kissing number problem

For an integer $d\ge 1$, call a finite set $\{x_{1},\dots,x_{N}\}\subset\mathbb{R}^{d}$ admissible if

$$x_{n}^{\mathsf{T}}x_{n}=1\quad\text{for all }n,\qquad (x_{n}-x_{m})^{\mathsf{T}}(x_{n}-x_{m})\ge 1\quad\text{for all }m\neq n,$$

that is, if the $x_{n}$ are unit vectors whose pairwise angular separations are all at least $60^{\circ}$. Let

$$\tau_{d}=\max\{N\in\mathbb{N}: \text{there exists an admissible set of } N \text{ vectors in } \mathbb{R}^{d}\},$$

which is finite by compactness of the unit sphere. Equivalently, $\tau_{d}$ is the greatest number of closed balls of radius $1$ in $\mathbb{R}^{d}$ with pairwise disjoint interiors that can all be tangent to one fixed closed ball of radius $1$: after translating the fixed ball to be centred at the origin and rescaling the centres of the tangent balls by $1/2$, tangency to the central ball becomes $\|x_{n}\|=1$ and disjointness of interiors becomes $\|x_{n}-x_{m}\|\ge 1$.

Determine the value of $\tau_{d}$ for every integer $d\ge 1$.