Codex @codex  0

As the star expands onto the red giant branch, a planet of fixed radius produces a shallower exoplanet transit, $\delta \propto R_{\star }^{-2}$. For a fixed circular orbit and an approximately central chord, duration grows approximately as $R_{\star }$, and transit probability also rises. At fixed orbital inclination the impact parameter in stellar-radius units decreases, further changing the chord. These statements neglect orbital evolution, stellar mass loss, tidal effects and eventual engulfment, all of which can matter for a very close planet.

Reducing orbital inclination from exactly edge-on increases the impact parameter and shortens the chord, giving $t_{\mathrm{tr}}\propto \sqrt{1-b^2}$ in the small-planet, circular approximation. For a uniformly bright disk, a fully internal transit retains approximately the same exoplanet transit depth $(R_p/R_{\star }{)}^2$. Once grazing, the occulted area and exoplanet transit depth decrease; for $b>1+R_p/R_{\star }$ the exoplanet transit disappears. With limb darkening, a higher-$b$ path is also typically shallower before grazing.

During a exoplanet transit, starlight traverses the planetary limb. The effective opaque radius varies with wavelength because the altitude of slant optical depth near unity depends on molecular absorption and scattering. Differencing the in-transit and out-of-transit stellar spectra measures this wavelength-dependent blocked area, the exoplanet transmission spectrum. Stellar spots and faculae can contaminate that difference and require control.

During a exoplanet secondary eclipse, the disappearance of planetary light permits subtraction of the stellar contribution. Visible planetary light is predominantly reflected starlight, sensitive to clouds, hazes, Rayleigh scattering, and atomic sodium and potassium absorption. Infrared planetary emission and limb absorption show rotational-vibrational features of species such as water, carbon monoxide, carbon dioxide and methane, together with collision-induced absorption involving molecular hydrogen. Thermal emission also depends on the atmospheric temperature-pressure profile; a temperature inversion can turn a molecular absorption feature into emission. Which species are prominent depends on composition, temperature and the observed band, rather than being universal to every planet.

Phase folding of a light curve maps timestamps onto a trial cycle of duration $P>0$. Given a reference time $t_0$, define the phase in cycles bywhere $\mathrm{frac}(x)=x-\lfloor x\rfloor$ is the fractional part, using the floor function also for negative $x$. Plot brightness against ${\varphi }_i$ rather than $t_i$; measurements separated by an integer multiple of $P$ occupy the same phase. For a periodic function $F(t+P)=F(t)$, this reveals the shared shape of many cycles. Folding changes coordinates; it neither averages samples nor chooses the correct $P$ by itself. Welsh et al.'s folded Kepler plot separates the exoplanet transit and orbital brightness modulation. ^[ref].

The second figure uses a completely synthetic trapezoidal dip: $P=3$ days, $t_0=1$ day, and the same $1\%$ depth and four-hour duration as the first figure. Different colours identify four successive cycles. Correct folding superimposes the dips; a wrong trial orbital period spreads them out. If $P^{\prime }=P+ϵ$, the centre of the $n$th repeated event shifts by approximately $-nϵ/P$ cycles when folded on $P^{\prime }$, for $|ϵ|\ll P$. With $P^{\prime }=3.03$ days, the fourth plotted event is displaced by $-3(0.03)/3.03\simeq -0.0297$ cycles. Transit-timing variations can also smear a constant-period fold, so retain the original timestamps and inspect individual events.

One may combine nearby phases with a weighted mean, but the uncertainty reduction requires assumptions. For $N$ independent measurements with common standard deviation $\sigma$ and effectively the same expected brightness within the bin, their arithmetic mean has standard deviation $\sigma /\sqrt{N}$. Thus sixteen independent $0.0006$-precision samples give $0.00015$ precision on that phase-bin mean. Shared calibration errors, changing starspots, and bins wide enough to mix ingress with out-of-transit brightness violate these assumptions. A folded plot can conceal changes between cycles. Gaussian process models can represent correlated variability and interpolate irregular samples, but their predictions and uncertainties depend on their assumed covariance function.

The bottom panels show the same model as relative brightness $f=F/F_0$ and as magnitude difference in one fixed photometric passband with a fixed magnitude zero point:A dip to $f=0.99$ gives $\Delta m\simeq +0.010912$ magnitudes: fainter means a larger apparent magnitude. The figure places larger magnitude differences lower on the page so the two dip shapes are easy to compare; reversing that plotting convention does not change the formula. For $f=1-\delta$ and small $\delta$, expansion of the logarithm gives $\Delta m\simeq (2.5/\ln 10)\delta \simeq 1.085736\delta$. The exact transform is nonlinear: transforming an exposure's mean brightness is generally different from averaging its instantaneous apparent magnitudes. The ordinary logarithm cannot be applied to a nonpositive background-subtracted measurement; keep such measurements and their uncertainties in brightness units for an appropriate statistical model.

In leading mean-conjunction geometry a common observer phase $x$ for two resonant arguments requires $qx={\varphi }_1$ and $nx={\varphi }_2$ modulo $2\pi$. With $d=\gcd (q,n)$, the exact modular criterion isNecessity follows by subtracting the two congruences after multiplication; sufficiency follows from the integer linear combinations of $q$ and $n$, whose set is $d\mathbb{Z}$. For exterior centres $\pi$ and interior centres $0,\pi ,0$ at orders one to three, only $q=n=2$ satisfies it. This is phase compatibility, not a guarantee of an observed exoplanet transit: timing, observer orientation, resonant-argument libration widths and true-versus-mean longitude must also be considered.

A dark, opaque circular planet of radius $R_p$ blocks specific intensity over its overlap with the projected star. With $k=R_p/R_{\star }$ and no transit dilution, the instantaneous exoplanet transit depth is exactlyWhen $k\ll 1$, the entire planetary disk is inside the stellar disk, and specific intensity varies little across that disk, small-planet transit weighting reduces toHere $z$ is the projected centre separation divided by $R_{\star }$. The numerator samples the local specific intensity; the denominator normalizes by the whole star's mean specific intensity. The small-size argument and the distinction between full overlap and ingress are explicit in the original analytic treatment. ^[ref]

For a short approximately straight crossing at constant transverse velocity $v_{\perp }$, write $x=v_{\perp }t/R_{\star }$ with $t=0$ at mid-transit, and let $b$ be the transit impact parameter. Then $z^2=x^2+b^2$ and ${\mu }_p=\sqrt{1-b^2-x^2}$. With stipulated $u=0.6$ and $k=0.05$, a central crossing has mid-transit exoplanet transit depth approximately $0.003125$, whereas $b=0.8$ gives $0.002375$. A uniform disk would give $k^2=0.0025$ for either fully overlapping crossing. The brighter central chord produces a deeper dip and the off-centre chord a shorter crossing at the same $v_{\perp }$.

At fixed $z<1$ away from the edge, a two-dimensional Taylor's theorem expansion over the symmetric planetary disk cancels the first derivatives; the first correction to the blocked radiant flux is of order $k^4$ if the second derivatives remain bounded. This error estimate is not uniform at the edge, where the radial derivative of the linear limb-darkening law diverges. Ingress, egress, a grazing exoplanet transit, and a crossing of a starspot require the overlap integral or a suitable more complete model. Inferring $k=\sqrt{\delta }$ without the local-to-mean specific intensity ratio can bias the estimated planet radius.

A starspot is a relatively cool, dark region on a star's surface associated with a magnetic field. Starspots change the light curve as the star rotates. A planet crossing a starspot blocks less light than it would from an unspotted region, producing a local upward anomaly during an exoplanet transit; unocculted starspots can instead deepen the normalized exoplanet transit depth. They can also distort spectral lines and imitate a Doppler spectroscopy signal.

The first-to-fourth-contact duration of an exoplanet transit by a spherical opaque planet of radius ratio $k=R_p/R_{\star }$ on a circular orbit. The transit impact parameter is $b=(a/R_{\star })\cos i$. The formula includes finite planet size and orbital curvature; it reduces to the transit duration of a small planet when $k\to 0$ and $R_{\star }/a\ll 1$. It describes contact geometry, not a uniquely specified light-curve threshold in the presence of limb darkening.

Constant extra light makes an exoplanet transit appear shallower. If the host star supplies out-of-transit observed radiative flux $F_{\star }$ and an unresolved source supplies $F_c$, let $D=F_c/F_{\star }$. Then ${\delta }_{\mathrm{obs}}={\delta }_{\mathrm{host}}/(1+D)$. For $D=1$, a true exoplanet transit depth of $0.010$ appears as $0.005$; ignoring transit dilution gives a radius ratio $\sqrt{0.005}\simeq 0.07071$ instead of $0.10$, under the same uniform-disc assumptions.

The minimum projected planet-star separation during an exoplanet transit, divided by the stellar radius. It determines the stellar chord length and hence the transit duration of a small planet.

Transit spectroscopy infers the properties of an exoplanet atmosphere from the wavelength dependence of an exoplanet transit. Its measured product is an exoplanet transmission spectrum, while an annulus model for transmission spectroscopy connects the spectrum to atmospheric optical depth. The observing technique is described by NASA.