Branch of physics

Astrophysics applies physical laws to stars, planets, galaxies, radiation, and the universe.

Radiation heats gas sufficiently to drive an escaping thermal outflow. It can remove material from protoplanetary disks, planetary atmospheres and irradiated clouds. In a heated Keplerian disk, the photoevaporative gravitational radius compares the sound speed with the orbital speed. The mass-loss rate depends on the heating, density and wind geometry rather than radiation pressure alone.

When a protoplanetary disk's inward viscous supply approaches its wind-loss rate, photoevaporation can prevent replenishment of the inner disk. That region drains on a viscous timescale, leaving a gap or hole. A negative density in a formal steady state calculation signals the breakdown of that approximation, not a physical continuation. An exposed outer edge can subsequently disperse rapidly.

For a sink $W=W_0(r/r_g{)}^{-5/2}$ outside $r_g$, mass conservation gives $d{F}_{\mathrm{acc}}/dr=2\pi rW$ in steady state. Remote feeding gives $F_{\mathrm{acc}}={\dot{M}}_d-{\dot{M}}_w$ inside $r_g$, and ${\dot{M}}_d-{\dot{M}}_w\sqrt{r_g/r}$ outside, where ${\dot{M}}_w=4\pi W_0{r}_g^2$. The inward flux decreases towards the star because mass is diverted into the wind. A finite feeding radius or outer truncation supplies an additional boundary condition.

With constant kinematic viscosity and zero inner viscous torque in an accretion disk, $\nu \Sigma \sqrt{r}=(6\pi {)}^{-1}{\int }_0^r{F}_{\mathrm{acc}}(r^{\prime })/\sqrt{r^{\prime }}d{r}^{\prime }$. For the steady photoevaporating-disk mass flux, the density is $({\dot{M}}_d-{\dot{M}}_w)/(3\pi \nu )$ inside $r_g$ and the displayed logarithmic profile outside. At equal feeding and wind rates the inner density vanishes, and the outer normalized profile rises from zero with zero slope and tends to one. Its near-edge expansion is $(r/r_g-1{)}^2/8$.

The thermal binding scale defined by equating heated gas sound speed to circular Keplerian orbit speed. It is $r_g=GM/c_s^2$. The circular-orbit binding energy is $GM/(2r)$ per unit mass, while the gas specific enthalpy is of order $c_s^2$. A heated disk surface at radii comparable to or larger than $r_g$ can therefore drive photoevaporation. The actual launching radius can differ by a factor of order unity; $c_s$ is not the ballistic escape speed.

Gravitational collapse is contraction driven by a system's own gravitational attraction or an enclosing potential. Pressure, angular momentum and other stresses can oppose it. A free-fall timescale measures the dynamical contraction rate when pressure support is removed; gas cooling determines whether thermal support can be removed fast enough.

A pressureless uniform sphere initially at rest collapses under Newtonian gravity in a time depending only on its initial mass density. Solving the radial shell equation gives $\sqrt{3\pi /(32G\rho )}$. It is commonly used as a dynamical-time benchmark for cooling and star formation, with the gravitating density rather than gas density alone when other matter supplies gravity.

A terrestrial planet with a thin present atmosphere and evidence of a different past volatile and climate history.

An ice giant with substantial heavy material beneath its H/He envelope. Its low intrinsic luminosity makes efficient, fully mixed cooling an imperfect universal assumption.

A gas giant with a substantial H/He envelope. Intrinsic cooling, contraction and helium differentiation contribute to its thermal evolution.

Local sidereal time is the right ascension on the upper celestial meridian. A sidereal day is approximately $1436.07$ minutes; its angular rate is $2\pi /T_{\mathrm{sid}}$. Mean and apparent sidereal time must be paired with corresponding equatorial coordinate systems.

The horizontal coordinate system describes a direction by its altitude above the astronomical horizon and its azimuth. Always specify the origin and orientation of azimuth.

For a spherical Earth, geographical latitude is the angle of the local vertical above the equatorial plane. It equals the altitude of the north celestial pole in the northern hemisphere. Precision astrometry distinguishes geodetic and astronomical latitude.

Azimuth is the angle about the local vertical measured in the astronomical horizon. A common convention increases eastward from north; some astronomical formulae use westward from north. The convention changes the sign in the hour angle transformation.

Altitude is the angle of a direction above the astronomical horizon, ranging from $-90^{\circ }$ to $+90^{\circ }$.

The zenith is the direction of the local upward vertical on the celestial sphere. Azimuth is undefined there.

The astronomical horizon is the great circle perpendicular to the local vertical. It is a geometric horizon, distinct from a terrain boundary or an apparent refracted horizon.

The celestial sphere is the sphere of viewing directions around an observer; distances to objects are suppressed. Great circles define coordinate reference planes.

An observer’s celestial meridian is the great circle containing the zenith and the celestial poles. Its upper half passes through the local north and south directions.

The celestial poles are the two intersections of the rotation axis of Earth with the celestial sphere. The northern pole has declination $+90^{\circ }$.

The celestial equator is the great circle perpendicular to the rotation axis of Earth; declination measures angular displacement from it.

Growth of an astronomical object's mass by capture of surrounding matter. During planetary scattering, a collision can lead to accretion if material is retained by the planet; not every collision retains all material. An accretion disk is one particular geometry supplying material.

An equatorial coordinate system locates directions on the celestial sphere using right ascension and declination, referred to a specified equator, equinox and epoch.

The hour angle of a direction is the westward angle from the upper celestial meridian to its hour circle. In a consistent equinox convention, $h=\mathrm{LST}-\alpha$ modulo one turn, where $\alpha$ is right ascension and LST is local sidereal time.

The parallactic angle is the angle between the great circles from an observed direction toward the celestial pole and toward the zenith. Its change gives sky rotation in pupil-stabilized observations.

Declination is angular latitude north of the celestial equator, ranging from minus to plus 90 degrees.

Right ascension is the eastward angular coordinate measured around the celestial equator from the reference equinox. It is often quoted in hours, with 24 hours corresponding to a full turn.

A light curve is the brightness of an astronomical source as a function of time, expressed as flux or apparent magnitude. A periodic light curve repeats after a characteristic time; Gaussian process models can reconstruct irregularly sampled light curves and propagate observational uncertainty.

Luminosity is the total radiative energy emitted by an astronomical source per unit time. It differs from the received radiative flux, which also depends on distance and geometry.

An astronomical luminosity function gives the number density of objects per luminosity interval, often also per comoving volume. Integrating luminosity times the function gives their luminosity density.

Astronomical magnitude is a logarithmic measure of the brightness of an astronomical source; a smaller magnitude denotes a brighter source.

Apparent magnitude measures how bright an astronomical source appears to an observer.

A magnitude zero point fixes the reference measurement $I_0$ in a specified photometric passband. Relative to a zero-magnitude reference, the apparent magnitude is $m=-2.5{\log }_{10}(I/I_0)$. Both measurements must use the same passband and calibration convention.

For identical luminosities, uniform spatial density, and no extinction, the inverse-square law gives limiting distance $r\propto F_{\min }^{-1/2}$. The number in the enclosed volume is therefore $N\propto F_{\min }^{-3/2}\propto 10^{0.6m_{\lim }}$. Increasing the sample by a factor $b$ needs a magnitude-limit increase $(5/3){\log }_{10}b$. This relation need not hold with evolving source populations, dust, or cosmological distance effects.

A photometric system specifies photometric passbands and magnitude zero points, allowing calibrated brightness comparisons.

A photometric passband is the wavelength-dependent response used for a brightness measurement. An energy-weighted measurement is $I=\int {\eta }_{\lambda }{F}_{\lambda }d\lambda$, with spectral flux density $F_{\lambda }$. Photon-counting conventions can instead include a factor $\lambda /(hc)$; the calibration must use the same convention as the measurement.

The traditional near- and thermal-infrared photometric bands in the 1–5 micrometre region are $J,H,K,L,M$, with approximate centers $1.25,1.65,2.2,3.5,4.8$ micrometres. Variants such as $K_s$, $L^{\prime }$ and $M^{\prime }$ have different detailed responses. They largely follow atmospheric windows; transmission, sky brightness, and detector response determine practical band edges. The ISAAC filter documentation records instrument-specific variants, including an $L$ filter closer to $L^{\prime }$ near 3.8 micrometres.

Absolute magnitude is the apparent magnitude that an astronomical source would have at a conventional reference distance. It separates intrinsic luminosity from distance-dependent dimming.

A standard candle is an astronomical source whose calibrated absolute magnitude or luminosity allows its apparent magnitude or flux to determine a distance. A population with unknown common absolute magnitude only determines relative distances without an external luminosity calibration. Type Ia supernova cosmology models this distinction explicitly.

The distance modulus is the difference between apparent magnitude and absolute magnitude. For distance $d$ in parsecs,

An exoplanet is a planet orbiting a star other than the Sun.

A super-Earth is an exoplanet more massive than Earth but below the ice-giant mass range. The label does not guarantee a rocky exoplanet interior, a specific radius, or habitability. Radius-based usage can overlap the sub-Neptune population.

A circumbinary planet orbits both members of a binary star rather than just one. If its orbital radius greatly exceeds the binary separation, the stellar contributions to irradiation add approximately as luminosities at a common distance.

For blackbody stars of radii $R_A,R_B$ and temperatures $T_A,T_B$, a distant circumbinary planet with Bond albedo $A_B$ and uniform global reradiation has planetary equilibrium temperatureThis neglects internal cooling, eccentricity, and corrections from finite binary separation. The stars heat the planet through the sum of their absorbed bolometric fluxes, not an average of their temperatures.

An exoplanet atmosphere controls the planet's spectrum, heat redistribution, chemistry, clouds, and atmospheric escape.

A secondary planetary atmosphere is supplied or replenished after the original nebular-envelope stage, through outgassing, volatile delivery and processing. The present atmospheres of Earth, Venus and Mars are predominantly secondary. The distinction from a primary planetary atmosphere concerns origin, not merely current molecular weight.

A primary planetary atmosphere is a retained gas envelope acquired from the protoplanetary nebula, commonly rich in molecular hydrogen and helium. The solar-system giant planets retain such envelopes, while escape and subsequent processing can substantially change their composition.

The lower part of a planetary atmosphere where temperature generally decreases outward and convection can carry intrinsic or surface heat. Its upper boundary is a tropopause. Stellar irradiation, composition gradients and clouds affect the extent of this region; pressure alone does not universally identify it.

The transition above a planetary troposphere, usually near a temperature minimum or the change to a weakly varying or outward-increasing temperature profile. The overlying planetary stratosphere is stable against ordinary thermal convection. Its pressure is planet- and location-dependent.

For $P_0>0$, $\alpha \ne 0$, $T\ge T_0$ and physical $T>0$, the atmospheric pressure-temperature profile $P=P_0\exp (\alpha \sqrt{T-T_0})$ is equivalent toThe branch restriction is essential: squaring does not create a second physical branch. Its temperature gradient is $dT/d\log P=2\sqrt{T-T_0}/\alpha$. Thus, in hydrostatic equilibrium, $\alpha <0$ gives an atmospheric thermal inversion, while $\alpha >0$ gives outward cooling. At $T=T_0$ the inverse branch has zero derivative; $\alpha =0$ gives constant pressure and cannot describe a nontrivial hydrostatic vertical interval.

For a homogeneous ideal gas of constant specific-heat ratio $\gamma >1$, the Schwarzschild criterion applied to the square-root exponential atmospheric profile givesFor $T_0>0$ and $\alpha >0$, put $s=\sqrt{T-T_0}$ and $A={\nabla }_{\mathrm{ad}}$. The quadratic inequality $A\alpha s^2-2s+A\alpha T_0<0$ has solutions precisely when $A\alpha \sqrt{T_0}<1$, between $s_{\pm }=[1\pm \sqrt{1-(A\alpha {)}^2{T}_0}]/(A\alpha )$. This is a local instability of the imposed temperature gradient; actual convection adjusts that gradient and the prescribed profile need not survive.

The atmospheric lapse rate is the decrease of temperature with altitude, $\Gamma =-dT/dz$. In an ideal gas with constant gravity, a constant lapse rate produces a power-law relation between temperature and pressure.

An irradiated planetary atmosphere receives stellar energy from above while internal heat emerges from below. Their competition creates radiative, convective, and sometimes thermally inverted regions.

An atmosphere that absorbs and reemits thermal radiation can reduce outgoing energy at a given surface temperature, requiring a warmer surface to balance absorbed stellar energy. The planetary equilibrium temperature refers to the radiation budget, not the resulting surface temperature. Water, carbon dioxide and clouds can alter this balance.

A greenhouse gas absorbs and emits planetary thermal radiation and therefore contributes to the greenhouse effect. Its climatic impact depends on absorption spectrum, concentration, lifetime and chemical effects. Carbon dioxide, methane and water vapour are important examples.

Global warming potential compares the time-integrated radiative forcing of equal-mass emission pulses of a greenhouse gas and carbon dioxide, over a stated time horizon $H$. It is not an instantaneous temperature ratio, and its numerical value depends on the horizon and included chemical and climatic responses.

A single-layer greenhouse model represents an atmosphere by one uniform layer that absorbs and emits a fraction of thermal radiation. With infrared emissivity $ϵ$, the layer emits $ϵ\sigma T_a^4$ both upward and downward.

For a planet of Bond albedo $A_B$ orbiting a blackbody star of radius $R_{\ast }$ and temperature $T_{\ast }$, complete global redistribution and negligible internal heat giveUniform dayside-only reradiation replaces $4a^2$ by $2a^2$. The planetary equilibrium temperature describes absorbed stellar power; including an intrinsic cooling flux gives a total effective temperature $T_{\mathrm{eff}}^4=T_{\mathrm{int}}^4+T_{\mathrm{eq}}^4$, distinct from $T_{\mathrm{eq}}$. Neither quantity is necessarily the greenhouse-warmed surface temperature.

A semi-grey atmosphere uses separate mean visible and infrared opacities. Their ratio $\gamma ={\kappa }_{\mathrm{vis}}/{\kappa }_{\mathrm{IR}}$ controls where stellar energy is deposited.

The irradiation temperature is defined from the incident stellar flux, commonly by $T_{\mathrm{irr}}=T_{\ast }\sqrt{R_{\ast }/a}$ before redistribution and albedo factors are applied.

The internal effective temperature parametrizes the intrinsic cooling luminosity of a planet through $F_{\mathrm{int}}=\sigma T_{\mathrm{int}}^4$.

An atmospheric thermal inversion is a region where temperature increases outward. In a strongly irradiated atmosphere it can arise when shortwave opacity exceeds thermal infrared opacity and stellar energy is absorbed high in the atmosphere.

An upper atmospheric region above the tropopause in which shortwave heating can produce an atmospheric thermal inversion. Ozone on Earth, and methane and aerosols on giant planets, are examples of absorbers that help heat it. Its positive altitude temperature gradient makes it stable against homogeneous-gas thermal convection.

For the usual semi-grey irradiated atmosphere with downward thermal optical depth $\tau$, an inversion occurs where $f_r{T}_{\mathrm{irr}}^4({\gamma }^2-1)e^{-\sqrt{3}\gamma \tau }>T_{\mathrm{int}}^4$, with $\gamma$ the visible-to-thermal opacity ratio. At negligible intrinsic flux the threshold reduces to $\gamma >1$.

A strong-opacity line forms higher than the nearby continuum. Under the Eddington-Barbier approximation, an outward-decreasing temperature therefore makes an absorption line, while an atmospheric thermal inversion makes the line-forming layer hotter and reverses it into emission.

The radiative-convective boundary is where radiative transport first requires a temperature gradient steeper than the adiabatic gradient, causing convection below.

This model solves a vertical atmospheric pressure-temperature profile consistent with heating, radiative transfer and a convective adjustment where the radiative gradient exceeds the adiabatic temperature gradient. Irradiation, intrinsic flux, gravity and opacity determine the radiative-convective boundary. A one-dimensional model describes a column or average rather than resolving all atmospheric winds.

A constant radiative-diffusion coefficient and constant gravity give $T^4=T_0^4+4AP/g$. Thus ${\nabla }_{\mathrm{rad}}=AP/(g{T}^4)<1/4$ for $T_0>0$. A diatomic ideal gas has adiabatic temperature gradient $2/7$, so this idealized profile never becomes convective. A physical radiative-convective boundary requires additional opacity, flux, or thermodynamic variation.

The radiative temperature gradient is the logarithmic pressure gradient ${\nabla }_{\mathrm{rad}}=d\log T/d\log P$ required to carry the flux by radiation. Convection begins when it exceeds the relevant adiabatic gradient.

The adiabatic temperature gradient is ${\nabla }_{\mathrm{ad}}=(\partial \log T/\partial \log P{)}_s$. For a perfect gas with constant adiabatic index, ${\nabla }_{\mathrm{ad}}=(\gamma -1)/\gamma$.

For a homogeneous dry ideal gas in a hydrostatic adiabat, $dT/dz=-g/C_p$, where $z$ increases upwards and $C_p$ is the specific heat capacity at constant pressure per unit mass. It follows from $dT/dP=1/(\rho C_p)$ and hydrostatic equilibrium. The altitude version of the Schwarzschild criterion is $dT/dz<-g/C_p$ for instability; equality is marginal. Moisture, composition gradients and nonideal equations of state modify the criterion.

A dry parcel in rapid pressure balance with an ambient ideal gas has $d{T}_g/dP=T_g\mathcal{R}/(P{C}_p)$. Ambient hydrostatic equilibrium gives $dP/dz=-Pg/(\mathcal{R}{T}_{\mathrm{env}})$, hence $d{T}_g/dz=-(g/C_p)T_g/T_{\mathrm{env}}$. At a temperature-matched launch point this equals the dry adiabatic lapse rate, supplying the local linear stability test. A substantially hotter parcel does not exactly follow the ambient density's hydrostatic adiabat.

During secondary eclipse the planet passes behind its star. The lost system light measures the planet-star flux ratio from reflected starlight and planetary thermal emission.

Geometric albedo compares a body's full-phase brightness with that of a flat, perfectly diffusing reference disc of the same cross-section.

The orbital variation of a planet's reflected stellar light as the visible illuminated area changes. In a simple spherical model, $F_p/F_{\ast }=A_g(R_p/a{)}^2\Phi (\alpha )$, where $\Phi (0)=1$ and $\alpha$ is the star-planet-observer angle. Exoplanet clouds can alter the geometric albedo and the angular scattering law; this observable differs from an exoplanet thermal phase curve.

The phase integral converts a monochromatic geometric albedo into the corresponding reflected power fraction. The bolometric Bond albedo is the incident-spectrum-weighted average of $q_{\lambda }{A}_{g,\lambda }$.

Bond albedo is the fraction of total incident power reflected in all directions and over all wavelengths.

A Lambertian surface has direction-independent radiance. A perfectly reflecting Lambertian patch receiving normal flux $F$ emits specific intensity $I=F/\pi$ into its outward hemisphere.

For blackbody star and planet, define the planet-star contrastA complete exoplanet secondary eclipse removes the planet's light from the combined out-of-eclipse light. Its exact normalized depth is $d_{\lambda }=r_{\lambda }/(1+r_{\lambda })$. The common approximation $d_{\lambda }\simeq r_{\lambda }$ requires $r_{\lambda }\ll 1$. This thermal model omits reflected light and spatial variations.

For blackbodies with $T_p<T_s$, the contrast $r_{\lambda }=(R_p/R_s{)}^2(e^{x_s}-1)/(e^{x_p}-1)$ increases with wavelength, where $x_i=hc/(\lambda k_B{T}_i)$. Indeed $d\log r/d\log \lambda =G(x_p)-G(x_s)>0$, with $G(x)=x/(1-e^{-x})$ increasing for $x>0$ because $1-(1+x)e^{-x}>0$. The contrast approaches $(R_p/R_s{)}^2{T}_p/T_s$ in the Rayleigh-Jeans law limit. This behavior differs from the peaked absolute blackbody flux.

For blackbody star and planet observed at one common wavelength sufficiently long for the Rayleigh-Jeans law to hold for both,This follows by substituting $B_{\lambda }=2c{k}_{B}T/{\lambda }^4$ in the thermal eclipse depth. A radiative flux ratio alone does not determine the radius without a planetary brightness temperature and a stellar temperature. Spectral peaks interpreted with Wien's displacement law must refer to $F_{\lambda }$, not $F_{\nu }$ or $\lambda F_{\lambda }$.

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.

An exoplanet transmission spectrum measures the wavelength-dependent area blocked as starlight passes tangentially through the atmosphere during transit.

Take a planet with opaque radius $R_p$ and an atmosphere of thickness $z\ll R_p$, projected against a uniform stellar disc of radius $R_{\star }$. If the atmospheric annulus has one representative slant optical depth ${\tau }_{\lambda }$, its additional transit depth isThe general exoplanet transmission spectrum instead uses an integral over impact parameter $b$,The annulus model replaces this varying slant opacity by an effective constant. The atmospheric signal scales with optical depth when optically thin and saturates at the annular area when opaque.

An exoplanet cloud deck divides the projected atmospheric annulus into cloudy and clear radial zones. Their contributions to the exoplanet transmission spectrum add as projected areas multiplied by $1-e^{-{\tau }_{\lambda }}$ for each zone's slant optical depth.

For an isothermal ideal-gas atmosphere, $H=k_{B}T/(\mu m_{H}g)$. A strong transmission feature spanning $N_H$ scale heights has approximate transit-depth amplitude $2R_p{N}_{H}H/R_{\ast }^2$.

For an isothermal thin atmosphere with number density $n(z)=n_0{e}^{-z/H}$ and extinction cross-section ${\sigma }_{\lambda }$, a tangent ray near radius $R_p+z$ has height $z+{\ell }^2/(2R_p)$ at distance $\ell$ along the ray. The resulting Gaussian integral gives slant optical depthThe effective transit height is near ${\tau }_{\lambda }\sim 1$, so $z(\lambda )=H\log {\sigma }_{\lambda }+\mathrm{constant}$ if composition is constant. This explains the scattering slope of a transmission spectrum.

If an isothermal thin atmosphere has surface radius $R_p$, surface pressure $P_0$, and nearly constant gravity, the radius of a pressure level $P$ is

The column mass between pressures $P_0$ and $P_{\mathrm{top}}$ is $(P_0-P_{\mathrm{top}})/g$. A thin atmosphere over a spherical planet therefore haswhen its top pressure is negligible.

If extinction scales as $\sigma \propto {\lambda }^{\gamma }$, an isothermal hydrostatic atmosphere has $d{R}_p/d\log \lambda =\gamma H$. The measured continuum slope can therefore estimate atmospheric temperature when gravity and mean molecular mass are known.

For $\delta =A+B(1-e^{-\tau })$ and $\tau \propto {\lambda }^{-p}$, the logarithmic slope is $-pB\tau e^{-\tau }/\delta$. The opaque planetary baseline generally dilutes the slope. In the optically thin limit, the atmospheric excess $\delta -A$ has logarithmic slope $-p$.