Sensitivity of stellar electron-capture rates to parent neutron number: A case study on a continuous chain of twenty Vanadium isotopes

Gamow-Teller (GT) strength distributions (B(GT)) in electron-capture (EC) daughters stemming from the parent ground state are computed with the shell-model in the full pf-shell space, with QRPA in the formalism of Krumlinde and M$\backslash $"oller and with an Approximate Method for assigning an effective B(GT). These are compared to data available from decay and charge-exchange (CE) experiments across titanium isotopes in the pf-shell from A$=$43 to A$=$62, the largest set available for any chain of isotopes in the pf-shell. The present study is the first to examine B(GT) and the associated EC rates across a particular chain of isotopes with the purpose of examining rate sensitivities as neutron number increases. EC rates are also computed for a wide variety of stellar electron densities and temperatures providing concise estimates of the relative size of rate sensitivities for particular astrophysical scenarios. This work underscores the need for CE experiments in inverse kinematics on neutron-rich nuclei at future RIB Facilities.