Multi-epoch observations with the Advanced Camera Survey and WFC3 on the Hubble Space Telescope provide a unique and comprehensive probe of stellar dynamics within 47 Tucanae. We confront analytic models of the globular cluster with the observed stellar proper motions that probe along the main sequence from just above 0.8–0.1M ⊙ as well as white dwarfs younger than 1 Gyr. One field lies just beyond the half-light radius where dynamical models (e.g., lowered Maxwellian distributions) make robust predictions for the stellar proper motions. The observed proper motions in this outer field show evidence for anisotropy in the velocity distribution as well as skewness; the latter is evidence of rotation. The measured velocity dispersions and surface brightness distributions agree in detail with a rotating anisotropic model of the stellar distribution function with mild dependence of the proper-motion dispersion on mass. However, the best-fitting models underpredict the rotation and skewness of the stellar velocities. In the second field, centered on the core of the cluster, the mass segregation in proper motion is much stronger. Nevertheless the model developed in the outer field can be extended inward by taking this mass segregation into account in a heuristic fashion. The proper motions of the main-sequence stars yield a mass estimate of the cluster of at a distance of 4.7 kpc. By comparing the proper motions of a sample of giant and subgiant stars with the observed radial velocities we estimate the distance to the cluster kinematically to be 4.29 ± 0.47 kpc.

Our goal in this paper is to derive a carbon-star luminosity function that will eventually be used to determine distances to galaxies at 50–60 Mpc and hence yield a value of the Hubble constant. Cool N-type carbon stars exhibit redder near-infrared colours than oxygen-rich stars. Using Two Micron All Sky Survey near-infrared photometry and the Gaia Data Release 2, we identify carbon stars in the Magellanic Clouds (MC) and the Milky Way (MW). Carbon stars in the MC appear as a distinct horizontal feature in the near-infrared ((J − Ks)0, MJ) colour–magnitude diagram. We build a colour selection (1.4 < (J − Ks)0 < 2) and derive the luminosity function of the colour-selected carbon stars. We find the median absolute magnitude and the dispersion, in the J band, for the Large and the Small Magellanic Clouds (LMC/SMC) to be, respectively, ($\bar{M_J} = -6.284~\pm ~0.004$ and σ = 0.352  ±  0.005) and ($\bar{M_J} = -6.160~\pm ~0.015$ and σ = 0.365  ±  0.014). The difference between the MC may be explained by the lower metallicity of the SMC, but in any case it provides limits on the type of galaxy whose distance can be determined with this technique. To account for metallicity effects, we developed a composite magnitude, named C, for which the error-weighted mean C magnitude of the MC are equal. Thanks to the next generation of telescopes (JWST, ELT, and TMT), carbon stars could be detected in MC-type galaxies at distances out to 50–60 Mpc. The final goal is to eventually try and improve the measurement of the Hubble constant while exploring the current tensions related to its value.

We introduce a new distance determination method using carbon-rich asymptotic giant branch stars (CS) as standard candles and the Large and Small Magellanic Clouds (LMC and SMC) as the fundamental calibrators. We select the samples of CS from the ((J − Ks)0, J0) colour–magnitude diagrams, as, in this combination of filters, CS are bright and easy to identify. We fit the CS J-band luminosity functions using a Lorentzian distribution modified to allow the distribution to be asymmetric. We use the parameters of the best-fitting distribution to determine if the CS luminosity function of a given galaxy resembles that of the LMC or SMC. Based on this resemblance, we use either the LMC or SMC as the calibrator and estimate the distance to the given galaxy using the median J magnitude ($\overline{J}$) of the CS samples. We apply this new method to the two Local Group galaxies NGC 6822 and IC 1613. We find that NGC 6822 has an ‘LMC-like’ CS luminosity function, while IC 1613 is more ‘SMC-like’. Using the values for the median absolute J magnitude for the LMC and SMC found in Paper I we find a distance modulus of μ0 = 23.54 ± 0.03 (stat) for NGC 6822 and μ0 = 24.34 ± 0.05 (stat) for IC 1613.

By examining the diffusion of young white dwarfs through the core of the globular cluster 47 Tucanae, we estimate the time when the progenitor star lost the bulk of its mass to become a white dwarf. According to stellar evolution models of the white-dwarf progenitors in 47 Tucanae, we find this epoch to coincide approximately with the star ascending the asymptotic giant branch (3.0 ± 8.1 Myr before the tip of the AGB) and more than ninety million years after the helium flash (with ninety-percent confidence). From the diffusion of the young white dwarfs we can exclude the hypothesis that the bulk of the mass loss occurs on the red-giant branch at the four-sigma level. Furthermore, we find that the radial distribution of horizontal branch stars is consistent with that of the red-giant stars and upper-main-sequence stars and inconsistent with the loss of more than 0.2 solar masses on the red-giant branch at the six-sigma level.

Using images from the Hubble Space Telescope Wide-Field Camera 3, we measure the rate of diffusion of stars through the core of the globular cluster 47 Tucanae using a sample of young white dwarfs identified in these observations. This is the first direct measurement of diffusion due to gravitational relaxation. We find that the diffusion rate κ ≈ 10 − 13 arcsecond 2 Myr −1 is consistent with theoretical estimates of the relaxation time in the core of 47 Tucanae of about 70 Myr.