abundance of chemical species .our formalism also shows that these effects are a common signature of all surfaces " , which create a sharp boundary below which stellar radiation is cut - off .we have also introduced the concept of a surface " cross section which is the minimum mean atmospheric cross section of a homogeneous atmosphere which can be observed given the location of the surface " .this allows us to quickly determine the expected spectral modulation of a species , with less than 0.5 scale height error , given the species s abundance and its peak in cross section . recently proposed several indices linked to the spectral modulation of sodium , potassium , and water features to quantify the cloudiness of exoplanets .however , he also points out that they lack universality because each of them are wavelength - specific , and may be affected differently by clouds , presumably because of opacity differences with wavelength .our concept of a surface " cross section is more general precisely because it is wavelength - independent .although much of the discussions in this paper assumes a constant slant optical depth scale height atmosphere , we also developed a numerical recipe to generalize our formalism to atmospheres with non - 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Abstract: although the formalism of is extremely useful to understand what shapes transmission spectra of exoplanets , it does not include the effects of a sharp change in flux with altitude generally associated with surfaces and optically thick clouds . recent advances in understanding the effects of refraction in exoplanet transmission spectra have , however , demonstrated that even clear thick atmospheres have such a sharp change in flux due to a refractive boundary . we derive a more widely applicable analytical formalism by including first - order effects from all these surfaces " to compute an exoplanet s effective radius , effective atmospheric thickness , and spectral modulation for an atmosphere with a constant scale height . we show that the effective radius can not be located below these surfaces " and that our formalism matches s in the case of a clear atmosphere . our formalism explains why clouds and refraction reduce the contrast of spectral features , and why refraction decreases the rayleigh scattering slope as wavelength increases , but also shows that these are common effects of all surfaces " . we introduce the concept of a surface " cross section , the minimum mean cross section that can be observed , as an index to characterize the location of surfaces " and provide a simple method to estimate their effects on the spectral modulation of homogeneous atmospheres . we finally devise a numerical recipe that extends our formalism to atmospheres with a non - constant scale height and arbitrary sources of opacity , a potentially necessary step to interpret observations .   [ firstpage ]  atmospheric effects methods : analytical planets and satellites : atmospheres radiative transfer .


multivariate calculus and its applications to extremal problems are commonplace in undergraduate physics and mathematics . a function of variables is stationary at a point if all partial derivatives of that function vanish at that point : for . whether that point is a local maximum , minimum , or neither is determined by the symmetric second - derivative matrix at that point . thus , the problem of locating the stationary points of a function is equivalent to the problem of solving equations in variables , and can be tackled analytically or numerically .a complementary approach for locating the stationary points is to look at the global landscape of the function in .this approach is typically emphasized  only for when it is straightforward to obtain the graph ( ) or the surface plot ( ) of the function using ubiquitous software packages . graphical exploration of the function in the vicinity of a stationary point also complements the standard technique used to determine whether the stationary point is a maximum , minimum , or a saddle point .however , this process becomes increasingly cumbersome when and is therefore not emphasized in such cases .the hamilton principle  ( also known as the principle of least action  ) is an integral part of advanced undergraduate physics .this principle states that for a particle in one dimension , among all world - lines that connect two specified events and , the hamilton action =\int_{t_1}^{t_2 } l(x(t),\dot{x}(t),t ) dt\ ] ] is stationary for the classical world - line that obeys newton s second law . here , is the lagrangian that , in general , depends upon the position and velocity of the particle , and may also explicitly depend upon time .since the action  ] is the dimensionless time variable .since the lagrangian for this system is , the action for the classical world - line is given by = s_{fp}[1 + 2a - a^2/3] ] where corresponds to the classical world - line for any .note that as increases the time the particle spends in the region increases ; hence the action for the trial path decreases and becomes negative when .( b ) the right panel shows the dimensionless action as a function of the trial world - line parameter for various values of .for ( dotted line and solid line ) , we see that the action has a minimum value of + 1 at , whereas for ( dashed line ) , the action has a minimum value of at . in each case , the action increases quadratically as deviates from the classical world - line value .,width=302 ]   ( color online)(a ) the left panel shows trial world - lines for a uniformly accelerated particle $ ] where corresponds to the classical world - line for any .note that as increases the time the particle spends in the region increases ; hence the action for the trial path decreases and becomes negative when .( b ) the right panel shows the dimensionless action as a function of the trial world - line parameter for various values of . for ( dotted line and solid line ), we see that the action has a minimum value of + 1 at , whereas for ( dashed line ) , the action has a minimum value of at . in each case, the action increases quadratically as deviates from the classical world - line value .,width=302 ]   ( color online ) typical contour plot of the action , eq.([eq : actionalphabeta ] ) , with trial world - lines for a uniformly accelerated particle with .the action shows a clear minimum at the classical world - line .we note that as the value of is increased , the anisotrophy in the - plane in the vicinity of the classical world - line increases.,width=453 ]   ( color online ) typical plots of the action for a simple harmonic oscillator for trial world - lines , eq([eq : shoomega ] ) .the action is plotted in units of absolute value of the classical action and the frequency is in units of the natural frequency of the system . for all values of and , the action shows the minimum at . for a fixed as increased from ( solid line ) to ( dotted line ) , is reduced , and hence the action minimum changes its value from + 1 ( solid line ) to ( dotted line).,width=453 ]

Abstract: the hamilton action principle , also known as the principle of least action , and lagrange equations are an integral part of intermediate and advanced undergraduate mechanics . although the hamilton principle is oft stated as the action for any nearby trial world - line is greater than the action for the classical world - line " , the landscape of action in the space of world - lines is rarely explored . here , for three common problems from introductory physics - a free particle , a uniformly accelerating particle , and a simple harmonic oscillator - we present families of trial world - lines , characterized by a few parameters , that evolve continuously from their respective classical world - lines . with explicit analytical expressions available for the action , they permit a graphical visualization of the action landscape in the space of nearby world - lines . although these trial world - lines form only a subset of the space of all nearby world - lines , they provide a pedagogical tool that complements the traditional lagrange equation approach and is well - suited for advanced undergraduate students .