  Solid states of strange-cluster matter called strangeon matter can form
strangeon stars that are highly compact. We show that strangeon matter and
strangeon stars can be recast into dimensionless forms by a simple
reparametrization and rescaling, through which we manage to maximally reduce
the number of degrees of freedom. With this dimensionless scheme, we find that
strangeon stars are generally compact enough to feature a photon sphere that is
essential to foster gravitational-wave (GW) echoes. Rescaling the dimension
back, we illustrate its implications on the expanded dimensional parameter
space, and calculate the GW echo frequencies associated with strangeon stars,
showing that the minimum echo frequency is $\sim 8$ kHz for empirical parameter
space that satisfies the GW170817 constraint, and can reduce to $\mathcal
O(100)$ Hertz at the extended limit.

High Energy Astrophysical Phenomena;General Relativity and Quantum Cosmology;High Energy Physics - Phenomenology;Nuclear Theory