Codex @codex  0

Absolute magnitude $M$, in the stellar convention used here, is the apparent magnitude a source would have at $10$ parsecs, with interstellar extinction removed. The hypothetical observation uses the specified photometric passband and magnitude zero point, preserving the source's emitted spectrum and luminosity. Moving every source to this common distance lets us compare intrinsic brightness without confusing it with distance. ^[ref].

For a fixed source with isotropy of emission, in transparent space with negligible relativistic and cosmological effects, the inverse-square law gives the hypothetical measurement $F_{10}$ and absolute magnitudeThe factor applies in the same photometric passband, with unchanged emitted spectrum. Thus absolute magnitude compares luminosity in that response, rather than automatically the total bolometric luminosity over all wavelengths.

Consider a toy source with $M=0$. Its apparent magnitude is $m=0$ at $10$ parsecs, $m=5$ at $100$ parsecs, and $m=10$ at $1000$ parsecs, neglecting interstellar extinction. Its absolute magnitude remains $M=0$ throughout: moving the observer changes received radiative flux, not emitted luminosity. Another toy source with $M=-5$ has $100$ times its passband output under the same isotropy and calibration assumptions. Negative absolute magnitudes are allowed too; "absolute" names the reference-distance convention, not an absolute value.

A calibrated standard candle reverses this reasoning: its known absolute magnitude and measured apparent magnitude give a distance modulus. This is why estimating intrinsic brightness is so useful for measuring astronomical distances. ^[ref].

In Type Ia supernova cosmology, the predicted apparent magnitude depends on the common absolute magnitude $M_0$ and the Hubble constant only through $\beta =M_0-5{\log }_{10}(H_0/H_{\mathrm{ref}})$, after choosing a fixed reference $H_{\mathrm{ref}}>0$. A flat improper prior on $M_0$ makes the likelihood integrated over $M_0$ independent of $H_0$, by translation of the integration variable. If the other prior factors are independent and the posterior is proper, the marginal posterior of $H_0$ equals its prior. An external calibration of $M_0$ can break this lack of identifiability.

Apparent magnitude $m$ is a calibrated description of the observed radiative flux received from an astronomical source. It answers how bright the source appears in a specified photometric passband, rather than how much luminosity the source emits. Define $F_{\mathrm{ref}}>0$ as the measurement assigned zero apparent magnitude by the chosen magnitude zero point. ThenHere $F$ is the positive passband measurement, with the same units and response convention as $F_{\mathrm{ref}}$. For an energy-weighted detector this measures received radiative flux through its response; for a photon-counting detector use the corresponding calibrated count rate for both quantities. Actual photometric passbands include the instrument's response across wavelength, so naming the photometric system is part of reporting an apparent magnitude. ^[ref]. ^[ref].

As toy examples in one photometric passband, $F=F_{\mathrm{ref}}$ gives $m=0$, $F=100F_{\mathrm{ref}}$ gives $m=-5$, and $F=0.01F_{\mathrm{ref}}$ gives $m=+5$. A negative apparent magnitude therefore means more received radiative flux than the reference; it is physically allowed. Zero apparent magnitude does not mean zero radiative flux. The ordinary logarithm formula requires $F>0$.

A small apparent magnitude alone cannot distinguish a nearby faint source from a distant source with large luminosity: this is the flux-distance degeneracy. Stellar parallax supplies independent distance information, while a calibrated standard candle supplies intrinsic brightness. Interstellar extinction also changes the received radiative flux and hence the measured apparent magnitude.

Astronomical magnitude expresses brightness on a logarithmic scale: equal steps in astronomical magnitude represent equal ratios of received radiative flux. The scale runs backwards, so a brighter source has a smaller astronomical magnitude. A difference of five magnitudes means exactly a factor of $100$ in flux; one magnitude means a factor of $100^{1/5}\simeq 2.511886$, rather than exactly $2.5$. This compresses the enormous range of astronomical brightnesses into manageable numbers. ^[ref].

Apparent magnitude describes received brightness, while absolute magnitude compares sources at a shared reference distance. Both require a specified photometric passband and magnitude zero point. A magnitude difference turns a ratio of positive radiative fluxes into a difference of numbers; it does not measure an additive change in emitted luminosity.

A logarithmic luminosity or measured radiative flux scale using radiation integrated over all wavelengths, in contrast to a measurement in one observing band. Absolute magnitude uses the flux the object would have at ten parsecs; apparent magnitude uses its measured flux. Neglecting extinction, $m_{\mathrm{bol}}-M_{\mathrm{bol}}=5{\log }_{10}(d/10\mathrm{pc})$.

The pulsation period of a Cepheid variable is correlated with its absolute magnitude in a specified photometric band. Calibrating the relation with stars of known distance allows an observed period and corrected apparent magnitude to give a distance. Extinction and metallicity corrections matter for a precise cosmic distance ladder.

A Cepheid variable is a pulsating evolved star whose regular changes in stellar radius and temperature produce a repeating light curve. The kappa mechanism supplies driving in a suitable part of the instability strip. Large stellar radius gives a long dynamical time, connecting the pulsation period to the star's mean mass density.

The practical importance is the calibrated Cepheid period-luminosity relation. Measuring a period predicts an absolute magnitude within a specified population and spectral band; comparison with apparent magnitude then constrains distance. Extinction, composition and unresolved companions affect that comparison. Thus the variable brightness acts as a physical ruler only after calibration, rather than every pulsating star having one universal luminosity at a given period.

A colour index is the difference of an object's measured apparent magnitudes in two photometric passbands, such as $B-V$. Since magnitude is logarithmic, this difference measures a flux ratio after the respective magnitude zero points are accounted for. It describes spectral shape rather than the object's overall brightness, and changes with both intrinsic radiation properties and interstellar extinction.

The distance modulus compares received brightness with intrinsic brightness through $\mu =m-M$, the difference between apparent magnitude and absolute magnitude in the same photometric passband and magnitude zero point. With no interstellar extinction, fixed emitted spectrum and luminosity, isotropy of emission, and the local inverse-square law, it determines Euclidean distance:The ratio inside the logarithm is dimensionless. To see the factor $5$, use the inverse-square law in the magnitude difference formula:For example, $m=5$ and $M=0$ give $\mu =5$, hence $d=10\mathrm{pc}\times 10=100\mathrm{pc}$. A tenfold increase in distance adds five magnitudes; an increase of one magnitude multiplies distance by $10^{1/5}\simeq 1.584893$. The distance modulus is a logarithmic scale for distance, not a distance in length units. ^[ref].

At $10$ parsecs, $\mu =0$ and $m=M$. Within $10$ parsecs, the distance modulus is negative; for example, $\mu =-5$ gives the positive distance $d=1\mathrm{pc}$. A negative distance modulus does not imply negative distance. At cosmological distances, the inference needs luminosity distance and a K correction; the simple local formula must not silently absorb these effects.

Interstellar extinction increases apparent magnitude. The observed distance modulus then includes both geometric dimming and attenuation; extinction bias in photometric distance explains how to separate them.

Let $A\ge 0$ be the interstellar extinction in magnitudes in the specified photometric passband. Compared with the same unobscured source at the same distance, its radiative flux is multiplied by $10^{-0.4A}$, so its apparent magnitude increases by $A$. Consequently,Here ${\mu }_0$ is the extinction-corrected distance modulus. The correction subtracts $A$ before using the distance formula. Caltech's NED distance-method documentation explicitly distinguishes an apparent distance modulus from an extinction-corrected distance modulus. ^[ref].

Ignoring positive interstellar extinction attributes too much dimming to distance, producing an overestimate:As an illustrative toy, put the source with $M=0$ at $100$ parsecs behind $A=1$ magnitude of interstellar extinction. The measured apparent magnitude is $m=6$. Setting $A=0$ incorrectly gives $d_{\mathrm{inferred}}=10\mathrm{pc}\times 10^{6/5}\simeq 158.489\mathrm{pc}$, a factor $1.584893$ too large. Subtracting the known $A=1$ recovers $100$ parsecs. The diagram's fixed $A=1$ offset is a controlled illustration, not a model of how actual interstellar extinction grows with distance, and none of its values are star catalogue measurements.

A light curve records an astronomical source's received brightness against time, in a specified photometric passband. Its measured form is a time series of timestamps, brightness estimates, and uncertainties; a smooth $F(t)$ is a model of that record. Brightness can be reported as calibrated observed radiative flux, a detector count rate, a dimensionless ratio to a reference level, or apparent magnitude. These quantities require different units and calibration conventions. A light curve concerns what reaches the observer: a dip during an exoplanet transit does not imply that the star's emitted luminosity fell.

The appeal of a light curve is that an unresolved point of light can reveal motion and structure. In exoplanet transit photometry, repeated dips constrain the orbital period; under the uniform-disc, dark-planet, full-overlap assumptions, the exoplanet transit depth is the blocked area fraction $(R_p/R_{\star }{)}^2$. A toy one-percent dip therefore corresponds to a radius ratio of $0.10$. Its duration and shape add information about the crossing geometry, with limb darkening, transit impact parameter, and transit dilution affecting the interpretation. See exoplanet transit photometry for the geometry. The first figure instead follows the measurement: an instrumental trend and sky contribution can obscure a small physical dip.

An instructive real observation is the early Kepler record of the already known exoplanet HAT-P-7b. Welsh and colleagues analysed 33.5 days from 13 May to 15 June 2009, containing 15 complete exoplanet transits at an orbital period of about $2.20473$ days. The CCDs were read roughly every six seconds and coadded to approximately one-minute samples: each point measures accumulated signal over a finite interval, not instantaneous brightness. After removing 21 positive outliers attributed to cosmic rays, 49,023 samples remained. The authors estimated a scatter of about $150$ parts per million per one-minute sample. A gradual $0.13\%$ count decrease, suspected to arise from focus drift with a fixed extraction aperture, was removed with a cubic polynomial fitted while masking the exoplanet transits. They normalized brightness at mid-exoplanet secondary eclipse, when the planet was hidden and only starlight remained. ^[ref].

What does the instrument actually measure? A CCD collects photoelectrons from incoming photons; pixel values first need electronic-offset, dark current, readout-smear, and sensitivity corrections. ^[ref]. Aperture photometry then sums a chosen group of pixels around the star, with background subtraction removing the estimated sky brightness. A count rate is a response-weighted brightness measurement, not automatically a bolometric observed radiative flux. Calibration must respect the photometric passband and, for apparent magnitude, its magnitude zero point. Kepler's pipeline also used engineering information to remove trends associated with pointing, focus, and thermal changes; its output included brightness uncertainties and image positions. ^[ref].

Calibration and interpretation need separate checks. A blended eclipsing binary can mimic an exoplanet transit, with transit dilution making a deep stellar eclipse appear shallow. The Kepler pipeline paper describes comparing odd and even dip depths and examining changes in the image centroid: an off-target fading source can move the combined image position. These tests help reject false positives but do not by themselves establish a companion's mass; complementary Doppler spectroscopy and host-star constraints remain valuable. ^[ref].

Even a real exoplanet's signal can have several contributors. Welsh and colleagues found brightness modulation from the star distorted by tidal forces as well as the planet; assigning all orbital brightness changes to planetary emission would misinterpret this light curve. ^[ref]. Starspots, limb darkening, and transit dilution likewise belong in the relevant physical model. A flexible trend fitted through a dip could absorb that dip; the masked fit in the real example illustrates why the fitting procedure is part of the measurement. Exposure-averaged light curves and phase folding of a light curve make two further limitations explicit.

For two positive measurements $F_1,F_2$ in the same photometric passband, using the same magnitude zero point, units, and weighting convention, their apparent magnitudes obeyThe logarithm takes a dimensionless ratio. An energy-weighted measurement must be compared with another energy-weighted measurement, and a photon-counting measurement with another consistently calibrated photon-counting measurement. Changing photometric passband or magnitude zero point requires the corresponding calibration, as in a colour index. ^[ref]. ^[ref].

For example, $F_2=100F_1$ gives $\Delta m=-5$: source 2 is brighter. Conversely, $\Delta m=+5$ gives $F_2/F_1=0.01$, and $\Delta m=+1$ gives $F_2/F_1\simeq 0.398107$. Even fractional steps work: $\Delta m=+0.1$ gives $F_2/F_1\simeq 0.912011$, an approximately $8.80\%$ decrease. These are illustrative ratios, independent of any particular star catalogue.

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.

Compare the extinction-corrected apparent magnitudes of a cluster's main sequence with calibrated absolute magnitudes at matching colour or stellar parameters. The magnitude displacement estimates its distance modulus. Age, metallicity, helium abundance, reddening and unresolved binaries can move or broaden the sequence, so a colour-magnitude displacement is not uniquely a distance change.

Adopt circular orbits, negligible planetary masses, spherical bodies, uniform stellar brightness and no transit dilution. Interpret the measured duration as the total duration of a circular planetary transit, from first to fourth contact, $T_{14}$. These conventions matter especially for the off-centre crossing: finite planet size cannot automatically be ignored there. Unknown orbital eccentricity or limb darkening would require further information to infer unique stellar radii.

First convert the apparent magnitude difference into a fractional exoplanet transit depth, rather than treating a milli-magnitude as a part in a thousand:Both crossings are non-grazing because their transit impact parameters satisfy $b<1-k$, so the full projected planetary disc can block the star. Equal depths determine equal radius ratios, not equal absolute planet radii.

For orbital inclination $i$ measured from the sky plane, $b=(a/R_{\star })\cos i$. Choosing mid-transit as orbital phase zero, the projected separation obeysAt first and fourth contact, $z/R_{\star }=1+k$. The full crossing duration therefore isFor the numerical scale use solar radius $R_{\odot }=6.957\times 10^8\mathrm{m}$ and nominal solar mass parameter $(GM{)}_{\odot }=1.3271244\times 10^{20}{\mathrm{m}}^3{\mathrm{s}}^{-2}$, with one day equal to $86400\mathrm{s}$. These are stated conversion choices, not additional measurements of the two systems.^[ref]

Kepler's third law gives $a=[G{M}_{\star }{P}^2/(4{\pi }^2){]}^{1/3}$. For a solar-mass, solar-radius host, this gives $a/R_{\odot }=12.3051$ at five days and $6.68023$ at two days. Substitution in the contact formula yieldsThe transit duration of a small planet instead takes both $k\to 0$ and $R_{\star }/a\ll 1$, giving $T\simeq P{R}_{\star }\sqrt{1-b^2}/(\pi a)$. It predicts $3.10\mathrm{h}$ and $1.37\mathrm{h}$, respectively. The difference is appreciable for B because the fractional change in its crossing chord is larger. Durations defined by mid-ingress to mid-egress or by second to third contact should not be silently equated with $T_{14}$.

The original diagram uses a straight projected track to display the contact geometry; the duration formula above retains orbital curvature.

To infer the actual stellar radii, invert the contact formula before applying the supplied mass-radius scaling. Let $A=a/R_{\star }$ and $s=\sin (\pi T_{14}/P)$; thenThe measured durations imply $A_A=8.34661$ and $A_B=7.55743$. Normalize the stipulated constant mass-radius ratio to the Sun, so $M_{\star }/M_{\odot }=R_{\star }/R_{\odot }\equiv r_{\star }$. Kepler's third law then givesThus the stellar radius from transit duration and a linear mass-radius relation isThe same assumed scaling gives $M_A=1.79M_{\odot }$ and $M_B=0.831M_{\odot }$. A's longer crossing requires a lower mean stellar mass density, while B's shorter full-contact crossing requires a higher one. They are not both solar replicas under this model. Their planets have $R_{p,A}=0.1213R_{\odot }$ and $R_{p,B}=0.05633R_{\odot }$.

For a consistently applied point-planet approximation, the duration scales as $T\propto P^{1/3}{r}_{\star }^{2/3}\sqrt{1-b^2}$ and the inferred radii would instead be $R_A\simeq 1.98R_{\odot }$, $R_B\simeq 1.07R_{\odot }$. The latter is close to solar. Consequently the assertion of a substantial non-solar radius for B depends on which duration convention and finite-size approximation are intended. The contact-based values above use the measured depth rather than discarding it. If eccentric orbits are allowed, the transit speed introduces an additional factor depending on eccentricity and orientation, so the two durations alone cannot establish those radii uniquely.

For an outer coplanar planet around B, the common orbital inclination gives $b_2/b_B=a_2/a_B=(P_2/P_B{)}^{2/3}$. A point-planet centre must cross the stellar disc, requiring $b_2<1$. The usual coplanar transit limit is thereforeThis is a limiting value approached from below; a tangent path has no duration. For an outer planet of finite, unspecified radius ratio $k_2$, any overlap instead requires $b_2<1+k_2$, giving $P_2<P_B[(1+k_2)/b_B{]}^{3/2}$. If it had the same ratio $k_2=k$, that geometric limit would be $3.08$ days; requiring full overlap would give $2.52$ days. Because neither its radius nor a photometric detection threshold is supplied, no unique finite-size practical maximum period can be inferred beyond this conditional expression. The stellar radius cancels from the point-planet period-ratio bound.

For B's inner planet, a reflected-light planetary phase curve haswhere $A_g$ is geometric albedo and $\alpha$ is phase angle. Near exoplanet secondary eclipse the visible hemisphere is almost fully illuminated. Using the standard full-phase approximation and the contact-based $A_B$,The exact lost fraction of combined stellar and planetary light is $F_p/(F_{\star }+F_p)$, which changes the result negligibly at this contrast. The corresponding magnitude depth is about $0.0437$ milli-magnitudes. The circular orbit has the same impact distance at eclipse as at transit, and $b_B<1-k$, so the star can hide the entire planetary disc.

The inclination inferred for B is $83.92^{\circ }$, putting closest approach to full phase at $\alpha \simeq 6.08^{\circ }$. A fully precise prediction would therefore also require the reflected-light phase function, which the question does not specify; the boxed value uses $\Phi \simeq 1$. The consistent point-planet duration approximation yields about $56.7\mathrm{ppm}$ instead, through its different inferred $a/R_B$. It is important to keep the geometry used for the radius inference and the reflected-light calculation consistent.

Finally, the stellar radial-velocity semi-amplitude for a low-mass companion on a circular orbit isThe two stars' equal apparent magnitudes make a comparison at similar photon statistics plausible, though spectral-line widths, stellar variability and observing cadence can still differ. With the contact-based stellar estimates,If the planets have comparable mean densities, A is easier: their equal depth then gives $M_{p,A}/M_{p,B}\simeq (R_A/R_B{)}^3\simeq 9.99$, so $K_A/K_B\simeq 4.44$. The larger planetary mass outweighs A's longer orbital period and larger host mass. For equal planetary masses, B is easier by a factor $2.25$ in amplitude. The supplied transit data give planet radii, not masses, so the ranking is conditional on a planetary mass-radius assumption; a unique unconditional ranking would invent information.

Type Ia supernovae are standardizable candles. Repeated multiband photometry measures peak apparent magnitude, light-curve width and colour; spectra identify the supernova class and determine redshift, often from the host galaxy. Nearby objects calibrate the relation between light-curve shape, colour and absolute luminosity. Applying those relations produces a corrected apparent magnitude $m_{\mathrm{corr}}$, not an assumption that every event has identical luminosity. The distance modulus then givesA common uncertainty in $M$ is degenerate with the absolute distance scale, but does not erase the redshift-dependent shape of the Hubble diagram.

luminosity distance probes the expansion history. In a spatially flat model,For fixed low-redshift calibration, a negative $q_0$ increases the second-order distance relative to a decelerating model. Objects at a given redshift are therefore fainter. A Hubble diagram spanning enough redshift can distinguish this curvature from the common magnitude intercept. In matter-plus-cosmological-constant models, $q_0={\Omega }_m/2-{\Omega }_{\Lambda }$; fitting the distances can test whether $q_0<0$. The two original supernova teams found distances supporting an accelerating expansion in that framework.^[ref][ref]

The comparison requires matched rest-frame passbands and K corrections, correction for redshift time dilation in the light curves, a common photometric calibration, and explicit uncertainty propagation. Dust extinction can imitate dimming; multicolour and infrared observations test reddening and the adopted extinction law. Evolution in the progenitor population or host-galaxy dependence can alter standardization, so one compares matched populations and fits host-dependent terms where warranted. Flux-limited discovery introduces Malmquist bias and other selection effects, which must be modelled through the survey detection efficiency. peculiar velocities matter at low redshift, while gravitational lensing and intrinsic scatter broaden the distant sample. Classification and outlier tests reduce contamination by other transient types.

Independent data reductions, nearby anchors, spectroscopic checks and a fit over the full redshift range distinguish an expansion-history signal from one instrumental offset. Complementary cosmological observations help separate matter density, curvature and dark-energy parameters. The acceleration inference comes from the relative distance-redshift relation, conditional on standardization and an expansion model. Distant supernovae integrate the past expansion history; they do not directly measure the instantaneous second time derivative of today's scale factor without such modelling.

A Type Ia supernova can be standardized as a standard candle by correcting its peak brightness using the light curve shape and colour index. Its host spectrum gives the redshift; the corrected apparent magnitude and calibrated absolute magnitude give the distance modulusThe resulting Hubble diagram measures the shape of the luminosity distance as a function of redshift. A spatially flat matter-and-vacuum model, for example, predictsNonzero cosmological spatial curvature changes both the expansion rate and the conversion from radial to transverse comoving distance. Comparing the observed Hubble diagram with these predictions constrains the matter density parameter, cosmological constant and cosmological spatial curvature; a single low-redshift observation cannot separate them. The nearby sample calibrates the common intercept, while distant Type Ia supernovae supply the redshift-dependent shape. Relative faintness at a specified redshift means a greater inferred luminosity distance.

Type Ia supernova cosmology requires correction for interstellar extinction, possible evolution of the standardized luminosity, and selection bias. The absolute magnitude–Hubble constant degeneracy means that an uncalibrated sample constrains the shape rather than independently determining $H_0$.

Type Ia supernovae have similar peak luminosities that can be standardized using their light-curve width and color, rather than assumed to be identical without correction. Their redshifts and corrected apparent magnitudes give a distance modulus,Compare this Hubble diagram with the luminosity-distance prediction of a background cosmology. Restoring $c$, put $E(z)=H(z)/H_0$, $I(z)={\int }_0^{z}d{z}^{\prime }/E(z^{\prime })$ and define ${\mathcal{S}}_{{\Omega }_k}(I)$ as $\sinh (\sqrt{{\Omega }_k}I)/\sqrt{{\Omega }_k}$ for ${\Omega }_k>0$, $I$ for ${\Omega }_k=0$, and $\sin (\sqrt{-{\Omega }_k}I)/\sqrt{-{\Omega }_k}$ for ${\Omega }_k<0$. ThenNeglecting radiation at these redshifts, Friedmann equations and a separately conserved dark energy component with $w(z)=p/(\rho c^2)$ giveFor constant $w$, the last factor is $(1+z{)}^{3(1+w)}$; a cosmological constant has $w=-1$. Thus the redshift dependence of the measured distance constrains matter density, curvature and the dark-energy equation of state. At small redshift, $d_L=(cz/H_0)[1+(1-q_0)z/2+\cdots ]$, showing how departures from the linear Hubble law probe the deceleration parameter.

The early distant-supernova analyses found larger distances than matter-only decelerating models predicted, providing evidence for accelerated expansion; see the 1998 supernova analysis and the 1999 independent supernova analysis. This is a statement about the distance-redshift curve under the standardization and cosmological assumptions, not a direct measurement of a fluid pressure.

The absolute magnitude–Hubble constant degeneracy means uncalibrated supernovae constrain the shape of $d_L(z)$ without separately determining $H_0$ and the common absolute luminosity. The integral over $H(z)$ also makes different matter, curvature and $w(z)$ histories partly degenerate. Wide redshift coverage and independent cosmic microwave background and large-scale structure of the universe information help break these degeneracies. Dust extinction, source evolution, photometric calibration, sample selection and K corrections must be modeled, and constraints on a time-dependent equation of state depend on how that function is parameterized. Standardized supernovae test the expansion history; they do not by themselves uniquely determine every cosmological parameter.

Write $v={\sigma }_{\mathrm{int}}^2$, $r_s={\sigma }_{m,s}^2$ and $d_s=\mu (z_s;H_0,w,{\Omega }_M)$. The latent absolute magnitude and measurement error are independent normal variables, so their convolution is normal. After integrating out the individual intrinsic magnitudes, each apparent magnitude has distributionConsequently the likelihood function for the independent sample isThis is the marginal observational likelihood of a hierarchical Bayesian model, not the likelihood conditional on each unknown intrinsic magnitude. The individual errors remain heteroscedastic through their known $r_s$.

LetSubtracting the measured distance modulus from the measured apparent magnitude givesFor the Hubble flow, defineThe Hubble law, with the Hubble constant parametrized by $\theta =5{\log }_{10}h$, gives $r_i\sim N(M_0-\theta ,H_i)$. After integrating over each intrinsic absolute magnitude and each unobserved true distance modulus, independence therefore gives the likelihood function