April 19, 2011

• Awesome Day - June 7th, 2014

  Last Saturday, around midnight, I started a train of thought that kept me busy for a bit...

  FourSquare celebrated Four Squared day on April 16th, because 4^2 = 16. Each day is a numerical day of the calendar year. For example, January 1st is the 1st day of the year. December 31st is the 365th day of the year. This lead me to ask is there ever a day where the value of the day squared equals the value of the month squared?

  It is apparent that this happens only when the day and the month have the same number. So outside of the 1st of January (D = 1, M = 1, meaning 1 ^ 2 = 1 ^ 2, this never happens. And if you want to get technical, they don't because they're two parabolas increasing at different rates.

  • Day of the year (D) = x + 1
  • Month of the year is very roughly M = D / 30
  • D ^ 2 = M ^ 2
  • D ^ 2 = (D / 30) ^ 2

  That lead me to ask another question: how many days would need to be in a year before the square root of the day of the year is equal to the month squared? Since it is obvious that this will require more than twelve months, I am officially defining all months after the 12th month as always having 30 days. And later we'll be presuming that days are always equivalent to Earth days.

  • D ^ (1/2) = M ^ 2
  • (x + 1) ^ (1/2) = ((x + 1) / 30) ^ 2
  • When graphed we get an intersection at (92.216975, 9.6548)
  • 92.216975 months * 30 days per month = 2766.50295 days = ~2766 days

  We can celebrate this day on any planet that has an orbit of at least 2766 days. Specifically, we want to celebrate on the 6th day of the 92nd month. And as a side note, the days of the 92nd month are the range of the 2,761st to the 2,790th days of the year.

  Jupiter is the closest planet to the Sun that you could live on and celebrate this day we've been hunting for. It has a calendar year with at least 2766 days (specifically its orbit is 4,431.572 days, just shy of 12 years). An Earth year is 365.25 days and a Martian year is 686.96 days. Jupiter is the next door neighbor to Mars, so that should give you an idea of how far apart Mars and Jupiter are.

  Next I wanted to know what distance does an object have to orbit the sun to celebrate this day? When we find this distance, we're going to call this the Hardiman Orbital Distance. However, I really need to find two different values here, a theoretical distance and a practical distance. If you lived on a planet at this distance, after the first 12 months you'd then have 80 months in a row that have 30 days. Then the last month of the year would be 6 days long (this will be our theoretical number). I'm going to guess that those 6 days would be assimilated in to some of the prior months out of practicality, so I want to figure a full 30 day 92nd month as well (this will be our practical number).

  Actual Hardiman Orbital Distance (AHOD) - The distance at which an object has to orbit the sun to have a 92.216975 month year.
  Practical Hardiman Orbital Distance (PHOD) - The distance at which an object has to orbit the sun to have a full 92 month year.

  Converted to days, we're looking for:

  • AHOD of 2766 days (92.216975 months)
  • PHOD of 2790 days (exactly 92 full months)

  Quickly to the astronomy equations! (Note: In the following equations p = period and a = semi-major axis)

  A quick estimate in advance is that the AHOD and PHOD should fall within the range of the average distance of Mars from the Sun (1.38 AU) and the average distance of Jupiter from the Sun (5.2 AU).

  • p = number of days the orbit takes / 365.25 days per Earth year
  • p ^ 2 = a ^ 3
  • a = (p ^ 2) ^ (1/3)
  • a = p ^ (2/3)

  AHOD = (2766 / 365) ^ (2/3) = 3.8580946 AU = 577,163,236 km = 358,632,608 mi
  PHOD = (2790 / 365) ^ (2/3) = 3.8803797 AU = 580,497,042 km = 360,704,139 mi

  Both values fall within the estimated range.

  I want to celebrate this on Earth, so I need to know when every 2766th day occurs. Since January 1st, 1 A.D. to April 18th, 2011 there have been 734,610 days. I determined this using MySQLs TO_DAYS function.

  • select TO_DAYS('2011-04-18');

  The most recent Earth-equivalent of this day was the 265th celebration.

  • 734,610 / 2766 = 265.585

  The next Earth-equivalent day will be the 266th celebration, which will be the 735,756th day since January 1st, 1 A.D.

  • 2766 * 266 = 735,756
  • From MySQL: select FROM_DAYS(735756);

  The next Earth-equivalent day will be on June 7th, 2014.

  Final conclusions:

  The technical name for the 2766th days is Meidiespenitus, which is Latin for "My Day Inside".
  The technical name for the 92nd month is Tuimater, which is Latin for "Your Mother".
  Therefore, on the 2766th day you can celebrate the "Your Mom" of astronomy jokes: "My Day Inside Your Mother" day.

  Out of practicality, this day is called the Awesome Day or Hardiman-Deffenbaugh Day. And the Month is called Fennec, because Fennec foxes are awesome.

  Actual Hardiman Orbital Distance is 3.8580946 AU.
  Practical Hardiman Orbital Distance is 3.8803797 AU.

  Any object with an orbit less than AHOD is known as Non-Awesome.
  Any object with an orbit of AHOD is known as Partially Awesome.
  Any object with an orbit of PHOD is known as Fully Awesome.

  We can celebrate the next Awesome Day on June 7th, 2014.

  • 7:40 pm
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