One day of solar motion, one band edge
The Sun advances a shade under a degree every day, and the ten degree bands inside a sign are therefore about ten days wide. Any given date lands somewhere inside one of them, and only a handful of dates land close enough to an edge for the year to matter. This is one of those dates, and its neighbour on the twelfth is the other half of the same story.
Because the range starts at nineteen point eight and ends at twenty one point two, the honest statement is that the thirteenth is in the third decan of Leo in most years and in the second in a few. Writing it as one or the other without qualification loses the only part of the answer that varies.
The reason the neighbouring dates behave differently is simply arithmetic. Each date carries a range of roughly one and a third degrees across the century, and the ranges of successive dates overlap heavily, so a line at twenty degrees passes through two of them rather than one. The twelfth is mostly below the line and the thirteenth is mostly above it, which is as precise as a date can be about the matter.
Mars above the line
From twenty degrees to the Virgo boundary, the band is Mars's. That is where the Chaldean order lands, counting the seven planets by apparent speed from the opening of Aries, and it is also where the triplicity scheme lands, since the fire signs run Aries, Leo, Sagittarius and the final band of Leo therefore falls to Aries and its ruler.
The tradition treats these degrees as the assertive end of Leo, the part more inclined to contest than to preside. Below the line the same tradition reads Jupiter's warmth and expansiveness instead. Neither reading is derived from an observation; both are conventions with long paper trails, and it is fair to use them as long as they are labelled for what they are.
Why the spread is as small as it is
A solar year is about a quarter of a day longer than a calendar year of three hundred and sixty five days. Left alone, the Sun's position on a fixed date would slide steadily backwards. The leap day exists to stop that, and the result is a four year sawtooth: three years of forward drift, then a jump back.
Across a century the sawtooth keeps the whole spread on this date inside a band of roughly one and a third degrees, with a very slow secular drift on top of it. The stability is real, and it is what makes a decan edge inside the range worth flagging rather than hand waving.
Everything the date does not settle
Leo is fixed fire ruled by the Sun, and that is true of every birthday for a month either side. What distinguishes charts is not the sign but everything the sign does not cover.
The Moon shifts twelve to fifteen degrees in a day and can be in either of two signs on this date depending on the hour. The Ascendant can be any of the twelve, since a full circle rises in a day, and its timing depends on how far north or south the birth took place. The widget above uses the year, hour and birthplace to give the exact solar degree, to say which decan that degree lands in, and to report the distance to the nearest sign boundary.
Questions people ask about this
Is 13 August in the third decan of Leo?
In most years, yes. The noon position runs from just under twenty degrees to just over twenty one across the last century, and twenty is where the third decan opens, so the low years fall just short.
Who rules the last decan of Leo?
Mars, under both schemes in general use. The Chaldean count reaches it by planetary speed and the triplicity scheme reaches it through Aries, so there is no disagreement to resolve here.
Is 13 August ever not Leo?
No. The Sun does not reach Virgo until around the twenty second of August, so this date has been Leo in every modern year regardless of hour or time zone.
How much does the position change between years?
A little over a degree in total. Leap years pull it back and the following years push it forward, which is enough to move this date across a decan edge but nowhere near enough to move it across a sign boundary.
Does the decan change how the sun sign is read?
It narrows it. Leo covers thirty degrees, the band covers ten, and the tradition attaches a different colouring to each band without altering the sign underneath.
What should I do if my year is one of the low ones?
Read the second decan of Leo instead, under Jupiter. The widget above will tell you which side you fall on once it has your hour and birthplace.