A radiometric model for demonstration of exoplanets detection by the transit method

This work describes an exact radiometric model for experimental demonstrators of the detection of exoplanets by the transit method. This model generalises the calculation of the depth of occultation from the standard transit method to the case of a finite size demonstrator apparatus. Results show that, for demonstrator apparatuses of moderately small sizes, a significant accuracy improvement in the determination of the size of a planet model can be achieved using the proposed method in comparison to using the formula from the standard transit method. The radiance distribution of the star model is found to be of crucial importance, as deviations from a Lambertian radiance distribution can lead to significantly different results.


Introduction
Since the discovery of the first exoplanet orbiting around a Sun-like star in 1995 [1], the search for exoplanets became one of the most active topics in Astronomy and Astrophysics, having led to the discovery of nearly 5550 exoplanets so far, as of 3 December 2023 [2].Research on exoplanets at different stages of evolution provides clues for understanding the dynamics of planetary formation [3].Additionally, exoplanets found in habitable zones are potential candidates in the search for extraterrestrial life.
Several classroom activities have been proposed to highlight the basic physical principles involved in the detection of exoplanets through several methods.Some of these activities use real or synthetic orbital data to allow students to calculate exoplanet mass, radius, and density [4][5][6].In other cases, an experimental approach is adopted through demonstrator apparatuses which allow students to conduct real measurements and determine the size of exoplanet models, using the transit [7][8][9] and radial velocity [10,11] methods.
So far, works on demonstrators of the transit method for detection of exoplanets [7][8][9] had relied on the formula used by astronomers to determine the size of real exoplanets.However, this formula assumes that the distances from the point of observation to the exoplanet and its host star are much larger than their sizes, which is not true for a demonstrator of moderately small size.Moreover, real stars are, to a good approximation, Lambertian light emitters, which cannot be absolutely ascertained for a star model used in a demonstrator.
In this work, we describe a radiometric model for demonstrators of the transit method which takes properly into account the finite distances involved.We also report the importance of the radiance distribution of the star model by analysing a second radiometric model where the radiance has a non-Lambertian distribution.

The standard transit method
In the standard transit method [12], the size of an exoplanet can be estimated from the variation in irradiance (E; unit: W m −2 ) at the observing telescope caused by the partial occultation of the host star during the planet transit.Let R p and R s be the radii of the exoplanet and its host star, respectively.Let E and E o , be the irradiance values originated from the star under normal conditions and maximum partial occultation, respectively.Apart from limb-darkening effects [13], a real star can be considered a blackbody, and as such has a Lambertian radiance.Then, because the astronomical observation distances of the star (D s ) and planet (D p ) are much larger than their dimensions, and also larger than the planet's orbital radius, the relative change in irradiance, also known as depth of occultation, can be directly related to the change in the star's apparent radiant disc area, giving: In the case of a demonstrator apparatus, however, the observation distances are of the same order of magnitude as, or just slightly larger than, the dimensions of the star and planet models used.Therefore, equation ( 1) is generally only a crude approximation in this context.Moreover, star models are real light sources that may not, in general, have a Lambertian radiance.

Radiometric models of a demonstrator apparatus
Describing the partial occultation produced by the transit of a planet model on its host star model in a demonstrator apparatus is a complex problem that can only be addressed by making several simplifying assumptions.

Assumption 1.
The orbit of the planet model and the line of observation are confined to the equatorial plane of the host star model.Due to increased complexity, calculating the depth of occultation for arbitrary orientation of the planet's orbit or observation line is beyond the scope of this work.Assumption 2. The relative sizes and distances between the planet and star models, as well as their proximities to the detector, are arranged in such a way that the resulting occultation is only partial.Otherwise, if the occultation was complete, apart from any stray light contribution, irradiance would vanish, E o = 0, resulting in ∆E/E = 1, which is insensitive to the planet's model size.
Assumption 3.Both the star and planet models are spherical, as significant deviations from spherical shape render the problem extremely complex.Assumption 4. The star model emits radiation with a predefined and known radiance distribution.In the following subsections, we will consider two special types of radiance distributions, viz Lambertian and locally isotropic distributions.For a more complex type of light emitter, a full experimental radiometric characterisation would be needed.
Given the axial symmetry with respect to the line joining the centres of the star and detector (see figure 1), the relevant surface elements contributing to irradiance at the detector are infinitesimal spherical segments of width R s dθ and perimeter 2π R s sin θ.The corresponding area is therefore dA s = 2π R 2 s sin θdθ.We denote by r (θ) the distance of the surface element to the detector and by β the incidence angle.These two quantities are given by: Figure 2 depicts the partial occultation caused by the transit of a planet model when it crosses the line of sight of its host star model.The maximum angles of view of the star (θ s, max ) and planet (θ p,max ) models are given by θ s,max = acos (c s ) and θ p,max = acos (c p ), respectively, where c s ≡ R s /D s and c p ≡ R p /D p .The light reaching the detector is emitted by the spherical segment defined by the angular range θ s,min ⩽ θ ⩽ θ s,max , with θ s,min , the star's model minimum angle of view, satisfying the condition: Solving equation ( 3) for cos θ s,min , gives: This result is valid under the condition c p ⩽ c s , which also ensures that θ s,min ⩽ θ s,max .A useful expression for c 2 p can be obtained from equation ( 4): The total irradiance under non occultation conditions (E) can be derived by integrating a proper radiometric function f (u) of a variable u (θ) from u (0) up to u(θ s,max ).As for the total irradiance under partial occultation (E o ), the integration limits are u(θ s,min ) and u(θ s,max ).Therefore, the relative change in irradiance caused by the star's model partial occultation can be expressed as This result can be further detailed invoking the antiderivative To determine the radiometric function, we will assume that the star model is a spherical light source with radiance L (unit: W m −2 sr −1 ).By definition, radiance is the radiant flux or luminosity (ϕ ; unit: W) per unit projected area of the light source and per unit solid angle at the detector: Here, dA proj s = dA s cos α with α = θ + β (see figure 1) and dΩ d = dA d cos β/r 2 , where dΩ d and dA d are the infinitesimal solid angle and area at the detector.Since irradiance at the detector is the radiant flux per unit detector area, the contribution of a source surface element dA s to irradiance is given by: To proceed with the calculation of the radiometric function, we need to specify the angular dependence of the star's model radiance.

Lambertian radiance
In this model, we assume that the star model behaves approximately like a blackbody, as a real star does [13], and therefore can be considered a Lambertian emitter.A Lambertian emitter has a uniform radiance across its surface given by L = M s /π , where M s is the emittance (radiant flux per unit source area; unit: W m −2 ), also known as brightness.As the star's model luminosity is given by ϕ s = M s 4π R 2 s , the irradiance on a surface oriented perpendicularly to the incident light at a distance D s can be written as Under partial occultation, irradiance is reduced by an amount ∆E corresponding to the light emitted from the angular range θ ⩽ θ s,min and whose propagation is obstructed by the planet model.This quantity can be easily calculated from equation (10) considering that it also corresponds to the irradiance that would be produced by the planet model if it was a Lambertian emitter with radiance L: Accordingly, we get for the depth of occultation the expression: In the limit of astronomic distances (D s , D p ≫ R s , R p ), we have D s /D p → 1, and therefore equation ( 12) reduces to equation (1), as expected.This result has a simple interpretation, as the quantity R ′ s = (D p /D s ) R s represents the projected equivalent star's model radius at distance D p .We can similarly define the quantity R ′ p = (D s /D p ) R p as the projected equivalent planet's model radius at distance D s .In terms of these quantities, equation ( 12) assumes a generalized version of equation (1): To confirm this result using the formalism leading to equation ( 9), we can start by detailing the expression for the radiance: Combining equations ( 9), ( 14), (2a and b), we obtain, through algebraic and trigonometric simplifications, where By making the change of variable u = 1 − b cos θ in equation (15), we get the Lambertian radiometric function: The corresponding antiderivative is given by, up to a constant: This allows us to calculate the total irradiation produced at the detector by the star model under non occultation conditions, using the limits which is the expected result for an isotropic spherical light source.Combining equations ( 7), (18) and using u(θ s,min ) = 1 − b cos θ s,min , the relative change in irradiance under partial occultation is given by Taking into account equation ( 5), we get the final result which confirms equations ( 12) and ( 13): Provided the values of R s , D s and D p are known and the depth of occultation ∆E/E is measured experimentally, an estimate for the size, R p , of the planet model can be obtained from equation (21):

Locally isotropic radiance
By symmetry, the irradiance from a uniform spherical source depends only on the sourcedetector distance, through the inverse-square law.This does not mean, however, that every surface element of the exposed hemisphere has the same contribution to irradiance at the detector and emits isotropically.Actually, it is the sum of all contributions that is independent of the source-detector direction.Therefore, isotropic irradiation from a uniform spherical source can be observed even for locally anisotropic emission, as was exemplified for a spherical Lambertian source (see equation ( 19)).In this section, we consider instead a locally isotropic model, in which every area element of the star's model surface emits light isotropically over the hemisphere above its tangent plane.This radiometric model is used only for the purpose of assessing to what extent different radiance models can affect the depth of occultation.The star's model brightness is given by ) and any infinitesimal surface element of the star model contributing to irradiance at the detector is an isotropic hemispherical emitter.The contribution of such surface element of area dA s to irradiance is therefore given by: It should be pointed out that this equation can also be derived directly from equation ( 14) by assuming the relation L = M s / (2π cos α), which, besides allowing a crucial simplification for the determination of the radiometric function, has no otherwise obvious meaning.Substituting the expression of M s in equation ( 23), and opting this time for using no change of integration variable, we obtain the radiometric function, along with its antiderivative, Proceeding as in section 3.1, the same result of equation ( 19) for the total irradiance under non occultation conditions is recovered, whereas the depth of occultation for the current model is given by: In the limit of astronomical distances, equation (25) reduces to which clearly differs from equation (1).Even for a small planet to star size ratio (R p /R s → 0), we get ) .The numerical determination of R p according this model can be done using a three-step procedure.First, equation ( 25) is solved for cos θ S,min using the quadratic formula and by selecting its positive root.Then c p is calculated using equation ( 5) and finally R p is determined from the values of c p and D p .

Model comparison results
In figure 3, we show the dependence of the depth of occultation on the ratio D p /R s , according to the standard transit model, and the demonstrator Lambertian and locally isotropic models.It can be seen that, for Dp Rs < 6, both demonstrator models predict significantly larger values of ∆E/E than the standard transit model.For sufficiently large values of D p /R s the standard transit model becomes an increasingly good approximation for the case of a demonstrator Lambertian star model.In the case of a locally isotropic star model, however, the standard transit model still presents a large asymptotic error, as can be inferred from equation (26).Interestingly, as shown in figure 3, there exists a specific value of D p /R s ≈ 7.2 for which this model yields the same result for the depth of occultation as the standard transit method.Furthermore, numerical calculations indicate that this value of D p /R s slightly increases with the ratio R p /R s .
To further illustrate the differences between the three models, we plot in figure 4 the corresponding variations of the occultation depth with 2 regardless of the value of D p /R s .Therefore, testing the validity of this model for a real demonstrator would be extremely challenging.
For a Lambertian star model, the relative error in the determination of R p using the standard transit model can be obtained combining equations (1 and 22): It should be noticed that this result can also be derived using simple geometric arguments, as illustrated in figure 5, considering ε Rp,L ≡ ∆R p /R p , where ∆R p = R ′ p − R p represents the difference between the projected equivalent planet's model radius at distance D s and the true planet's model radius.According to equation ( 27), the relative error in R p is exactly equal to the relative (positive) difference between the distances D s and D p , and therefore is independent of R p .Moreover, for a given value of D s , the quantity ε For R s = 5 cm and R p = 1 cm, e.g. to ensure an error bound of 1% (x = 0.01), a distance D p = 6 m would be required.It is clear, therefore, that it is inadequate to determine R p using the standard transit model unless an exceedingly large demonstrator is a viable option.
For a locally isotropic star model, the relative error in the determination of R p using the standard transit model, ε Rp, LI , can be numerically calculated using equations (1 and 25).In figure 6, we show the variation of the relative errors ε Rp, L and ε Rp, LI with the ratio R s /D p , for different values of R p /R s .It is noteworthy that, for fixed R s , unlike Rp, L , ε Rp, LI depends on R p and exhibits a nonlinear dependence on D −1 p , particularly for low values of D p .Consistent with the earlier discussion on the depth of occultation, the value of D p for which ε Rp, LI = 0 increases with R p .
Since both radiometric models introduced in this work provide significantly different results for the occultation depth, except under full occultation conditions, an accurate determination of the size of a planet model is strongly dependent on the degree of approximation of the star model to a Lambertian emitter.

Conclusion
A real demonstrator of the detection of exoplanets by the transit method is an excellent tool to attract and involve students in a classroom interactive activity.It is perhaps one of few cases where astronomical observations can be effectively simulated through a real experiment.Students are able not only to fully understand this method but also can apply it to estimate the size of an exoplanet model.
In this work, we showed that estimating R p using the standard transit method, which was originally developed in the context of real astronomical observations, leads, in general, to a large error.Minimising this error requires using a demonstrator apparatus of large dimensions, which may be a strong limitative factor for a classroom activity.
To cope with this difficulty, we developed an exact radiometric model describing the transit of an exoplanet across a Lambertian star.This model corrects the relation between the planet size and the depth of occultation of the standard transit method by considering finite distance effects.Such improvement allows the accurate determination of R p even for a moderately small exoplanet transit demonstrator.Furthermore, measuring the depth of occultation as a function of the planet's model radius allows the testing of the validity of this model.However, as the second developed radiometric model showed, the radiative properties of the light source used as star model play an important role, since deviations from Lambertian emission can lead to significantly different results.In practice, this problem can be minimised provided a commercial Lambertian spherical light source can be found to serve as star model.Otherwise, a radiometric characterisation of the star model may be needed.

Figure 1 .
Figure 1.Diagram depicting the geometric elements involved in the calculation of the irradiance at the detector originated by an (unocculted) infinitesimal spherical segment of the star's model surface.

Figure 2 .
Figure 2. Schematic representation of the partial occultation caused by the transit of a planet model when it crosses the line of sight of its host star model.Angles θ s,min and θs,max delimit the spherical segment of the surface of the star model radiating all light received at the detector.Angle θp,max defines the planet model maximum angle of view from the detector.

Figure 3 .
Figure 3. Variation of the occultation depth with Dp/Rs for the standard transit model, and the demonstrator Lambertian and locally isotropic models.For the selected values of Rp/Rs = 0.2 and Ds/Rs = 10, the region of full occultation is defined for Dp/Rs ⩽ 2.

Figure 4 .
Figure 4. Variation of the occultation depth with (Rp/Rs) 2 for Ds/Rs = 20 and several values of Dp/Rs, according to the standard transit model, and the demonstrator Lambertian (L) and locally isotropic (LI) models.Each LI curve intersects its corresponding L line at their common full occultation point.(Rp /R s )2 for different values of D p /R s .As expected from equation (1), the standard transit method yields a straight line with unit slope, regardless of the value of D p /R s .The Lambertian model generates a set of straight lines whose slopes, as predicted by equation (21), are given by (D s /D p ) 2 and therefore approach unit as D p /R s increases.This suggests that measuring the occultation depth as a function of the planet's model radius in a real demonstrator can provide an excellent test of the validity of the Lambertian model.Regarding the locally isotropic model, the occultation depth changes nonlinearly with (R p /R s )2 regardless of the value of D p /R s .Therefore, testing the validity of this model for a real demonstrator would be extremely challenging.For a Lambertian star model, the relative error in the determination of R p using the standard transit model can be obtained combining equations (1 and 22):

Figure 5 .
Figure 5. Geometric derivation of ε Rp, L based on the difference ∆Rp between the projected equivalent planet's model radius at distance Ds (R ′ p ) and the true planet's model radius (Rp).

Figure 6 .
Figure 6.Dependence of the relative errors ε Rp, L and ε Rp, LI on the ratio Rs/Dp, for different values of Rp/Rs.Note that all curves of ε Rp, L overlap within common ranges of Rs/Dp, resulting in a continuous linear trend.Each ε Rp, LI curve intersects the ε Rp, L line at the corresponding full occultation point.