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In 1962, Riccardo Giacconi and colleagues flew three Geiger counters to search for lunar X-ray fluorescence. Different mica thicknesses tested penetration; lampblack blocked ultraviolet. Each counter sat in an anticoincidence detector using a scintillation counter to reject cosmic rays. There was no mechanical collimator. Rotation scanned the sky; optical sensors supplied orientation.^[ref]

Counts were sorted by azimuth; a faulty counter and tumbling descent data were excluded. A directional peak lay above a diffuse component. Window, altitude and angular responses tested X-rays against charged particles; the Moon's nondetection checked optical leakage. The authors argued for an origin outside the Solar system, while residual cosmic ray background was not directly measured and source size, position and spectrum remained assumption-dependent.^[ref]

A simple original forward problem makes the distinction precise. Let $\varphi$ be the scan azimuth, let $g(\varphi )$ be a dimensionless directional response with maximum one, let $F$ be a source's band-integrated photon arrival rate per unit area, let $A_0$ be its on-axis X-ray effective area, and let $b$ be a constant accepted background rate. The expected rate isFor a broad illustrative response $g(\varphi )=\max (0,\cos \varphi )$, the full width at half maximum is $120^{\circ }$: $\cos \varphi =1/2$ at $\varphi =\pm 60^{\circ }$. For a one-dimensional ideal collimator model $g(\varphi )=\max (0,1-|\varphi |/{\theta }_c)$, the full width at half maximum is ${\theta }_c$. Choosing ${\theta }_c=20^{\circ }$ narrows the response sixfold without focusing any additional photons. These are teaching response functions, not reconstructions of the rocket's window transmission or geometry. A narrower scan peak can reflect the instrument response even when the source is unresolved.

An exoplanet is a planet outside the Solar system; here the focus is on planets orbiting stars other than the Sun, including pulsars. Most exoplanets are detected indirectly: a star's changing Doppler shift reveals its motion, or a small dip in its light curve reveals an exoplanet transit. The striking idea is that an unresolved point of light can tell us about an otherwise unseen world's mass, radius, and orbit.

The measurements answer different questions. Exoplanet transit photometry primarily gives a radius ratio, while Doppler spectroscopy constrains mass together with orbital inclination. Combining them with the host star's properties gives a mean mass density, ${\bar{\rho }}_p=3M_p/(4\pi R_p^3)$, useful for exoplanet interior models. Even this combination does not uniquely determine composition: see exoplanet interior-composition degeneracy. Transit spectroscopy and an exoplanet secondary eclipse then probe an exoplanet atmosphere.

The discovery history has two distinct milestones. In 1992 Aleksander Wolszczan and Dale Frail reported two planet-mass companions to the pulsar PSR B1257+12 from periodic changes in pulse arrival times; their paper also suggested an additional companion. These were the first confirmed exoplanet discoveries, around a neutron star. ^[ref]. Wolszczan's contemporary conference account, archived by NASA, likewise distinguishes the two reported companions from a possible third. ^[ref].

In 1995 Michel Mayor and Didier Queloz discovered 51 Pegasi b through Doppler spectroscopy, the landmark first confirmed exoplanet around a Sun-like star. This was a hot Jupiter, not the first exoplanet of every kind: the earlier pulsar discoveries belong to a different host category. ^[ref]. NASA's contemporary account describes the confirmation and the approximately four-day orbital period. ^[ref].

The conventional Kelvin-Helmholtz cooling time is of order $G{M}^2/(RL)$. For the monatomic uniform-density stellar model, its precise accessible total-energy reservoir is $|E|=3G{M}^2/(10R)$, soUsing $M_{\odot }\simeq 1.9885\times 10^{30}\mathrm{kg}$, $R_{\odot }\simeq 6.957\times 10^8\mathrm{m}$ and $L_{\odot }\simeq 3.828\times 10^{26}\mathrm{W}$, the usual order-of-magnitude normalization without the structural factor is $t_{\mathrm{KH}}\simeq 3.1\times 10^7\mathrm{yr}$. The full gravitational binding reservoir $|\Omega |/L$ is about $1.9\times 10^7\mathrm{yr}$, but that overcounts the radiatable energy in virial equilibrium.

The Sun and the Solar system are approximately $4.6\times 10^9\mathrm{yr}$ old, much older than any of these contraction estimates. Gravitational contraction cannot sustain the Sun's long-lived present power. The dominant long-term source is stellar nuclear fusion. Contraction can still supply transient luminosity, especially in a pre-main-sequence star. The timescale comparison rules out a contraction-only explanation over the observed age; it does not assert that all gravitational energy release is absent.

An atmospheric thermal inversion is an altitude interval with $dT/dz>0$. Absorption of incoming stellar radiation above the usual thermal-emitting layers can heat the upper gas faster than it cools, producing an inversion. In a semi-grey irradiated atmosphere, a large shortwave-to-infrared opacity ratio favors such high-altitude energy deposition; local infrared emitters and the intrinsic flux also matter.

In the Solar system, Earth and all four giant planets have well-known stratospheric inversions. The ozone layer absorbs ultraviolet sunlight on Earth; methane and photochemical hydrocarbons absorb solar radiation in the giant planets, with aerosols contributing. The giant planets are Jupiter, Saturn, Uranus and Neptune. This refers to their stratospheric temperature rise, not to the gradient at every atmospheric level.

For hot Jupiters, influential factors include the stellar flux and spectrum; the abundances of high-altitude absorbers such as titanium monoxide and vanadium monoxide; atmospheric metallicity of a giant planet and atmospheric carbon-to-oxygen ratio; thermal dissociation, atmospheric photochemistry and condensation; a atmospheric cold trap or atmospheric condensate rainout that removes absorbers; replenishment by vertical mixing; exoplanet clouds and atmospheric hazes; and heat redistribution by circulation. The ratio of visible heating to infrared cooling, rather than a single chemical species in isolation, determines whether an inversion persists.

To escape the Solar system in the single-Sun approximation, the heliocentric specific orbital energy must reach zero. At Earth's orbital radius, the required heliocentric escape speed is $\sqrt{2}{v}_E$. The smallest change from Earth's initial velocity is prograde, parallel to its motion:Combining the terrestrial and solar energy requirements givesThe asymptotic heliocentric trajectory is a parabolic Kepler orbit; larger launch speeds can give positive solar orbital energy.

The mature experiment uses channel-template comparison, pulsar dedispersion, GPS clock-offset transfer, Solar system barycenter corrections and connected pulse counts.^[ref]

For a geometric illustration, let a perfectly uniform source emit pulse $n$ at $t_n=t_0+n{P}_s$, with spin period $P_s$, and let $z_1$ increase away from the observer. At leading order in orbital velocity divided by the speed of light, an extra path length $z_1$ adds ${\Delta }_R=z_1/c$. Consecutive arrivals then obeyA receding source has $v_{\parallel }>0$ and a longer apparent period. This derivation isolates the geometric Doppler effect; a precision relativistic clock model also needs gravitational redshift and time dilation.