When distance and inclination move a whole curve

The same angle can span a larger physical radius

Write the source distance as $D_0$, a candidate distance as $D$, and their ratio as $d=D/D_0$. Keeping positions on the sky fixed makes physical radius proportional to distance. Luminosity inferred from the received flux scales as distance squared, so baryonic mass at fixed mass-to-light ratio does too.

For the same angular shape, lengths scale as $d$ and masses as $d^2$. The baryonic acceleration stays unchanged, its speed-squared contribution scales as $d$, and its equivalent speed scales as $\sqrt d$. This fixed-shape homology does not recalculate a warp or perform a new photometric deprojection.

Reading: Lelli, McGaugh & Schombert (2016), SPARC, AJ152,157 · Independent saved-observation rotation-curve calculation

A more face-on disk hides more of its motion

Inclination $i$ is measured from the disk normal to our line of sight. A face-on disk has zero inclination; an edge-on disk is near a right angle. In this global circular-motion mapping, after the sky-position geometry is handled, line-of-sight motion contains the factor $\sin i$. A smaller inclination therefore requires a larger circular speed for the same observed motion.

Let $i_0$ be the source inclination and $V_{0,j},\epsilon_{0,j}$ its deprojected speed and error. A candidate inclination multiplies both by $k=\sin i_0/\sin i$. Every radius shares that same factor. It is not a new random error drawn independently for each point.

Changing from $63^{\circ}$ to $60^{\circ}$ multiplies speed and its plotted uncertainty by $1.02885$, an increase of about 2.9 percent. Changing distance from $3.16$ to $3.32\,\mathrm{Mpc}$ multiplies radius and baryonic speed squared by $1.05063$; equivalent speed changes by its square root.

Reading: Lelli, McGaugh & Schombert (2016), SPARC, AJ152,157 · Independent saved-observation rotation-curve calculation

Choose a fixed data coordinate for the likelihood

A likelihood asks how common the observed data would be under a specified model and error law. It is a probability density of data viewed as a function of model parameters. It is not itself a probability distribution for those parameters.

Fix $y_j=V_{0,j}\sin i_0$ and $\sigma_{y,j}=\epsilon_{0,j}\sin i_0$. At a candidate inclination the model predicts $\mu_j=v_{c,j}\sin i$. The uncertainty $\sigma_{y,j}$ belongs to the fixed reference-projected coordinate and does not change with candidate inclination.

For an initial calculation, assume independent Gaussian errors. Multiply their probability densities and take a natural logarithm. This produces a residual term and a density normalizer. Native SPARC error bars do not supply a complete covariance across radii or certify this Gaussian law; the resulting fit remains conditional on that working assumption.

The residual sum is $\chi^2=\sum_j[(y_j-\mu_j)/\sigma_{y,j}]^2$. A reduced chi-square divides it by the number of data points minus the number of fitted parameters. Our later example has $73$ points and $3$ fitted parameters, so its denominator is $70$. These are summaries under the stated error law; their numerical values do not by themselves validate that law.

Reading: Independent saved-observation rotation-curve calculation · Li et al., Fitting the radial acceleration relation, arXiv:1803.00022v1

Changing coordinates also changes density height

Suppose a calculation instead uses candidate-dependent circular coordinates $kV_0$ and errors $k\epsilon_0$. The residual divided by its error can stay the same, but the probability density height changes. Stretching each of $N$ data coordinates by $k$ contributes $k^{-N}$ to the density. A Jacobian restores this change of measure.

Call the candidate-coordinate likelihood $L_{\rm scaled}$, the original deprojected-coordinate likelihood $L_{V_0}$, and the fixed-projected-coordinate likelihood $L_y$. Restore the inclination-dependent scale first, then account for the fixed reference sine.

A useful counterexample makes the model fit the data perfectly at every inclination. Its likelihood in a fixed error coordinate should be unchanged. Omitting the correction gives $72^{\circ}$ an artificial log-likelihood advantage of about $7.764$ over $54^{\circ}$ for $48$ points, solely because the candidate error coordinates shrank.

Reading: Independent saved-observation rotation-curve calculation

More points do not remove a common offset

Covariance describes how errors at two positions vary together. If nine speeds have independent noise plus one common calibration offset, averaging reduces the independent contribution to the mean. The same offset still shifts all nine measurements together.

Take independent standard deviation $2.5\,\mathrm{km\,s^{-1}}$ and common-offset standard deviation $3\,\mathrm{km\,s^{-1}}$. The mean variance is $2.5^2/9+3^2=9.6944\,(\mathrm{km\,s^{-1}})^2$. Treating the common term as independent would give $(2.5^2+3^2)/9=1.6944$, a much smaller uncertainty.

Distance and inclination generally produce radius-dependent changes rather than a uniform velocity shift. This small example explains what “shared” means. The tutorial separately recalculates distance and inclination as common parameters of an entire curve.

Reading: Independent saved-observation rotation-curve calculation

A profile displays compensating parameter choices

Fix one stellar mass-to-light ratio and re-optimize the halo density and scale. Repeat at another ratio. The sequence of conditional best likelihoods is a profile. It displays compensating choices that a single optimum can conceal.

The tutorial tries six starting points for each halo. Some starts for the synthetic cored fit stop at a poorer local minimum. “Optimization succeeded” means that a stopping condition was met; compare the objective values, inspect parameter bounds and save the predictions and residuals.

A profile is not automatically a calibrated interval. Converting its height into a confidence statement requires the error law, parameter conditions and coverage to be checked. Here the plotted result remains the conditional profile itself.

Reading: Independent saved-observation rotation-curve calculation

Keep the large residuals of the real object

For the fixed $73$-point NGC 2403 record at its reference geometry, the three-parameter NFW and cored fits have working reduced chi-square values near $9.14$ and $10.70$. Both are far from the often-used reference value near one. The simple models with these point-error weights describe the data poorly.

Further work can examine geometry, stellar mass, noncircular motions and correlated errors, with each additional assumption stated. Keeping the original errors and every selected point exposes which discrepancy the next piece of evidence needs to address.

Reading: Lelli, McGaugh & Schombert (2016), SPARC, AJ152,157 · Independent saved-observation rotation-curve calculation