[Last year's post about Asarah be-Tevet is even more apt this year than last.]
Why the Tenth of Tevet?
As we know, while the Jews were still in exile in Babylonia, four dates were set in the Jewish calendar to commemorate the fall of Jerusalem and the destruction of the First Temple in 586 B.C.E. This is attested by the prophet Zechariah (7:1-5):
In the fourth year of King Darius, on the fourth day of the ninth month, Kislev, the word of the Lord came to Zechariah – when Bethel-sharezer and Regem-melech and his men sent to entreat the favor of the Lord, [and] to address this inquiry to the priests of the House of the Lord and to the prophets: “Shall I weep and practice abstinence in the fifth month, as I have been doing all these years?”
Thereupon the word of the Lord of Hosts came to me: Say to all the people of the land and to the priests: When you fasted and lamented in the fifth and seventh months all these seventy years, did you fast for my benefit?
According to the Sages,  this passage refers to four days during the months mentioned by Zechariah (see below) that had been set aside for fasting and prayer due to the troubles that had befallen the people on those days, as reported in Tosefta Sukkah, Lieberman edition, ch. 6, halakhah 10 (p.189):
Rabbi expounded:  Lo, the prophet says: Thus said the Lord of Hosts: The fast of the fourth month, the fast of the fifth month, the fast of the seventh month, and the fast of the tenth month ...(Zech. 8:19) The fast of the fourth month is the seventeenth of Tammuz, on which day the city walls were breached, ... the fast of the fifth month is the ninth of Ab, on which day the Temple was burned, ... the fast of the seventh month is the third of Tishri, on which day Gedaliah son of Ahikam was killed by Ishmael son of Nethanya, teaching us that the Omnipresent views the death of the righteous just as severely as the destruction of the Temple ... the fast of the tenth month is the tenth of Tevet, on which day the king of Babylonia laid hands on Jerusalem, as it is said, “In the ninth year, on the tenth day of the tenth month, the word of the Lord came to me: O mortal, record this date...” (Ezek. 24:1-2).
Three of these four fast days have a theme in common. To begin with, they commemorate tragic events whose results were immediate and calamitous: breaching the walls on the seventeenth of Tammuz marked the inevitable fall of the entire city of Jerusalem;  the First and Second Temple were destroyed on the ninth of Ab; on the Fast of Gedaliah, “Gedaliah son of Ahikam was killed and the last remaining ember of Israel was extinguished, sealing their fate that they be exiled.”  Secondly, these dates had the good fortune of being links in larger segments of the calendar – the seventeenth of Tammuz and the ninth of Ab belong to the three weeks known as Bein ha-Metzarim,” between the straits”  and the fast of Gedaliah was incorporated into the ten days of repentance.  Therefore these days became more prominent in the public consciousness.
The Odd Man Out
The situation is somewhat different for the fast of the tenth of Tevet. It is isolated on the calendar, not part pf any context that might strengthen awareness of the date and its importance.  At first glance, its content also appears different from the other fasts, since it marks neither the end of a process nor an event with immediate impact. The tenth of Tevet marks the date on which the siege of Jerusalem began. Hence we must ask, what in the events of that day moved the exiles themselves to proclaim a day of fasting and memorial over the beginning of the siege on Jerusalem?  What sort of trauma passed over the Jewish people due to the beginning of the siege?
Indeed, the Jews everywhere were deeply shaken to hear the bad tidings,  and this finds clear expression in the words of the prophet Ezekiel. Thus the exiled prophet on the Chebar Canal received the bitter news (Ez. 24:1-2):
In the ninth year, on the tenth day of the tenth month, the word of the Lord came to me: O mortal, record this date, this exact day; for this very day the king of Babylon has laid siege to Jerusalem.
The prophet’s trembling reverberates through his words. Thrice “this day” is repeated, as if to stress the intense significance of what befell Jerusalem that particular day. The reader feels as if the prophet is stunned and shaken, refusing to believe the news that the Holy One, blessed be He, brought him; therefore, the Lord emphasizes time and again that the siege indeed began “this very day.” The question arises: Did not Ezekiel know that Nebuchadnezzar was invading the land of Israel, heading for Jerusalem with the intention of conquering it? So why does he appear to have been taken by surprise?
A Sense of Security
The answer, I believe, is to be found in the feeling of the residents of Judea that Jerusalem and the Temple were immune to attack. This is expressed unequivocally in the famous oratory of the prophet Jeremiah (7:1-8):
The word which came to Jeremiah from the Lord: Stand at the gate of the House of the Lord, and there proclaim this word: Hear the word of the Lord, all you of Judah who enter these gates to worship the Lord! Thus said the Lord of Hosts, the God of Israel: Mend your ways and your actions, and I will let you dwell in this place. Don’t put your trust in illusions and say, “the Temple of the Lord, the Temple of the Lord, the Temple of the Lord are these [buildings].”... See, you are relying on illusions that are of no avail.
Jeremiah was warning and speaking out against the apparently widespread belief among the people that the Temple and the city of Jerusalem were totally immune to attack by the king of Babylonia.  The Lord, so they maintained, would not cause His Temple to be destroyed, nor would He make His land surrender and His people be exiled, and their moral and religious behavior would not change matters. Jeremiah sought to shatter their false illusions (Jer. 7:9-15):
Will you steal and murder and commit adultery and swear falsely, and sacrifice to Baal, and follow other gods whom you have not experienced, and then come and stand before Me in this House which bears My name and say, “We are safe”? – [Safe] to do all these abhorrent things! Do you consider this House, which bears My name, to be a den of thieves? As for Me, I have been watching – declares the Lord.
Just go to My place at Shiloh, where I had established My name formerly and see what I did to it because of the wickedness of My people Israel. And now because you do all these things – declares the Lord – and though I spoke to you persistently, you would not listen; and though I called to you, you would not respond – therefore I will do to the House which bears My name, on which you rely, and to the place which I gave you and your fathers just what I did to Shiloh. And I will cast you out of My presence as I cast out your brothers, the whole brood of Ephraim.
A Bitter Lesson
The reason for Ezekiel’s shock, it seems, is to be found herein. Clearly he knew what was going to happen, for he himself had a vision of the catastrophe for which the people were headed. But when the day actually arrived, he found it difficult to assimilate what was happening and therefore the Holy One, blessed be He, had to tell him emphatically, time and again, that this was indeed the reality.
Little wonder, therefore, that the rest of the people were even more deeply traumatized when Jerusalem was put under siege.  The stern lesson of the Tenth of Tevet – that Jerusalem was vulnerable on account of the nation’s corruption—was what led the people to include the day on which the siege of Jerusalem began among the days of mourning and commemoration for the destruction of the First (and later also the Second) Temple. 
In view of Jeremiah’s words, cited above, Maimonides’ remarks are even more poignant: 
There are days on which all of Israel fast and practice abstinence on account of the troubles that befell them on those days, in order to stir the heart to repentance, that this may remind us of our evil ways and the ways of our ancestors which were like our own ways now, so much so that it caused them and us the same troubles. For in remembering these things we will return to the good path as it is written (Lev. 26:40), “and they shall confess their iniquity and the iniquity of their fathers, in that they trespassed against Me, yea, were hostile to Me.”
In loving memory of my mother, Peshe bat Yosef, who passed away on the ninth of Tevet, 5751 (1990).
 See the discussion in J. Tabori, Moadei Yisrael be-Tekufat ha-Mishnah ve-ha-Talmud, Jerusalem 2000, p. 350 ff.
 According to Lieberman, Tosefta Ki-fshutah, Sotah-Kiddushin, p. 674, lines 187-189, read “Rabbi Akiva expounded.”
 According to Jeremiah, the city walls were actually breached by the Babylonians on the ninth of Tammuz and not on the seventeenth. On the custom to fast on the seventeenth of the month, the day on which the Romans entered the city prior to the destruction of the Second Temple, cf. Jerusalem Talmud, Ta’anit 4.5 (p.68c); Ritba, Rosh ha-Shanah 18b, s.v. “girsat ha-sefarim;” Shulhan Arukh Orah Hayyim 549.2.
 So Maimonides, Mishneh Torah, Laws of Fastdays 5.2. Regarding the Rambam’s original explanation as to why the fast was declared, see Isidore Twersky, Introduction to the Code of Maimonides (Mishneh Torah), New Haven, 1980, p. 429, note 182.
 Based on the verse, “Judah has gone into exile because of misery and harsh oppression; when she settled among the nations she found no rest; all her pursuers overtook her in the narrow places [Heb. bein ha-metzarim]” (Lament. 1.3).
 Cf. Y. D. Gilat, “Ta’anit be-Shabbat,” Tarbiz, 52 (1983), 1-15 (=Perakim be-Hishtalshelut ha-Halakhah, Ramat Gan 1092, p. 217 ff.), and D. Sperber, Minhagei Yisrael I, Jerusalem 1989, pp. 138-153.
 Acknowledgment of the somewhat shaky standing of the tenth of Tevet can be found in the fact that Rabbi Herzog suggested that this particular day be chosen as the day commemorating the Holocuast and general national mourning because he wished in a certain way to buttress the status of the day. See Irving Greenberg, The Jewish Way: Living the Holidays, New York 1988, 314-372.
 It is evident from the Lord’s response (“when you fasted and lamented in the fifth and seventh months all these seventy years, did you fast for my benefit?”) that initially the memorial days were set on the people’s initiative and only later received ratification.
 Compare Tosefta Sotah (loc. sit.), halakhah 11.
 Presumably they were relying on the miracle that G-d wrought when Sennacherib was repulsed from the city walls in the time of Hezekiah (I Kings 19).
 Note, not to imply any comparison, that there were similar reactions in the western world when Rome was pillaged by the Goths (cf. Jerome, Letter CXXVII and Augustin, Civitas Dei, Book I, ch. 1).
 It is interesting to note that at this stage Jerusalem had not yet fallen. However, see Tosefta cited above, note 9.
 Laws of Fastdays, loc. sit., halakhah 1.
'Thus saith the Lord of hosts: The fast of the fourth month, and the fast of the fifth, and the fast of the seventh, and the fast of the tenth, shall be to the house of Judah joy and gladness, and cheerful seasons; therefore love truth and peace. ' (Zekh. 8, 19)
במהרה בימינו אמן.