Jesus born at night 1

I've been preaching on Jesus's birth over 20 years and I know the relevant texts well. One question I've never asked is exactly when Jesus was born. My instinct says 'at night', probably quite late. Am I right?

Christmas hymns certainly suggest it. There is Gruber's famous 'Silent night, holy night!' and Philips Brooks' 'O little town of Bethlehem!' He speaks of how

Above thy deep and dreamless sleep the silent stars go by.

Yet in thy dark streets shineth the everlasting Light;

The hopes and fears of all the years are met in Thee tonight.

For Christ is born of Mary, and gathered all above,

While mortals sleep, the angels keep their watch of wondering love.

O morning stars together, proclaim the holy birth,

And praises sing to God the King, and peace to men on earth!

The idea is reinforced by hymns such as 'All my heart this night rejoices as I hear far and near Sweetest angel voices', 'on a cold winter’s night that was so deep' and 'it came upon the midnight clear' and, in a different way, 'the race that long in darkness pined'.
But what about Scripture itself? References to the wisemen are irrelevant as, contrary to tradition, they did not see Jesus until some time after his birth. In Luke 2, immediately after saying that Jesus was born the evangelist writes (2:8) And there were shepherds living out in the fields nearby, keeping watch over their flocks at night. This clearly points to the birth happening at night. Knowing John's fondness for such imagery I wonder if there is any significance in John 1:5 - The light shines in the darkness, but the darkness has not understood it. Given the state of the world, just as his being placed in a manger is surely significant, so also is his birth in the darkness.