What is our immediate association when we think of "Asarah be-Teves"? A fast day, no eating or drinking. Tzom Gedaliyah, the seventeenth of Tamuz and the ninth of Av trigger the same [unwelcome?] association. And while that association is obviously correct, it is woefully inadequate.
There are days on which the Jewish people fast on account of the tragedies that occurred thereupon, in order to arouse the hearts and open the pathways of teshuva ... these are the fast days: the third of Tishrei ... , and the tenth of Teves... [Rambam, Hilchos Ta'anios 5:1,2]
Undoubtedly we abstain from food and drink on the tenth of Teves. But fasting is not and end unto itself. Fasting induces frailty. Frailty, in turn, generates humility, the necessary mindset for feelings of remorse and contrition, and teshuva.
Our association with Asarah be-Teves is fasting. Our association should be repenting. While fasting is compulsory for all [with the obvious exception of those whom the halacha exempts], ta'anis without teshuva is hollow, a body sans soul.