Paris Observatory Entrance Exam Set

The question probes the understanding of the fundamental principles governing the observation and characterization of exoplanetary atmospheres, a core area of research at the Paris Observatory. The scenario describes a transit photometry observation of an exoplanet, where the planet passes in front of its host star. During this transit, starlight filters through the exoplanet’s atmosphere. Different chemical species in the atmosphere absorb specific wavelengths of light, causing a dip in the observed stellar spectrum at those wavelengths. This wavelength-dependent dimming is the signature of atmospheric composition. The key concept here is transmission spectroscopy. When the planet transits, the starlight that reaches us has passed through the exoplanet’s atmosphere. If the atmosphere contains a molecule like water vapor, it will absorb photons at specific wavelengths characteristic of water vapor’s molecular transitions. This absorption will manifest as a deeper dip in the observed stellar flux at those particular wavelengths compared to wavelengths where the atmosphere is transparent. Therefore, by analyzing the variations in the transit depth across the electromagnetic spectrum, astronomers can infer the presence and abundance of various atmospheric constituents. The correct answer, therefore, is the detection of wavelength-dependent variations in the transit depth. This directly relates to how transmission spectroscopy works. The other options are plausible but incorrect in this context. Measuring the overall transit duration or the precise orbital period relates to orbital mechanics and planet size, not atmospheric composition. Similarly, analyzing the star’s radial velocity variations primarily reveals the star’s wobble caused by the planet’s gravitational pull, which is crucial for determining the planet’s mass, but not its atmospheric makeup. The intensity of the reflected starlight from the planet’s surface would be relevant for albedo studies, but transmission spectroscopy, as implied by the transit scenario, focuses on light passing *through* the atmosphere.

The question probes the understanding of observational biases in astronomical surveys, a core concept for aspiring researchers at the Paris Observatory. The scenario describes a survey targeting luminous, nearby galaxies. The key issue is that such a survey will inherently overrepresent massive galaxies and those with active star formation or galactic nuclei, as these are typically brighter and more easily detected. Conversely, it will underrepresent less luminous, fainter galaxies, including those that are smaller, have lower star formation rates, or are more distant but intrinsically faint. This selection effect, known as Malmquist bias (though not explicitly named, the concept is tested), means the observed sample is not representative of the general galaxy population. Therefore, inferring properties of the *entire* galaxy population from this biased sample would lead to an inaccurate picture, likely overestimating average stellar mass and star formation rates. The correct option must reflect this fundamental limitation of observational selection.

The question probes the understanding of observational biases in astronomical surveys, specifically concerning the detection of exoplanets via the transit method. The transit method relies on observing the periodic dimming of a star’s light as a planet passes in front of it. The depth of this dimming is directly proportional to the ratio of the planet’s radius to the star’s radius (\( \Delta F / F_0 \approx (R_p / R_s)^2 \)). Consequently, larger planets produce deeper transits, making them easier to detect. Furthermore, the duration of a transit is influenced by the planet’s orbital period and its impact parameter (how close to the stellar center the transit occurs). Shorter orbital periods mean more frequent transits, increasing the chances of detection within a given observation window. However, the question specifically asks about factors influencing the *detectability* of a planet, implying the signal strength and frequency. A planet with a larger radius will cause a deeper dip in stellar flux, thus a stronger signal. A shorter orbital period means more opportunities to observe a transit, increasing the probability of catching one. The combination of a larger radius and a shorter orbital period leads to a higher probability of detection. While stellar activity can introduce noise, and the inclination of the orbit is crucial for transits to occur at all, the intrinsic properties of the planet and its orbit that directly enhance the signal-to-noise ratio and observation frequency are its size and orbital period. Therefore, a planet that is both larger and orbits more quickly will be more readily detectable through the transit method.

The question probes the understanding of observational biases in astronomical surveys, specifically concerning the selection of targets for detailed study. When a survey aims to characterize a specific population of celestial objects, such as exoplanet-hosting stars, but employs a detection method that is inherently more sensitive to certain properties (e.g., stellar mass, orbital period), the resulting sample will not be representative of the true underlying distribution. This leads to a selection bias. For instance, radial velocity surveys are more sensitive to massive planets in close orbits. If a researcher then uses this biased sample to infer general properties of exoplanetary systems, their conclusions will be skewed. The concept of “completeness” in surveys is crucial here; a survey is complete for a given set of parameters if it is guaranteed to detect all objects meeting those parameters. However, most astronomical surveys have varying completeness limits across their parameter space. Therefore, to obtain a more unbiased understanding of exoplanet demographics, it is essential to account for these detection biases by applying appropriate statistical corrections or by designing surveys with more uniform completeness. This is a fundamental consideration in astrophysical research, particularly at institutions like the Paris Observatory, which are at the forefront of exoplanet discovery and characterization. Understanding these biases is critical for drawing accurate scientific conclusions about planetary formation and evolution.

The question probes the understanding of observational biases in astronomical surveys, specifically concerning the detection of exoplanets. The transit method, a primary technique for exoplanet discovery, is sensitive to the orbital period and inclination of a planet relative to the observer’s line of sight. A planet must transit its host star for detection. This means its orbital plane must be aligned such that it passes in front of the star from our perspective. Planets with very short orbital periods (days to weeks) have a higher probability of transiting because they complete many orbits within a typical observational campaign. Conversely, planets with very long orbital periods (years or decades) are less likely to be observed during a transit, even if their orbital plane is favorably inclined, simply due to the limited duration of most surveys. Furthermore, the transit depth, which is proportional to the square of the planet’s radius divided by the square of the star’s radius (\(\Delta F/F \approx (R_p/R_*)^2\)), is a key factor in detectability. However, the question focuses on the *frequency* of detection based on orbital characteristics, not the signal strength of a single detection. Therefore, while planet size and stellar type influence the *signal-to-noise ratio* of a transit, they don’t inherently bias the *probability* of a planet having an orbit that *allows* for transit detection in the same way orbital period and inclination do. The inclination of the orbit is crucial; an inclination of exactly 90 degrees (edge-on) is required for a transit. Planets with inclinations far from 90 degrees will not transit. Shorter orbital periods mean more opportunities to observe a transit within a given time frame. Thus, surveys are inherently biased towards detecting planets with shorter periods and favorable inclinations. The explanation emphasizes that the transit method’s efficacy is intrinsically linked to the geometric alignment and the temporal frequency of orbital events. This leads to an overrepresentation of short-period planets in detected exoplanet populations, a well-documented selection effect in exoplanet demographics.

The question probes the understanding of observational biases in astronomical surveys, specifically concerning the detection of exoplanets via the transit method. The transit method relies on detecting the slight dimming of a star’s light as a planet passes in front of it. The magnitude of this dimming is directly proportional to the ratio of the planet’s radius to the star’s radius, and inversely proportional to the square of the star’s radius (since the star’s flux is proportional to its surface area, \(F \propto R_{star}^2\)). The depth of the transit, \(\Delta F / F\), is approximately \( (R_{planet}/R_{star})^2 \). Therefore, for a fixed planet size, a smaller star will produce a deeper, more easily detectable transit signal. Conversely, for a fixed transit depth, a smaller star would imply a smaller planet. However, the question asks about the *detection probability* given a fixed planet size. A smaller star has a larger angular size for a given orbital period and inclination, meaning the planet’s orbital path is more likely to intersect the star’s disk from our perspective. Furthermore, the signal-to-noise ratio of the transit is higher for smaller stars because the intrinsic stellar variability is often lower, and the relative dimming is more pronounced. Therefore, surveys are biased towards detecting planets around smaller, dimmer stars.

The question probes the understanding of observational biases in astronomical surveys, a critical concept for any aspiring researcher at the Paris Observatory. The scenario describes a survey targeting exoplanets using the transit method, which inherently favors the detection of planets with specific orbital characteristics. Planets with shorter orbital periods are more likely to transit their host stars within the observation window of a given survey. Furthermore, planets with larger radii produce a more significant dip in stellar brightness, making them easier to detect. The transit method is also sensitive to the planet’s orbital inclination; only those transiting within a narrow range of angles relative to our line of sight are observable. Therefore, a survey employing this method will naturally overrepresent planets that are closer to their stars (shorter periods), larger in size, and whose orbits are favorably aligned. The concept of “completeness” in astronomical surveys refers to the fraction of the true population of objects that are detected. In this case, the survey’s methodology limits its completeness for planets with longer orbital periods, smaller radii, and less favorable inclinations. The bias towards detecting planets with shorter orbital periods and larger radii is a direct consequence of the transit method’s sensitivity. This is not a calculation but a conceptual understanding of how observational techniques shape our perception of astrophysical populations.

The question probes the understanding of observational biases in astronomical surveys, specifically concerning the detection of exoplanets via the transit method. The transit method relies on observing the dip in a star’s brightness as a planet passes in front of it. The magnitude of this dip is directly proportional to the ratio of the planet’s radius to the star’s radius, \( \frac{R_p}{R_s} \). A larger planet relative to its star will produce a deeper, more easily detectable dip. Conversely, a smaller planet will cause a shallower dip. Therefore, surveys employing the transit method are inherently biased towards detecting larger exoplanets, as their transits are more pronounced and less likely to be masked by stellar variability or instrumental noise. This bias is a fundamental consideration in interpreting the demographics of exoplanet populations derived from such surveys. The Paris Observatory, with its significant contributions to exoplanet research and instrumentation, emphasizes the critical need for candidates to understand these observational limitations.

The question probes the understanding of how observational biases can affect the interpretation of cosmological data, specifically in the context of the Paris Observatory’s research in extragalactic astronomy and cosmology. The core concept is the Malmquist bias, a selection effect that arises when the observed sample of objects is flux-limited. Brighter objects are detectable at greater distances than fainter objects. Consequently, a flux-limited sample will tend to be biased towards intrinsically brighter objects at larger distances. This means that the average luminosity of objects in a distant subsample will be higher than the average luminosity of objects in a nearer subsample, even if the underlying population of objects has a uniform luminosity distribution. This can lead to an overestimation of the cosmic expansion rate (Hubble constant) if not properly accounted for. For instance, if one were to analyze a sample of quasars selected based on their apparent brightness, the more distant quasars would necessarily be intrinsically more luminous than the nearer ones simply to be detectable. This systematic overestimation of luminosity at greater distances can skew distance modulus calculations and, consequently, estimates of cosmological parameters. Understanding and mitigating such biases is crucial for the precise measurements of cosmic distances and the evolution of the universe, a key area of study at the Paris Observatory.

The question probes the understanding of observational biases in astronomical surveys, specifically concerning the detection of exoplanets. The transit method, a primary technique for exoplanet discovery, is sensitive to orbital inclination. An exoplanet’s transit is only observable if its orbital plane is aligned with the observer’s line of sight. Therefore, surveys employing the transit method will inherently undercount planets with orbital inclinations significantly deviating from \(90^\circ\) (edge-on). This leads to a bias where the detected population of exoplanets is skewed towards those with near-perfect alignment. Consequently, the inferred frequency of exoplanets, particularly those with wider orbits or those that might not transit at all due to their inclination, will be an underestimate of the true population. This is a fundamental concept in exoplanet demographics and is crucial for interpreting survey results at institutions like the Paris Observatory, which actively contributes to exoplanet research. The bias is not related to the planet’s mass, luminosity, or the star’s spectral type directly, but rather to the geometric probability of observing a transit.

The question probes the understanding of observational biases in astronomical surveys, specifically concerning the detection of exoplanets. The transit method, a primary technique for exoplanet discovery, is sensitive to the orbital period and inclination of a planet. A planet with a very long orbital period (e.g., decades or centuries) would require an exceptionally long observation baseline to detect even a single transit. Furthermore, for a transit to be observable from Earth, the planet’s orbit must be nearly edge-on with respect to our line of sight. If the orbital inclination deviates significantly from 90 degrees, the planet will not transit its host star, or the transit will be partial and potentially missed. Therefore, surveys designed to detect transiting exoplanets are inherently biased against systems with very long orbital periods and those whose orbital planes are not favorably aligned. This bias means that the exoplanet populations detected by such surveys may not accurately represent the true distribution of exoplanet properties in the galaxy. The Paris Observatory, with its significant contributions to exoplanet research, emphasizes the critical need to understand and account for these observational limitations when interpreting survey results and formulating theories about planet formation and evolution.

The question probes the understanding of how observational biases can affect the interpretation of astronomical data, specifically in the context of exoplanet detection. The transit method, a primary technique for finding exoplanets, is more sensitive to planets that orbit close to their stars and have larger radii relative to their host star’s size. This is because closer orbits result in more frequent transits, and larger planets create a more significant dip in stellar brightness during transit, making them easier to detect. Therefore, a survey employing the transit method would inherently over-represent planets with shorter orbital periods and larger physical sizes. This selection effect means that the observed distribution of exoplanet properties might not accurately reflect the true underlying distribution in the galaxy. For instance, the prevalence of “hot Jupiters” (gas giants in close orbits) in early transit surveys was a direct consequence of this observational bias, rather than an indication that such planets are overwhelmingly common. Understanding these biases is crucial for developing accurate models of planet formation and evolution, a core area of research at institutions like the Paris Observatory. Recognizing that the observed exoplanet population is a filtered version of the true population allows for more robust statistical inferences and a deeper comprehension of planetary system architectures.

The question probes the understanding of how observational biases can affect the interpretation of astronomical data, specifically concerning the detection of exoplanets. The primary bias in transit photometry, the most successful exoplanet detection method, is the “edge-on orbit” bias. This is because a transit can only be observed if the planet’s orbit is aligned such that it passes directly in front of its host star from our perspective. Planets with orbits significantly inclined relative to our line of sight will not produce a detectable dip in stellar brightness. Consequently, surveys employing transit photometry are inherently more likely to discover planets with orbital inclinations close to 90 degrees. This leads to an underestimation of the true frequency of exoplanets with a wider range of orbital inclinations. The other options represent valid astronomical concepts but are not the primary bias directly linked to the *detection method* of transit photometry. Stellar activity (like flares) can introduce noise but doesn’t inherently bias the *detection* of planets with specific orbital parameters. The Doppler wobble method is sensitive to inclination but in a different way; it directly measures the radial velocity, and the derived mass is a minimum mass unless inclination is known. The atmospheric composition of exoplanets is studied *after* detection and doesn’t influence the initial detection bias. Therefore, the most significant bias in transit surveys is the preference for detecting planets with near-edge-on orbits.

The question probes the understanding of stellar evolution and the observational consequences of different stellar masses. A star with a mass of approximately 8 solar masses (\(M_\odot\)) is at a critical juncture in its life cycle. Such stars, upon exhausting their core hydrogen, will evolve through helium burning and potentially carbon burning stages. The core will eventually become dense enough to support electron degeneracy pressure. When the core mass exceeds the Chandrasekhar limit (approximately \(1.44 M_\odot\)), it can no longer support itself against gravitational collapse. For stars in the 8 \(M_\odot\) range, the core collapse will trigger a Type II supernova. This event expels the outer layers of the star into space, enriching the interstellar medium with heavier elements. The remnant left behind after a Type II supernova is typically a neutron star, a highly dense object composed primarily of neutrons. White dwarfs, on the other hand, are the remnants of lower-mass stars (up to about 8 \(M_\odot\)) that undergo a gentler process of mass loss and do not experience core collapse supernovae. Black holes are formed from the collapse of even more massive stars, where the remnant core’s mass exceeds the Tolman-Oppenheimer-Volkoff limit (around 2-3 \(M_\odot\)), leading to a gravitational collapse that even neutron degeneracy pressure cannot halt. Therefore, a star of 8 \(M_\odot\) is most likely to end its life as a neutron star after a Type II supernova. This understanding is fundamental to astrophysics and directly relates to the study of nucleosynthesis and the formation of compact objects, key areas of research at the Paris Observatory.

The question probes the understanding of observational biases in astronomical surveys, specifically concerning the detection of exoplanets. The transit method, a primary technique for exoplanet discovery, is sensitive to the orbital inclination of a planet relative to the observer’s line of sight. For a transit to be observable, the planet’s orbit must be edge-on from Earth’s perspective. Planets with orbits significantly inclined or face-on will not produce a detectable dip in stellar brightness. Therefore, surveys employing the transit method inherently favor the detection of planets with shorter orbital periods and those whose orbits are nearly coplanar with the star’s equator, as these are more likely to present an edge-on configuration. This leads to a selection effect where planets with larger orbital separations (and thus longer orbital periods) are statistically less likely to be detected by transit surveys, even if they are abundant. The concept of “completeness” in a survey refers to the fraction of the true population of objects that are detected. A transit survey’s completeness is significantly reduced for planets with long orbital periods because the probability of their orbital plane aligning with our line of sight decreases with increasing orbital semi-major axis. This is a fundamental limitation that influences our understanding of exoplanet demographics.

The question probes the understanding of how observational biases can affect the interpretation of astronomical data, particularly in the context of exoplanet detection. The radial velocity method, a primary technique for finding exoplanets, is more sensitive to massive planets orbiting close to their stars. This is because massive planets exert a stronger gravitational pull, inducing a larger and more easily detectable wobble in the star. Furthermore, planets in closer orbits complete their orbits faster, leading to more frequent observations of the stellar wobble, thus increasing the statistical significance of the detection. Therefore, a survey employing the radial velocity method would inherently be biased towards discovering massive planets with short orbital periods. This bias is crucial to acknowledge when assessing the demographics of exoplanetary systems and avoiding overgeneralizations about planet formation and evolution. Understanding such biases is fundamental for any aspiring researcher at the Paris Observatory, as it underpins the rigorous interpretation of observational results and the design of future, more comprehensive surveys.

The question probes the understanding of observational biases in astronomical surveys, specifically concerning the detection of exoplanets. The transit method, a primary technique for exoplanet discovery, is sensitive to the orbital period and inclination of a planet relative to the observer’s line of sight. Planets with orbital periods that are too long will not complete a full orbit within the observation window of a survey. Similarly, planets whose orbital planes are significantly inclined relative to our line of sight will not transit. Therefore, surveys that observe for a limited duration are inherently biased against detecting exoplanets with longer orbital periods. This bias is a fundamental consideration in interpreting the demographics of exoplanetary systems and designing future observational strategies. The Paris Observatory, with its focus on observational astronomy and exoplanet research, would expect candidates to grasp these nuances of detection methodologies and their implications for scientific conclusions. Understanding these limitations is crucial for developing more comprehensive and unbiased exoplanet catalogs, which is a key objective in modern astrophysics.

The question probes the understanding of stellar evolution and the observational consequences of different stellar masses. A star with a mass of \(1.5 \times 10^{31}\) kg is significantly more massive than our Sun (approximately \(2 \times 10^{30}\) kg). Such a massive star, upon exhausting its core hydrogen, will evolve rapidly through its main sequence and subsequent stages. Unlike lower-mass stars that might end their lives as white dwarfs or neutron stars, stars in this mass range (typically above 8-10 solar masses) are destined to undergo a Type II supernova. This explosive event is characterized by the core collapse of the star, leading to the formation of a compact remnant. Given the initial mass, the most likely remnant after a supernova is a neutron star. A black hole typically forms from even more massive progenitors (above approximately 20-25 solar masses). A white dwarf is the remnant of stars with masses up to about 8 solar masses. The formation of a planetary nebula is characteristic of the end stages of low-to-intermediate mass stars, not massive stars undergoing core collapse. Therefore, the most appropriate outcome for a star with \(1.5 \times 10^{31}\) kg is the formation of a neutron star. This understanding is crucial for astrophysical studies conducted at institutions like the Paris Observatory, which investigate the life cycles of stars and the extreme physics of compact objects. The ability to predict stellar fates based on initial mass is a fundamental concept in stellar astrophysics.

The question probes the understanding of observational biases in astronomical surveys, specifically concerning the detection of exoplanets via the transit method. The transit method relies on observing the dip in a star’s brightness as a planet passes in front of it. The magnitude of this dip is directly proportional to the ratio of the planet’s radius to the star’s radius, \( \frac{R_p}{R_s} \). A larger planet relative to its star will produce a deeper, more easily detectable dip. Conversely, smaller planets, or planets orbiting smaller stars, will produce shallower dips. The probability of observing a transit from a given planetary system is also dependent on the orbital inclination, which must be close to \( 90^\circ \) for a transit to be visible from Earth. Systems with planets in orbits significantly inclined to our line of sight will not produce observable transits. Therefore, surveys that aim to detect exoplanets using this method are inherently biased towards finding planets that are larger relative to their host stars and systems with favorable orbital alignments. This means that the detected exoplanet population might not accurately represent the true distribution of planet sizes and orbital inclinations in the galaxy, as smaller planets or those in misaligned orbits are less likely to be found. This bias is crucial for interpreting the results of exoplanet surveys conducted by institutions like the Paris Observatory, which contribute to our understanding of planetary system formation and evolution.

The question probes the understanding of observational biases in astronomical surveys, specifically focusing on how the selection criteria for cataloging celestial objects can influence our perception of their distribution and properties. The Paris Observatory Entrance Exam emphasizes a deep conceptual grasp of astronomical methodologies and their inherent limitations. When considering the detection of exoplanets using the radial velocity method, a significant bias emerges: more massive planets orbiting closer to their host stars exert a stronger gravitational pull, leading to larger stellar radial velocity shifts. These larger shifts are easier to detect and measure accurately, especially with current instrumentation. Consequently, surveys employing this method are inherently more sensitive to massive planets in short-period orbits. This does not mean that low-mass planets or planets in wide orbits are absent; rather, they are less likely to be detected by such surveys. Therefore, a catalog compiled primarily through radial velocity measurements would disproportionately represent massive, close-in exoplanets, leading to an overestimation of their prevalence relative to other types of planetary systems. This understanding is crucial for interpreting survey results and developing more comprehensive detection strategies, aligning with the rigorous analytical approach fostered at the Paris Observatory.

The question probes the understanding of observational biases in astronomical surveys, specifically concerning the detection of exoplanets. The transit method, a primary technique for exoplanet discovery, is sensitive to the orbital inclination of a planet. For a transit to be observed, the planet’s orbit must be aligned such that it passes directly in front of its host star from the observer’s perspective. This alignment requirement introduces a selection effect. Planets with orbital inclinations significantly deviating from edge-on will not produce detectable transits, even if they exist. Therefore, surveys employing the transit method are inherently biased towards detecting planets with near-zero inclination relative to the line of sight. This bias means that the detected population of exoplanets may not accurately represent the true distribution of orbital inclinations in the galaxy. Understanding this bias is crucial for interpreting survey results and for developing more comprehensive exoplanet detection strategies that might mitigate such effects, such as radial velocity or direct imaging, which are less sensitive to inclination. The Paris Observatory, with its extensive research in exoplanetary science and observational astronomy, emphasizes the critical evaluation of observational methodologies and their inherent limitations.

The question probes the understanding of observational biases in astronomical surveys, specifically concerning the selection function of a hypothetical survey designed to detect exoplanets. The survey’s detection limit is defined by a minimum signal-to-noise ratio (SNR) of 5. The orbital period of an exoplanet is a crucial factor influencing the SNR achievable during transit observations. Shorter orbital periods generally lead to more frequent transits, allowing for more data points to be accumulated and averaged, thus increasing the effective SNR for a given observation time. Conversely, longer orbital periods mean fewer transits within a typical survey duration, making it harder to achieve a high SNR and increasing the likelihood of missing a detection, especially for planets with smaller radii or lower atmospheric transmission signals. Therefore, a survey with a fixed detection threshold will inherently be more sensitive to exoplanets with shorter orbital periods, as these planets provide more opportunities for robust signal confirmation. This bias towards shorter-period planets is a well-established characteristic of many exoplanet detection methodologies, including transit photometry and radial velocity measurements. Understanding this selection effect is fundamental for interpreting the demographics of exoplanet populations derived from surveys conducted by institutions like the Paris Observatory, which contribute significantly to this field. The ability to identify and account for such biases is a hallmark of rigorous astronomical research.

The question probes the understanding of observational biases in astronomical surveys, specifically concerning the detection of exoplanets. The transit method, a primary technique for exoplanet discovery, is sensitive to the orbital period and inclination of a planet relative to the observer’s line of sight. A planet must transit its host star for detection. This means its orbital plane must be aligned within a certain range of angles with respect to our line of sight. Consider a star system with a planet in a circular orbit of radius \(R\). The probability of observing a transit depends on the angle \(i\) between the orbital angular momentum vector and the line of sight. For a transit to occur, the planet’s path must cross the star’s disk. This happens when the angle \(i\) is such that the projected distance of the planet from the star’s center, as seen from Earth, is less than or equal to the star’s radius \(R_\star\). For a circular orbit, the maximum angular deviation from a perfectly edge-on orbit (\(i = 90^\circ\)) that still results in a transit is related to the ratio of the star’s radius to the orbital radius. Specifically, the condition for a transit is approximately \(R_\star \approx R \sin i\). If we consider the range of inclinations that lead to a transit, it is roughly from \(i = 90^\circ – \arcsin(R_\star/R)\) to \(i = 90^\circ + \arcsin(R_\star/R)\). However, the probability distribution of orbital inclinations for randomly oriented systems is uniform with respect to the cosine of the inclination angle, \(P(i) \propto \sin i\). The probability of a transit is proportional to the range of inclinations that allow it, normalized by the total possible range of inclinations. For a circular orbit, the probability of a transit is approximately \(P_{transit} \approx \frac{2 R_\star}{a}\), where \(a\) is the semi-major axis (which is equal to \(R\) for a circular orbit). This probability is inherently biased towards planets with shorter orbital periods because, for a given stellar mass, a shorter orbital period implies a smaller semi-major axis \(a\), thus increasing the transit probability \(2R_\star/a\). Therefore, surveys employing the transit method will naturally detect a higher proportion of planets with shorter orbital periods, even if the underlying distribution of orbital periods in the galaxy is different. This is because planets with longer orbital periods are less likely to have their orbital planes aligned favorably for transit observation. This selection effect is a fundamental aspect of exoplanet detection via transit photometry and is crucial for interpreting the demographics of discovered exoplanets. Understanding this bias is essential for researchers at institutions like the Paris Observatory, which contribute significantly to exoplanet characterization and the study of planetary system formation.

The question probes the understanding of observational biases in astronomical surveys, specifically concerning the detection of exoplanets. The transit method, a primary technique for exoplanet discovery, is sensitive to the orbital inclination of a planet relative to the observer’s line of sight. For a transit to be observable, the planet’s orbit must be nearly edge-on. Planets with orbits significantly inclined to our line of sight will not produce detectable transits, even if they are otherwise within the survey’s sensitivity range for mass and orbital period. This leads to an inherent bias in transit surveys, systematically underrepresenting planets with high inclinations. Therefore, the observed distribution of exoplanet properties, such as their orbital parameters and inferred masses, is not a true reflection of the underlying population but is shaped by the detection methodology. This selection effect is crucial for interpreting exoplanet statistics and understanding planet formation and evolution, a core area of study at the Paris Observatory.

The question probes the understanding of how observational biases can affect the interpretation of astronomical data, a crucial concept in observational astronomy and astrophysics, areas of focus at the Paris Observatory. Specifically, it addresses the Malmquist bias, a selection effect that arises when the observed sample of objects is flux-limited or magnitude-limited. In such samples, intrinsically brighter objects are more easily detected at greater distances than intrinsically fainter objects. This leads to an apparent correlation between luminosity and distance, where more distant objects in the sample tend to be intrinsically more luminous. This bias can lead to an overestimation of the average luminosity of a population if not properly accounted for. For instance, if one were to study the luminosity function of galaxies based on a magnitude-limited survey, the Malmquist bias would cause the inferred average luminosity to be higher than the true average luminosity of the entire galaxy population, as the fainter, more nearby galaxies would be underrepresented in the distant bins. Understanding and correcting for such biases is fundamental to accurate astrophysical research, including cosmological parameter estimation and the study of galactic evolution, which are core to the research conducted at the Paris Observatory.

The question probes the understanding of observational biases in astronomical surveys, specifically concerning the detection of exoplanets via the radial velocity method. The core concept is that the radial velocity method is most sensitive to massive planets in close orbits because their gravitational influence causes a larger and more easily detectable Doppler shift in the star’s spectrum. Conversely, low-mass planets or those in wide orbits exert a weaker gravitational pull, resulting in smaller radial velocity variations that are harder to distinguish from stellar noise or instrumental limitations. Furthermore, the duration of observation plays a crucial role; a planet in a wide orbit requires a longer observation baseline to complete an orbit and reveal its periodic signature. Therefore, a survey designed to detect planets with orbital periods of a few days is inherently biased towards finding massive planets in short orbits, as these are the easiest to detect with current technology and observation strategies. This bias is a fundamental consideration in interpreting exoplanet population statistics derived from such surveys.

The question probes the understanding of observational biases in astronomical surveys, specifically concerning the detection of exoplanets via the transit method. The transit method relies on observing the dip in a star’s brightness as a planet passes in front of it. The magnitude of this dip is directly proportional to the ratio of the planet’s radius to the star’s radius, \( \frac{R_p}{R_s} \). A larger planet relative to its star will produce a deeper, more easily detectable dip. Conversely, smaller planets or planets orbiting smaller stars will produce shallower dips. The probability of observing a transit also depends on the alignment of the planetary orbit with our line of sight, which is related to the orbital inclination. For a circular orbit, the probability of transit is approximately \( \frac{R_s}{a} \), where \( a \) is the semi-major axis of the orbit. However, the question focuses on *detectability* given a transit occurs. A planet with a larger radius will cause a more significant photometric signal (a deeper dip), making it more likely to be distinguished from stellar noise and instrumental fluctuations. Therefore, surveys using the transit method are inherently biased towards detecting larger planets or planets that produce larger transit depths. This is a fundamental aspect of exoplanet characterization and survey design at institutions like the Paris Observatory, where understanding these biases is crucial for interpreting survey results and planning future missions. The selection of a planet with a radius of 2 Earth radii orbiting a Sun-like star will result in a larger transit depth compared to a planet with a radius of 0.5 Earth radii orbiting the same star, assuming similar orbital parameters. This increased depth directly translates to a higher probability of detection against the background noise.

The question probes the understanding of the fundamental principles governing the detection of exoplanets using the transit photometry method, a cornerstone technique in modern observational astronomy and a key area of research at institutions like the Paris Observatory. The transit method relies on observing the periodic dimming of a star’s light as an exoplanet passes in front of it from our perspective. The magnitude of this dimming is directly proportional to the ratio of the exoplanet’s cross-sectional area to the star’s cross-sectional area. Specifically, the fractional decrease in stellar flux, denoted as \(\Delta F / F\), is given by the square of the ratio of the exoplanet’s radius (\(R_p\)) to the star’s radius (\(R_s\)): \(\Delta F / F = (R_p / R_s)^2\). To determine the exoplanet’s radius, we must first know the star’s radius. Without knowledge of the star’s radius, the observed dip in brightness alone is insufficient to uniquely determine the exoplanet’s size. For instance, a small planet transiting a large star could produce the same dimming as a larger planet transiting a smaller star. Therefore, accurately characterizing the host star’s properties, particularly its radius, is a prerequisite for deriving the exoplanet’s radius from transit data. This highlights the interconnectedness of stellar astrophysics and exoplanet science, a synergy fostered at the Paris Observatory. Furthermore, the duration of the transit and the orbital period are crucial for determining orbital parameters like semi-major axis and inclination, but they do not directly yield the exoplanet’s radius without the stellar radius. The spectral type of the star provides information about its temperature and luminosity, which can be used to estimate its radius if other stellar parameters are known, but it is the radius itself that is the direct missing piece of information for calculating the exoplanet’s radius from the transit depth.

The question probes the understanding of observational biases in astronomical surveys, specifically concerning the detection of exoplanets via the transit method. The transit method relies on observing the dip in a star’s brightness as a planet passes in front of it. The magnitude of this dip is directly proportional to the ratio of the planet’s radius to the star’s radius, \( \frac{R_p}{R_s} \). A larger planet relative to its star will produce a deeper, more easily detectable transit signal. Conversely, smaller planets, or planets orbiting smaller stars, will produce shallower transits. Therefore, surveys that aim to detect a broad range of exoplanets, particularly those with limited observational precision or duration, are inherently biased towards finding larger planets or planets orbiting smaller stars, as these yield more significant and detectable photometric variations. The ability to detect smaller planets, especially those similar in size to Earth, requires extremely high photometric precision and sustained observations, often over multiple orbital periods. This fundamental limitation means that surveys are not uniformly sensitive across all planet sizes. The Paris Observatory’s research often involves characterizing exoplanetary systems, necessitating an awareness of these detection biases to accurately interpret survey results and plan follow-up observations. Understanding this bias is crucial for developing a comprehensive census of exoplanets and for avoiding overestimation of the prevalence of larger planets.

The question probes the understanding of observational biases in astronomical surveys, specifically concerning the detection of exoplanets via the transit method. The transit method relies on detecting the slight dimming of a star’s light as a planet passes in front of it. This dimming is more pronounced and easier to detect for larger planets and for planets with shorter orbital periods (which result in more frequent transits). Therefore, surveys designed to detect exoplanets using this method are inherently biased towards finding larger planets that transit more frequently. Smaller, Earth-like planets with longer orbital periods are more challenging to detect because their transits are fainter and occur less often, making them statistically less likely to be captured by a survey of limited duration and sensitivity. This bias is a fundamental consideration when interpreting the demographics of exoplanets discovered through transit surveys, such as those conducted by missions like Kepler or TESS, and is a crucial concept for students at the Paris Observatory to grasp for accurate astrophysical interpretation.