  The nuclei $^{151}$Sm and $^{171}$Tm have been identified as attractive
candidates for the detection of the cosmic neutrino background. Both isotopes
undergo first-forbidden non-unique beta decays which inhibits a prediction of
their spectral shape using symmetries alone and this has, so far, obstructed a
definitive prediction of their neutrino capture cross sections. In this work we
point out that for both elements the so-called "$\xi$-approximation" is
applicable and this effectively reduces the spectral shape to deviate by at
most $1\%$ from the one that would arise if beta decays were of the allowed
type. Using measured half-lives we extract the relevant nuclear matrix element
and predict the neutrino capture cross sections for both isotopes at $1\%$
level, accounting for a number of relevant effects including radiative
corrections and the finite size of the nuclei. We obtained $(1.12\pm
0.01)\times 10^{-46}{\rm cm}^2$ for $^{171}$Tm and $(4.77\pm 0.01)\times
10^{-48}{\rm cm}^2$ for $^{151}$Sm. This method is robust as it does not rely
on the data points near the end-point of the beta spectrum which may be
contaminated by atomic physics effects, namely shake-up and shake-off. Finally,
we calculate the target mass which is necessary for cosmic neutrino discovery
and discuss several bottlenecks and respective solutions associated to the
experimental program. We conclude that the detection of cosmic neutrino
background by neutrino capture on $^{151}$Sm and $^{171}$Tm is achievable and
free from theoretical limitations but still subject to technical issues that
should be further investigated by the experimentalists in the context of the
proposed PTOLEMY project.
