<p>The <a href="https://en.wikipedia.org/wiki/Year_zero" rel="nofollow noreferrer">Year 0</a> does not exist in BCE/CE and BC/AD notation. It jumps from 1 BCE to 1 CE directly. So if we look at the sequence of years in BCE/CE notation, we have:</p>

<pre><code>3 BCE
2 BCE
1 BCE
1 CE
2 CE
3 CE
</code></pre>

<p>If you want to calculate the number of years between, say, January 1, 2 BCE and January 1, 2 CE, you can add the year numbers to get 4, but then you need to <em>subtract</em> 1 to remove year 0 from the total, giving a total of 3 years. In your particular case, this leads to <span class="math-container">$4713 - 1 + 2010 = 6722$</span> years.</p>

<p>In astronomy, we often use another year numbering system called the <a href="https://en.wikipedia.org/wiki/Astronomical_year_numbering" rel="nofollow noreferrer">Astronomical year numbering</a>, which includes a year 0 and makes things more convenient arithmetically when calculating time intervals between eras. It goes (-3, -2, -1, 0, 1, 2, etc.) To convert years BCE to the Astronomical year notation, we subtract 1 and add minus(-) in front of the year. For example, the year 4713 BCE is -4712 in Astronomical year notation. Your year interval is then calculated as <span class="math-container">$2010 - (\text{-}4712) = 2010 + 4712 = 6722$</span>. </p>

<p>For the leap years calculation, since -4712 is a leap year, this simplifies matters, but don't forget to take the smallest integer greater than or equal to your result (the ceiling function). So in this case <span class="math-container">$\frac{6722}{4} - 3 = 1677.5$</span>, and the ceiling is 1678, and that is the number of leap years between your years. Also, in your initial calculation, you had forgotten to subtract 3, because 6724 / 4 - 3 = 1678, not 1681. </p>

<p>Finally, the time interval between January 1 and February 28 is 58 days, not 59, because we do not count the last date as part of the interval. This gives a total of <span class="math-container">$5044 * 365 + 1678 * 366 - 10 + 58 + \frac{6 * 60 * 60 + 30 * 60}{24 * 60 * 60} = 2455256.27$</span>. </p>

<p>You state that the actual answer is 2455255.77, but in fact the answer really is 2455256.27. Perhaps you typed "06:30" instead of "18:30" in the converter. </p>

CC BY-SA 4.0
2019-02-26T04:48:17.587Z
2019-02-26T22:22:47.193Z
2019-02-26T22:22:47.193Z
FSimardGIS
<p>As an exercise, I am trying to calculate the JD from a given date in the Gregorian Calendar at a time given in UT. Furthermore, I want to do so from first principles without relying on any formula. Let's say I choose UT 18:30 on 2/28/2010. I am not able to obtain a result consistent with the U.S. Naval Observatory results. I am in need of a calendar guru to show me where my calculation is going astray.</p>

<p>First I find the number of years from 4713 BCE (Julian Date 0) and the target year making sure to add an extra one so as to include 0 CE. This is <span class="math-container">$4713 + 1 + 2010 = 6724$</span> years. The number of these years which are leap is <span class="math-container">$\frac{6724}{4} - 3 = 1681$</span>. This equation attempts to reflect that every fourth year is a leap year except for those years falling after 1582 (the year of the Gregorian Calendar reform) which are both divisible by <span class="math-container">$100$</span> and <em>not</em> by <span class="math-container">$400$</span>. There are three such years which need to be deducted from the total-- 1700, 1800, 1900. This leaves <span class="math-container">$6724 - 1681 = 5043$</span> non-leap years.</p>

<p>Now I convert these to days and subtract the <span class="math-container">$10$</span> days which were skipped as part of the reform from October 4 to October 15 in 1582. <span class="math-container">$5043*(365) + 1681*(366) - 10 = 2,455,928$</span>. Next, I find the number of days from January 1st to the target day in the target year (February 28th). This is <span class="math-container">$59$</span> days and so I have <span class="math-container">$2,455,987$</span>.</p>

<p>Finally, the time number of seconds afternoon on the target day is calculated as a fraction of the number of seconds in a 24 hour period. This works out to be <span class="math-container">$\frac{6(60)(60) + 30(60)}{24(60)(60)} \approx 0.27$</span>. Adding this to the earlier result yields <span class="math-container">$JD = 2455987.27$</span>.</p>

<p>The <em>actual</em> answer is <span class="math-container">$2455255.77$</span> which leaves a difference of <span class="math-container">$2455987.27 - 2455255.77 = 731.5$</span> days. This is nearly two years. Obviously, I have either misunderstood the calculation or have performed it incorrectly. Any help would be much appreciated!</p>

CC BY-SA 4.0
2019-02-26T03:40:29.103Z
[USNO Julian Date Converter](https://aa.usno.navy.mil/data/docs/JulianDate.php) returns JD 2455256.27 for 2010-02-28 18:30 UT. There is no year 0, so 1 CE - 1 BCE = 1 year; subtract 1 year instead of adding 1. [MJD](https://tycho.usno.navy.mil/mjd.html) is simpler to compute.
Mike G
CC BY-SA 4.0
2019-02-26T20:29:30.053Z
Wow this is a very interesting question!
Max0815
<time><celestial-mechanics>
Determine Julian Date from Gregorian without formula
Shadow43375
# Determine Julian Date from Gregorian without formula

(Asked by: username_0 on 2019-02-26)

As an exercise, I am trying to calculate the JD from a given date in the Gregorian Calendar at a time given in UT. Furthermore, I want to do so from first principles without relying on any formula. Let's say I choose UT 18:30 on 2/28/2010. I am not able to obtain a result consistent with the U.S. Naval Observatory results. I am in need of a calendar guru to show me where my calculation is going astray.

First I find the number of years from 4713 BCE (Julian Date 0) and the target year making sure to add an extra one so as to include 0 CE. This is $4713 + 1 + 2010 = 6724$ years. The number of these years which are leap is $\frac{6724}{4} - 3 = 1681$. This equation attempts to reflect that every fourth year is a leap year except for those years falling after 1582 (the year of the Gregorian Calendar reform) which are both divisible by $100$ and *not* by $400$. There are three such years which need to be deducted from the total-- 1700, 1800, 1900. This leaves $6724 - 1681 = 5043$ non-leap years.

Now I convert these to days and subtract the $10$ days which were skipped as part of the reform from October 4 to October 15 in 1582. $5043\*(365) + 1681\*(366) - 10 = 2,455,928$. Next, I find the number of days from January 1st to the target day in the target year (February 28th). This is $59$ days and so I have $2,455,987$.

Finally, the time number of seconds afternoon on the target day is calculated as a fraction of the number of seconds in a 24 hour period. This works out to be $\frac{6(60)(60) + 30(60)}{24(60)(60)} \approx 0.27$. Adding this to the earlier result yields $JD = 2455987.27$.

The *actual* answer is $2455255.77$ which leaves a difference of $2455987.27 - 2455255.77 = 731.5$ days. This is nearly two years. Obviously, I have either misunderstood the calculation or have performed it incorrectly. Any help would be much appreciated!

**Tags:** time, celestial-mechanics
> **Comments:**
> **username_1**: [USNO Julian Date Converter](https://aa.usno.navy.mil/data/docs/JulianDate.php) returns JD 2455256.27 for 2010-02-28 18:30 UT. There is no year 0, so 1 CE - 1 BCE = 1 year; subtract 1 year instead of adding 1. [MJD](https://tycho.usno.navy.mil/mjd.html) is simpler to compute.  
> **username_2**: Wow this is a very interesting question!  

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## 1 answers

### Answer by username_3 on 2019-02-26. Score: 4 | Accepted answer

The Year 0 does not exist in BCE/CE and BC/AD notation. It jumps from 1 BCE to 1 CE directly. So if we look at the sequence of years in BCE/CE notation, we have:
3 BCE
2 BCE
1 BCE
1 CE
2 CE
3 CE

If you want to calculate the number of years between, say, January 1, 2 BCE and January 1, 2 CE, you can add the year numbers to get 4, but then you need to subtract 1 to remove year 0 from the total, giving a total of 3 years. In your particular case, this leads to $4713 - 1 + 2010 = 6722$ years.
In astronomy, we often use another year numbering system called the Astronomical year numbering, which includes a year 0 and makes things more convenient arithmetically when calculating time intervals between eras. It goes (-3, -2, -1, 0, 1, 2, etc.) To convert years BCE to the Astronomical year notation, we subtract 1 and add minus(-) in front of the year. For example, the year 4713 BCE is -4712 in Astronomical year notation. Your year interval is then calculated as $2010 - (\text{-}4712) = 2010 + 4712 = 6722$. 
For the leap years calculation, since -4712 is a leap year, this simplifies matters, but don't forget to take the smallest integer greater than or equal to your result (the ceiling function). So in this case $\frac{6722}{4} - 3 = 1677.5$, and the ceiling is 1678, and that is the number of leap years between your years. Also, in your initial calculation, you had forgotten to subtract 3, because 6724 / 4 - 3 = 1678, not 1681. 
Finally, the time interval between January 1 and February 28 is 58 days, not 59, because we do not count the last date as part of the interval. This gives a total of $5044 * 365 + 1678 * 366 - 10 + 58 + \frac{6 * 60 * 60 + 30 * 60}{24 * 60 * 60} = 2455256.27$. 
You state that the actual answer is 2455255.77, but in fact the answer really is 2455256.27. Perhaps you typed "06:30" instead of "18:30" in the converter.


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