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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].

Astrometry measures the positions and motions of celestial objects. Repeated position measurements can distinguish stellar parallax, caused by the observer's motion, from proper motion of the source across the sky.

An astrometry measurement needs a date and a reference frame, as well as an angle. In an equatorial coordinate system, right ascension and declination label two viewing directions. A catalogue's reference date states when its quoted position applies; comparing entries at different dates without accounting for proper motion can make the same star appear misplaced. The observational problem is to infer a consistent history from successive incoming light directions, while accounting for the observer's changing position and measurement error.

To see the separation clearly, consider just one angular coordinate $\theta$, with $t$ the numerical elapsed time in years. Assume a simplified circular orbit of the Earth, a fixed source Euclidean distance, constant proper motion in this coordinate and no other accelerations. Choose the direction and the starting date so that the annual stellar parallax projection is the cosine termHere ${\theta }_0$ is the coordinate without the annual stellar parallax offset at the reference date; the observed $\theta (0)$ is ${\theta }_0+p$. The proper motion coefficient $\mu$ is quoted in arcseconds per year, and $p$ is the annual stellar parallax in arcseconds. This is an illustrative choice of a full-amplitude projection. General sky directions have different projected amplitudes and phases in their two coordinates; their apparent paths need not be circles. Real astrometry uses the observer's calculated position rather than imposing this single cosine. ^[ref].

Set ${\theta }_0=0$, $\mu =100\mathrm{mas}\mathrm{y}{\mathrm{r}}^{-1}$ and $p=50\mathrm{mas}$, where $1\mathrm{mas}$ is one thousandth of an arcsecond. The model gives $\theta (0)=50$, $\theta (0.5)=0$ and $\theta (1)=150\mathrm{mas}$. Same-season observations one year apart differ by $100\mathrm{mas}$ because the annual stellar parallax contribution repeats. Subtracting that fitted straight trend leaves an annual signal between $-50$ and $+50\mathrm{mas}$. The existing stellar parallax distance relation then gives $20$ parsecs.

Observing only at the same annual phase would recover the proper motion trend but leave $p$ indistinguishable from a constant offset. Sampling several seasons as well as several years separates the repeating signal from the accumulating displacement. The two signatures answer different questions using the same measurements.

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 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.

One observed radiative flux cannot determine both luminosity and Euclidean distance: for every $k>0$, replacing $(L,d)$ by $(k^{2}L,kd)$ leaves $F$ unchanged. In particular, $4\times 10^{26}\mathrm{W}$ at $40$ parsecs produces the same observed radiative flux as $10^{26}\mathrm{W}$ at $20$ parsecs. This is a concrete failure of identifiability in an inverse problem.

An independent stellar parallax fixes the distance and breaks this degeneracy; alternatively, a calibrated standard candle supplies the luminosity. Interstellar extinction can further reduce the observed radiative flux, and a photometric passband measures only a selected spectral range. At cosmological distances, use luminosity distance in the bolometric relation. Each inference must specify which radiative flux was measured and what propagation assumptions connect it to luminosity.

The Hubble constant is the present value $H_0=H(t_0)$ of the Hubble parameter. The word "constant" fixes the epoch; it does not mean $H(t)$ stays unchanged throughout cosmic time. It sets the present instantaneous proportionality in Hubble's law and the approximate nearby observed redshift-distance relation.

For illustration, stipulate $H_0=70\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$; this is an example input, not a claim about a measured consensus value. A million parsecs is $3.085677581\dots \times 10^{19}\mathrm{km}$, givingHere a year means $365.25$ days of $86400\mathrm{s}$, and $\mathrm{Gyr}$ denotes a billion such years. The parsec conversion follows ^[ref]. This Hubble time is a present expansion timescale; the age of an FLRW universe also depends on the earlier evolution of the scale factor and need not equal $H_0^{-1}$.

Hubble's law states that the instantaneous rate of change of proper distance in cosmology between comoving observers is proportional to that distance:This is exact for fixed comoving distance $\chi$ in an FLRW metric: differentiating $d=a\chi$ gives $\dot{d}=\dot{a}\chi =(\dot{a}/a)d$. The dot uses cosmic time, and both distances are compared on spatial slices at the same cosmic time. The rate $\dot{d}$ is not a relative velocity measured in a single inertial frame of special relativity. Thus the instantaneous relation is not an exact observed redshift-distance formula. The associated Hubble flow describes the ideal expansion; peculiar velocities describe departures from it.

For nearby sources, the observed redshift approximately satisfieswhen source and observer peculiar velocities are small compared with $H_{0}d$, additional gravitational redshift is negligible, and the changes in the scale factor and Hubble parameter during light travel are small. Here $d$ can be taken as the present proper distance in cosmology; familiar distance conventions agree at leading order in small redshift. Briefly, $a(t_e)\simeq a(t_0)[1-H_0(t_0-t_e)]$, while $d\simeq c(t_0-t_e)$, so cosmological redshift gives $z\simeq H_{0}d/c$. At greater redshift, use an expansion history and an explicit distance convention instead.

Stipulate the illustrative Hubble constant $H_0=70\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ and speed of light $c=299792.458\mathrm{km}{\mathrm{s}}^{-1}$. A $\mathrm{Mpc}$ is a million parsecs. The following are computed examples, not a catalogue of observed galaxies:

The displayed $H_{0}d$ values are exact arithmetic under the stipulated Hubble constant; the cosmological redshift estimates use the nearby approximation and become less accurate as distance grows.

Hubble's law turns a nearby observed redshift into an approximate distance indicator and makes the uniformity of cosmic expansion tangible: twice the proper distance in cosmology means twice the instantaneous separation rate. Recentring the ideal Hubble flow on another comoving observer preserves the same proportionality, as illustrated under scale factor, so recession in every direction does not select a central observer.

In his 1929 PNAS paper, Edwin Hubble presented a relation between the distances and radial velocities of extragalactic objects. His fits gave coefficients of $465$ and $513\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ for 24 individual objects and nine groups respectively, and he used about $500\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ as a rounded representative value. Those historical fits used the paper's distance estimates; they are distinct from the stipulated modern-scale illustration above. ^[ref].

Luminosity is emitted radiative power: the energy carried away by electromagnetic radiation per unit time, measured in watts. Bolometric luminosity includes all wavelengths; a measurement through one photometric passband captures only part of it. Spectral luminosity records how that power is distributed over frequency. Received observed radiative flux has different units, watts per square metre, because it measures what crosses an observer's collecting area. ^[ref].

For example, an ideal source with luminosity $10^{26}\mathrm{W}$ emits $10^{26}\mathrm{J}$ each second regardless of whether it is viewed from $20$ or $40$ parsecs. Its observed radiative flux changes with distance, while its luminosity remains fixed. Combining stellar parallax with observed radiative flux therefore lets us compare the actual outputs of stars rather than their apparent brightnesses.

For a steady point-like astronomical source with isotropy of emission, in transparent space with negligible relativistic or cosmological effects, conservation of energy spreads its luminosity over a sphere of surface area $4\pi d^2$. Its bolometric observed radiative flux isThis inverse-square law reflects geometric dilution: the same emitted power crosses increasingly large spheres. It does not require light to lose energy on the journey. For a fixed luminosity, doubling Euclidean distance quarters the observed radiative flux. ^[ref].

With $L_0=10^{26}\mathrm{W}$ and $d_0=20$ parsecs, the result is $F_0\simeq 2.09\times 10^{-11}\mathrm{W}{\mathrm{m}}^{-2}$. At $40$ parsecs, the same source gives $F_0/4\simeq 5.22\times 10^{-12}\mathrm{W}{\mathrm{m}}^{-2}$. An ideal collecting area of $1{\mathrm{m}}^2$ at $20$ parsecs intercepts approximately $2.09\times 10^{-11}\mathrm{W}$, before detector losses.

The parsec is a distance unit defined by the exact conventionThe astronomical unit is exactly $149597870700\mathrm{m}$; one arcsecond is exactly $\pi /648000$ radians. The IAU resolution explicitly records the parsec convention. A megaparsec is one million parsecs. ^[ref]. ^[ref].

The familiar description "the distance giving one arcsecond of stellar parallax" uses the small-angle approximation. Exact right triangle geometry instead gives $\mathrm{au}/\tan (\pi /648000)$, slightly shorter than the conventional parsec. The relative difference is approximately $(\pi /648000{)}^2/3\simeq 7.83\times 10^{-12}$. This is negligible for the examples here, but distinguishing an exact unit convention from an approximate geometric interpretation prevents a misleading equality.

Once swept-up matter dominates the ejecta and radiative cooling is negligible, the Sedov-Taylor blast wave has its shock radius fixed by dimensional analysis:Here $\xi$ is a dimensionless constant of order unity fixed by the adiabatic index. Treating the ambient matter as hydrogen gives ${\rho }_0\simeq 10^6{m}_H\simeq 1.67\times 10^{-21}\mathrm{kg}{\mathrm{m}}^{-3}$, while $t\simeq 3.16\times 10^{10}\mathrm{s}$. ThusFor a monatomic gas, the usual similarity normalization $\xi \simeq 1.15$ instead gives about $5.3\mathrm{pc}$. An estimate of a few parsecs is therefore appropriate. Negligible ejecta mass does not by itself guarantee this energy-conserving stage: substantial radiation losses or a strongly nonuniform ambient mass density would invalidate the assumed similarity law.

For the hypothetical Hubble flow, Hubble's law gives $v=H_{0}d$, where $d=4\times 10^5\mathrm{km}$ and $H_0$ is the Hubble constant. The accompanying constants sheet is not present in the supplied PDF, so retain its dependence explicitly. Taking $H_0=70\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ as an illustrative input, and using one million parsecs equal to $3.08568\times 10^{19}\mathrm{km}$,A year of $365.25$ days contains $3.15576\times 10^7\mathrm{s}$. ThereforeEquivalently, $H_0=100h\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ gives $v=4.09h\mathrm{cm}\mathrm{y}{\mathrm{r}}^{-1}$. This is a counterfactual recession rate for independently comoving bodies, rather than a prediction for the actual Earth-Moon orbit.

The Milky Way has a thin stellar galactic disk, radius roughly $15$–$20\mathrm{kpc}$ and scale height about $0.3\mathrm{kpc}$, containing gas, dust, young and old stars, spiral arms and most current star formation. Its stellar mass is a few $10^{10}{M}_{\odot }$. A thicker, older galactic disk has scale height around $1\mathrm{kpc}$ and stellar mass of order $10^{10}{M}_{\odot }$, with little cold gas.

The central galactic bulge and bar extend a few thousand parsecs and contain about $10^{10}{M}_{\odot }$, largely older stars. The central supermassive black hole has mass about $4\times 10^6{M}_{\odot }$ and is negligible in the total galactic budget. A diffuse old metal-poor stellar halo, extending tens to over $100\mathrm{kpc}$, contains globular clusters and roughly $10^9{M}_{\odot }$ in stars. An extended dark matter halo reaches roughly $200\mathrm{kpc}$ and dominates the total mass, of order $10^{12}{M}_{\odot }$. These are approximate component scales; the gaseous galactic halo also extends beyond the stellar galactic disk.

Evaluate the source's printed logarithm using the numerical magnitude value 21.0, with base-ten logarithms:This is the literal printed-fit convention identified above. Convert the mean visual surface brightness to a linear luminosity per projected area. At 10 parsecs, one square arcsecond subtends $100/(206264.806{)}^2\mathrm{p}{\mathrm{c}}^2$. The definition of absolute magnitude consequently givesThe constant $21.572=5{\log }_{10}(206264.806)-5$ is the conversion between the angular and physical projected area. Since the effective radius encloses half the total luminosity,For the stipulated stellar population of Solar-type stars, approximate the stellar mass-to-light ratio by $M_{\odot }/L_{V,\odot }$. Summing 100 such galaxies then givesThis assumes negligible interstellar extinction and that the supplied surface brightness is an intrinsic or already corrected value; an observed cosmological surface brightness requires additional dimming and passband corrections before this conversion.

For comparison only, if the empirical coefficients and intercept were instead meant for ${\log }_{10}({\mathcal{I}}_e/[L_{V,\odot }\mathrm{p}{\mathrm{c}}^{-2}])$, substituting $140.9$ in that fit would give $R_e=1.09\mathrm{kpc}$, $L_V=1.06\times 10^9{L}_{V,\odot }$, and $M_{\ast ,\mathrm{cluster}}=1.06\times 10^{11}{M}_{\odot }$. Those are conditional alternative results, not an unannounced correction of the printed units.

The Milky Way has a thin stellar galactic disk, radius roughly $15$–$20\mathrm{kpc}$ and scale height about $0.3\mathrm{kpc}$, containing gas, dust, young and old stars, spiral arms and most current star formation. Its stellar mass is a few $10^{10}{M}_{\odot }$. A thicker, older galactic disk has scale height around $1\mathrm{kpc}$ and stellar mass of order $10^{10}{M}_{\odot }$, with little cold gas.

The central galactic bulge and bar extend a few thousand parsecs and contain about $10^{10}{M}_{\odot }$, largely older stars. The central supermassive black hole has mass about $4\times 10^6{M}_{\odot }$ and is negligible in the total galactic budget. A diffuse old metal-poor stellar halo, extending tens to over $100\mathrm{kpc}$, contains globular clusters and roughly $10^9{M}_{\odot }$ in stars. An extended dark matter halo reaches roughly $200\mathrm{kpc}$ and dominates the total mass, of order $10^{12}{M}_{\odot }$. These are approximate component scales; the gaseous galactic halo also extends beyond the stellar galactic disk.

Proper motion describes the change in a celestial object's direction on the sky due to its motion relative to the observer. Its angular velocity differs from the annual stellar parallax signal caused by the observer's orbit.

Thus proper motion is an angular velocity, rather than a speed in kilometres per second. It describes the relative transverse motion of source and observer in a specified reference frame. A star moving towards us can have little proper motion, while a nearer star can show larger proper motion than a faster but more distant one. In the example above, the observed coordinate actually decreases over the first half-year, although $\mu$ is positive: the annual stellar parallax excursion temporarily outweighs the steady drift. A short observed displacement alone therefore need not estimate proper motion well.

Real catalogues give two proper motion components. For right ascension $\alpha$ and declination $\delta$, the angular velocities in perpendicular directions on the local sky areThe cosine factor accounts for the shorter east-west arc associated with a given right ascension increment away from the celestial equator. ESA's Gaia DR3 fields pmra and pmdec already contain these two components; pm is their total proper motion. Do not multiply pmra by another cosine. Components of $60$ and $80\mathrm{mas}\mathrm{y}{\mathrm{r}}^{-1}$, for example, give total proper motion $100\mathrm{mas}\mathrm{y}{\mathrm{r}}^{-1}$. ^[ref].

For nearby stars in the ordinary nonrelativistic approximation, Euclidean distance $d$ converts total proper motion into transverse speed $v_{\perp }$:The small-angle approximation gives transverse speed as Euclidean distance times angular velocity in radians per unit time. With arcseconds and parsecs, the parsec convention turns their product into astronomical units per year. One astronomical unit per Julian year of $365.25$ days is approximately $4.74047\mathrm{km}{\mathrm{s}}^{-1}$. The coefficient is a unit conversion, rather than a property of a particular star.

Taking the model's $100\mathrm{mas}\mathrm{y}{\mathrm{r}}^{-1}=0.100\mathrm{arcsec}\mathrm{y}{\mathrm{r}}^{-1}$ as the total proper motion, with the other component zero, gives $v_{\perp }=4.74047\times 0.100\times 20\simeq 9.48\mathrm{km}{\mathrm{s}}^{-1}$. At the same total proper motion, Euclidean distances of $10$, $20$ and $100$ parsecs instead give transverse speeds of $4.74$, $9.48$ and $47.40\mathrm{km}{\mathrm{s}}^{-1}$, respectively. Their stellar parallax values are $100$, $50$ and $10\mathrm{mas}$. The same annual angular displacement can therefore describe quite different journeys through space.

The transverse speed still omits the line-of-sight velocity, which can be measured through the Doppler effect in spectral lines. Combining these measurements permits a three-dimensional velocity estimate in a consistent reference frame. ^[ref].

Historically, Edmond Halley's 1718 paper compared contemporary and ancient stellar directions and argued that some supposedly fixed stars had moved. Its title, "Considerations on the change of the latitudes of some of the principal fixt stars", captures the conceptual surprise: long-separated measurements could reveal motion invisible in an ordinary night's watching. Modern astrometry retains that insight, using repeated precise measurements to turn the apparently static sky into a record of changing positions. ^[ref]. ^[ref].

The scale factor $a(t)>0$ measures how spatial lengths in an FLRW metric change with cosmic time $t$. Choose $a$ to be a dimensionless quantity with $a(t_0)=1$ at a reference epoch, usually today. For two comoving observers, their fixed radial comoving distance $\chi$ and their proper distance in cosmology $d(t)$ satisfyThe proper distance in cosmology is measured along a spatial slice at one cosmic time; it is not the distance that a travelling photon covers between emission and reception. Only ratios of the scale factor have a normalization-independent meaning: replacing $a$ by $Ca$ and $\chi$ by $\chi /C$ leaves $d$ unchanged.

For example, take $\chi =100\mathrm{Mpc}$, where $\mathrm{Mpc}$ means a million parsecs. At $a=0.5,1,1.5$, the same pair of comoving observers has proper distance in cosmology $50,100,150\mathrm{Mpc}$ respectively. Every such separation changes by the same factor, although more distant pairs gain more distance. The illustration treats the marked objects as ideal comoving observers; internal motions in galaxies need not follow this background scaling.

The beauty of the scale factor is that one function describes the changing spatial ruler for every comoving observer. Spatial homogeneity gives no preferred centre of cosmic expansion: in a flat spatial patch, relative positions obey $\mathbf{r}-{\mathbf{r}}_O=a(\mathbf{x}-{\mathbf{x}}_O)$, so recentering on any comoving observer preserves Hubble's law. The evolution of the scale factor is determined by the Friedmann equations, while its ratio between emission and reception determines cosmological redshift.

Stellar parallax is the apparent change in a star's viewing direction caused by the observer's changing position. Nearby stars shift more against a distant reference background. In an ideal circular Earth orbit, the conventional annual stellar parallax $p$ uses a baseline of one astronomical unit, approximately the Earth-Sun distance. Two opposite observing positions, about six months apart, are separated by approximately two astronomical units. Their maximum angular separation is $2p$, so $p$ is half that excursion, not the full January-to-July shift. The observed excursion depends on the projected baseline and observing dates. ^[ref].

For the perpendicular baseline in the diagram, a right triangle giveswhere $p$ in the final expression is in radians. Combining the small-angle approximation with the parsec convention givesThus $p=0.05$ arcseconds implies $d\simeq 20$ parsecs, while the ideal six-month excursion is $0.10$ arcseconds. Geometry supplies this distance without assuming a luminosity or a particular stellar atmosphere.

In practical astrometry, many observations separate the annual stellar parallax signal from proper motion, the star's own changing sky position. The Gaia DR3 catalogue reports stellar parallax and its uncertainty in milliarcseconds: $50\mathrm{mas}=0.05$ arcseconds. A catalogue angle is a fitted measurement with measurement error. ^[ref].

For a positive stellar parallax with small standard deviation ${\sigma }_p$, first-order propagation gives ${\sigma }_d/d\simeq {\sigma }_p/p$. Thus $p=50\mathrm{mas}$ with ${\sigma }_p=1\mathrm{mas}$ gives $d\simeq 20$ parsecs with ${\sigma }_d\simeq 0.4$ parsecs, assuming systematic offsets are negligible. This linear estimate requires ${\sigma }_p\ll p$; a reciprocal is nonlinear and does not generally preserve symmetric uncertainty intervals.

Stipulate $H_0=70\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ and $c=299792.458\mathrm{km}{\mathrm{s}}^{-1}$, so $c/H_0=4282.7494\mathrm{Mpc}$. A megaparsec is $10^6$ parsecs. Neglect radiation in cosmology. Compare an accelerating flat model $({\Omega }_{m,0},{\Omega }_{\Lambda ,0})=(0.3,0.7)$, an open matter-only model $(0.3,0)$ and the Einstein-de Sitter universe $(1,0)$. Their present deceleration parameters are respectively $-0.55$, $0.15$ and $0.5$. For these models,Here $\chi$ is dimensionless; the radial null geodesic in FLRW spacetime supplies its integral and spatial curvature of an FLRW universe supplies $S_k$. The observer-to-source comoving radial distance is $(c/H_0)\chi$, whereas the transverse distance uses $S_k$. Omitting the latter distinction would bias the open-model comparison.

At $z=0.5$, numerical integration gives $\chi =0.440984335$ for the flat accelerating model and $\chi =0.392227378$ for the open matter-only model. ThusThe accelerating model is $0.21559$ mag fainter than the open matter-only prediction at the same redshift and standardized absolute magnitude, corresponding to $1.10438$ times the luminosity distance and $0.81991$ times the radiative flux. These are model calculations, not observations from either collaboration.

The Milne model has $q=0$, ${\Omega }_{m,0}={\Omega }_{\Lambda ,0}=0$ and ${\Omega }_{k,0}=1$, giving $\chi =\log (1+z)$ and $d_L=(c/H_0)(z+z^2/2)$. The Einstein-de Sitter luminosity-distance relation independently checks the flat matter-only integral. At $z=0.5$, the accelerating model exceeds the Milne model by only $0.12317$ mag, illustrating the dependence of a residual on the chosen reference. Over a longer redshift interval, luminosity distance integrates past deceleration as well as recent acceleration; a residual need not increase forever.

Multiplying $H_0$ by $f>0$ at fixed density fractions divides every $d_L$ by $f$ and shifts every distance modulus by $-5{\log }_{10}f$. The predicted apparent magnitude is unchanged if $M_{B,0}$ simultaneously increases by $5{\log }_{10}f$. This is the absolute magnitude–Hubble constant degeneracy. For example, $70\to 77\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ shifts $\mu$ by $-0.20696$ mag and $M_{B,0}$ by $+0.20696$ mag. Relative curves at the same $H_0$ retain their separation; determining $H_0$ requires an external absolute magnitude calibration.