Calculating the Interval and Acceleration: Prep for Adjusting the Sidereal Time
Part 6: Chart Calculation Course, Time Conversion
Hi Everyone,
Here is part 6 of the chart calculation course. In this lesson, we'll use the Greenwich Mean Time for the Virgo New Moon that we worked out in part 4 to calculate two numbers, the interval and the acceleration on the interval. In the next lesson, after we determine the sidereal time, we will use these numbers to adjust it.
These numbers are not difficult to calculate, but I wanted to take extra time to go over them. Math is not my strong suit and when I first learned how to calculate charts, I didn’t understand all of the reasoning for the calculations, so, to help me and, hopefully, to also help you (especially those of you for whom math does not come naturally), I wanted to take time to explain my understanding so far.
In the next lesson, we'll determine the sidereal time and make all of the necessary adjustments to it. By the end of that lesson, we will have the correct, location-specific sidereal date and time we need to find the planet positions and house cusps for the chart.
Remember that you can always speed ahead by watching Dave Campbell’s video series on calculating charts by hand or reading Ken Ward’s article series. My article series is based on what I learned from those two resources, as well as a handout linked to on the American Federation of Astrologers’ website that was recommended by Campbell. In terms of where to start, I found Campbell’s videos easier to follow, so I recommend starting with those.1
With regard to this article series, I have included links at the end to everything that has been posted so far.
Calculating Time: Where We Left Off
In part 4, we converted the event date and time for the Virgo New Moon for this coming September to its Greenwich Mean Time (GMT) equivalent. Because ephemerides list planet positions for Greenwich Mean Time, to find the sidereal time, we needed to first convert the event date and time to their GMT equivalents so that we could know which date to look up.
Our ultimate aim is to adjust the Greenwich sidereal time to the event location. To get there, we first need to work out a couple of numbers from the Greenwich Mean Time for our event. As mentioned, the first is called the interval, and the second is called the acceleration on the interval. In this lesson, we’ll go over what these are and calculate them. Then, in the next lesson, we’ll determine the sidereal time for our GMT date and adjust it using the numbers we get in this lesson.
Where We Are So Far
The event date, time, and location with which we started for the Virgo New Moon for this upcoming September is:
September 14, 2023
18:40
Los Angeles, California
The GMT equivalent we worked out (in part 4) from this given data is:
September 15, 2023
01:40:00
We’ll use this time to calculate the numbers we’ll need to adjust the sidereal time.
Calculating the Interval
The first number we’ll calculate is called the interval. The planet positions listed in standard ephemerides are for noon or midnight, depending on the ephemeris. That means that the GMT equivalent we work out for an event will likely be different from either of those times. Because we want to find the planet positions for the time of our event, not the ephemeris time, we need to find the difference between the two so that we can then adjust the sidereal time accordingly. This difference is called the interval.
How the Interval Is Used
When we find the sidereal time for the GMT date of our event, we will find it for midnight or noon GMT, depending on which ephemeris we use. Because we want the sidereal time for our event time, not noon or midnight, we will add or subtract the interval to the sidereal time depending on the relationship between the event time and ephemeris time.
Calculating the Interval: Midnight Ephemeris
When using a midnight ephemeris, we want to find out how many hours, minutes, and seconds our event, converted to GMT, occurred after midnight. We do that by subtracting midnight (00:00:00) from the GMT equivalent for our event. Thus, to find the interval for the Virgo New Moon, we do the following:
GMT Equivalent for Event: 01:40:00
Minus Midnight Ephemeris Time: 00:00:00
Interval (Midnight Ephemeris): 01:40:00
This interval is telling us that the time difference between our event and the midnight ephemeris time is 1 hour and 40 minutes. It is also telling us that the event occurred after midnight. Thus, we will add the interval to the sidereal time given for midnight when we get to that step.
Calculating the Interval: Noon Ephemeris
I prefer to use a midnight ephemeris because the calculations are more straightforward. However, it is worth going over how to calculate the interval when using a noon ephemeris as the process is a little different. For times after 12 noon, we subtract noon from the event time for GMT. For times before noon, we do the opposite - we subtract the event time from noon. These calculations make sense as each will give us the difference from noon.
In our example, the event time for GMT is before noon, so we subtract our time from noon, as follows:
Ephemeris Time: 12:00:00
Minus GMT Time for Event: 01:40:00
Interval (Noon Ephemeris): 10:20:00
In this case, the interval is telling us that our event time, in GMT, is 10 hours and 20 minutes different from the noon ephemeris time. Specifically, it is saying that the event time occurred 10 hours and 20 minutes before noon. Thus, when we find the sidereal time for the GMT date of our event, we will subtract this difference, or interval, from the sidereal time to adjust it to the event time.
As explained in the AFA handout, if our GMT event time had been 13:40:00 instead of 01:40:00, we would subtract noon from this time, as follows:2
GMT Event Time: 13:40:00
Ephemeris Time: 12:00:00
Hypothetical Interval: 01:40:00
In this case, the interval is telling us that the event occurred 1 hour and 40 minutes after 12 noon. If the GMT equivalent for our event had indeed occurred after 12 noon, we would add this interval to the sidereal time we find in the noon ephemeris.
Calculating the Acceleration on the Interval
We usually think of acceleration as meaning speeding up. But, technically, it means a change in speed or direction. Thus, acceleration happens when an object slows down, too. It also happens when an object changes direction, even if the speed remains the same.3
Because the Earth is a sphere and is constantly rotating, it is also constantly changing direction. Thus, it is constantly accelerating.4 Based on this fact, the assumption I was making was that we calculate the acceleration on the interval to account for the distance traveled in the interval between the ephemeris time and the GMT event time. However, Ken Ward, in his chart calculation article series, writes:
“Because sidereal time is faster than regular time, we need to correct [the interval]...”5
He goes on to say that a sidereal day is shorter (or faster) than a tropical day by about 4 minutes.6 Thus, Ward is connecting acceleration to this difference in speed (or length) between the tropical and sidereal day rather than to the difference between GMT (tropical) time and ephemeris (tropical) time. His explanation makes sense: when we convert the GMT for the event to sidereal time, we are converting from tropical to sidereal time. It also makes sense in terms of the calculation we use for this step, which I’ll explain below.
Interval Equation
The equation Campbell gives to calculate the acceleration is to multiply the hours by 10 and to divide the minutes by 6. Ward indicates that the difference between the sidereal and tropical day is 10 seconds per hour, thus we need to multiply by 10.7 The result of this calculation gives us the acceleration in seconds. We will look at the reason for the difference in these two instructions below.
Reason for Interval Equation
To make sense of why both Campbell and Ward use the number 10 as the multiplier, we can look more closely at the difference of four minutes per day between tropical and sidereal time. Working with this number, we can first convert it to seconds by multiplying by 60. When we do, we get 240 seconds.
4 minutes x 60 seconds = 240 seconds8
This number tells us that the difference between sidereal time and tropical time is 240 seconds per day. To determine the hourly difference, we can divide the seconds by 24. When we do that, we get 10 seconds.
240 seconds / 24 = 10 seconds
From this formula, we see that the difference between sidereal time and tropical time is 10 seconds per hour, with sidereal time being 10 seconds faster (or shorter) than tropical time. Thus, when we multiply by 10, we are recognizing that tropical time is slower (or longer) than sidereal time and correcting accordingly.
Reason for the Difference in the Two Equations
If you think about the reasoning behind the step of dividing the minutes by 6 that Campbell recommends, it makes perfect sense. When, following Ward’s instruction, we multiply both hours and minutes by 10, the result for the minutes might be over 60. To take our example for the Virgo New Moon, when we multiply the minutes for the interval calculated above (40 minutes) by 10, we get 400.
40 minutes x 10 seconds = 400 seconds
(I don’t understand why multiplying minutes by 10 seconds gives us seconds. I just know that we take the result as seconds. If you know why, please leave a comment explaining!)
We want a number within 60, so we can divide 400 by 60 to get that, as follows:
400 / 60 = 6.66
Thus, our answer is 6.66 seconds.
Putting all the steps together that we just went through, we get this:
Minutes x (10 / 60)
Looking at 10 / 60, we can take the zero away from both (divide both by 10), and that gives us:
Minutes x (1 / 6)
We can then rework this formula, as follows:
(Minutes x 1) / 6
Thus, what we did when multiplying the minutes by 10 was to take the long way around to dividing them by 6. Or, we could say that, when we divide the minutes by 6, we are folding into one operation a complete set of steps that first involve multiplying by 10 and then dividing by 60.
When we take the shortened version of the formula and apply it to our example, we get the following:
40 / 6 = 6.66
Thus, using the shortened formula gives us the same result.
Midnight Ephemeris: Acceleration on the Interval
The interval we calculated in the previous step for the midnight ephemeris was:
01:40:00
Multiplying the hours (1) times 10 gives us 10 seconds:
1 x 10 = 10 seconds
As we saw above, dividing the minutes (40) by 6 gives us 6.66 seconds:
40 / 6 = 6.66 seconds
We can round 6.66 up to 7 and then add it to 10 to get the acceleration:
Acceleration: 10 + 7 = 17 seconds
Noon Ephemeris: Acceleration on the Interval
When calculating the interval for a noon ephemeris, we do the same thing we did for the midnight ephemeris.
The noon interval we got for our event was:
10:20:00
Multiplying the hours by 10, we get:
10 x 10 = 100 seconds
Dividing the minutes by 6, we get:
20 / 6 = 3.33 seconds
We can round the result for the minutes down to 3. Thus, we get an acceleration of:
100 + 3 = 103 seconds
Because our result is greater than 60 seconds, we can convert it to minutes and seconds by subtracting 60 from the seconds, as follows:
103 seconds
Minus 60 seconds
43 seconds
Remembering to add the 60 seconds as a minute, we get an acceleration of:
Acceleration: 1 minute, 43 seconds (or 00:01:43)
Adding the Interval and Acceleration
The final step for this lesson is to add the interval and acceleration together, as shown below. In the next lesson, we will use this combined number to adjust the sidereal. We will then convert that adjusted time to the local sidereal time for the event.
Interval Plus Acceleraton: Midnight Ephemeris
01:40:00
Plus 00:00:17
Interval + Acceleration: 01:40:17
Interval Plus Acceleration: Noon Ephemeris
Interval: 10:20:00
Plus Acceleration: 00:01:43
Interval + Acceleration: 10:21:43
Worksheets
To see all of the above calculations in one place, see the worksheets I’m creating in Noton. As we cover each step, I add a new table showing the steps and results. It might be easier to reference those tables than to search through the article to find what you’re looking for, though it works best to use both together as, at this time, I am not including instructions with the worksheets.
Here is a screenshot of the steps we went over today:
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Previous Articles in the Series
Part 1
Part 2
Part 3
Part 4
Part 5
References
Campbell, D. [Dave Campbell]. (2019, January 13). AFA Math How to Calculate Your Birth Chart, part 2 of 5 The American Federation of Astrologers [Video]. YouTube.
Campbell, D. [Dave Campbell]. (2013, January 13). How to Calculate Your Birth Chart by Dave Campbell AFA part 1 of 5 [Video]. Youtube.
Campbell, D. [Dave Campbell]. (2019, Jan 13). How to Calculate Birth Chart part 3 of 5 [Video]. YouTube.
Libretexts. (2020). 5.3: Velocity, acceleration, and force. Physics LibreTexts. https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_Physics_(Boundless)/5%3A_Uniform_Circular_Motion_and_Gravitation/5.3%3A_Velocity_Acceleration_and_Force#:~:text=As%20the%20object%20moves%20through,toward%20the%20axis%20of%20rotation.)
Procedure for Erecting a Natal Horoscope (pdf). (n.d.). American Federation of Astrologers. https://www.astrologers.com/userfiles/Handout%20for%20Calculation%20Videos.pdf
Ward, Ken. (n.d.). Astrology: Calculating Local Sidereal Time using the Midnight Ephemeris. Trans4mind.com. https://trans4mind.com/personal_development/astrology/Calculations/calcLocalSidTimeMidnight.htm
Ward, Ken. (n.d.). Astrology: Calculating Local Sidereal Time using the Noon Ephemeris. Trans4mind.com. https://trans4mind.com/personal_development/astrology/Calculations/calcLocalSidTime.htm
Ward, Ken. (n.d.). Astrology - Master Table of Contents. Trans4mind. https://trans4mind.com/personal_development/astrology/toc.htm
What is acceleration? (article). (n.d.). Khan Academy. https://www.khanacademy.org/science/physics/one-dimensional-motion/acceleration-tutorial/a/acceleration-article
The link to Campbell’s video series is to the first video. You can find links to all of them in the resources article for this series (part 3). The handout Campbell uses in his video series can be found on the AFA website. From the drop down menu for the Learn Astrology tab on the home page, choose Learning Videos. You will find the link there. The link to Ward’s articles is to the Master Table of Contents for astrology on his website, trans4mind.com. When you scroll down the page, you will find all the links to the chart calculation articles.
See the handout linked to on the AFA website. For where to find it on the website, see footnote 1.
See the Khan Academy article, What Is Acceleration?
See the lesson on LibreTexts, Rotational Angle and Angular Velocity.
See Ward’s article, Astrology: Calculating Local Sidereal Time using the Noon Ephemeris.
Ibid.
See Ward’s article, Astrology: Calculating Local Sidereal Time using the Midnight Ephemeris.
The formula for converting the difference in minutes to seconds between tropical time and sidereal time came up, initially, as plagiarized in one of the free plagiarism checkers I use. It did not come up as plagiarized in subsequent checks, but I thought I would explain how I came to the formula. First, I did not copy it from another site. Rather, I wrote it as the natural next step of what I had written before. I came up with the formula when piecing together bits and pieces of information I came across in both Ward’s articles and Campbell’s videos. At some point, it occurred to me to multiply 4 minutes by 60 seconds to see what the result would be - 240 seconds. (This is an example of how math is not my strong suit :). ) When I saw that, I understood where the 240 seconds comes from that Campbell uses in his third video, and also why we multiply the interval by 10. Given that the difference in time between a sidereal day and tropical day is well-known, I believe the formula for converting the minutes to seconds can be considered established, common knowledge within the field of astronomy. In addition to Ward’s article cited in footnotes 5 and 6, and Campbell’s third video mentioned above, see Campbell’s second video for his formula for calculating the acceleration.