Odd abundant number and Kepler's Laws: Difference between pages

Latest revision as of 22:12, 30 May 2017

Figure 1: Illustration of Kepler's three laws with two planetary orbits.
(1) The orbits are ellipses, with focal points ƒ₁ and ƒ₂ for the first planet and ƒ₁ and ƒ₃ for the second planet. The Sun is placed in focal point ƒ₁.

(2) The two shaded sectors A₁ and A₂ have the same surface area and the time for planet 1 to cover segment A₁ is equal to the time to cover segment A₂.

(3) The total orbit times for planet 1 and planet 2 have a ratio a₁^3/2 : a₂^3/2.

Kepler's laws of planetary motion are three laws that describe the motion of planets around the sun:

Planets move around the sun in elliptic orbits. The sun is in one of the two foci of the orbit.
A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

Johannes Kepler found these laws, between 1609 and 1619.

Comparison to Copernicus

Kepler's laws improve the model of Copernicus. If the eccentricities of the planetary orbits are taken as zero, then Kepler basically agrees with Copernicus:

The planetary orbit is a circle
The Sun at the center of the orbit
The speed of the planet in the orbit is constant

The eccentricities of the orbits of those planets known to Copernicus and Kepler are small, so the rules above give good approximations of planetary motion; but Kepler's laws fit the observations better than Copernicus's.

Kepler's corrections are not at all obvious:

The planetary orbit is not a circle, but an ellipse.
The Sun is not at the center but at a focal point of the elliptical orbit.
Neither the linear speed nor the angular speed of the planet in the orbit is constant, but the area speed is constant.

The eccentricity of the orbit of the Earth makes the time from the March equinox to the September equinox, around 186 days, unequal to the time from the September equinox to the March equinox, around 179 days. A diameter would cut the orbit into equal parts, but the plane through the sun parallel to the equator of the earth cuts the orbit into two parts with areas in a 186 to 179 ratio, so the eccentricity of the orbit of the Earth is approximately

[math]\displaystyle{ \varepsilon\approx\frac \pi 4 \frac {186-179}{186+179}\approx 0.015, }[/math]

which is close to the correct value (0.016710219) (see Earth's orbit). The calculation is correct when perihelion, the date the Earth is closest to the Sun, falls on a solstice. The current perihelion, near January 4, is fairly close to the solstice of December 21 or 22.

@@ Line 1: / Line 1: @@
-An '''odd abundant number''' is an [[odd number]] <math>n</math> that its [[sum-of-divisors|sum-of divisors]] greater than the [[double|twice]] of itself.
+[[File:Kepler laws diagram.svg|thumb|300px|Figure 1: Illustration of [[Johannes Kepler|Kepler's]] three laws with two planetary orbits.<br> (1) The orbits are ellipses, with focal points ''&fnof;''<sub>1</sub> and ''&fnof;''<sub>2</sub> for the first planet and ''&fnof;''<sub>1</sub> and ''&fnof;''<sub>3</sub> for the second planet. The Sun is placed in focal point ''&fnof;''<sub>1</sub>. <br><br> (2) The two shaded sectors ''A''<sub>1</sub> and ''A''<sub>2</sub> have the same surface area and the time for planet 1 to cover segment ''A''<sub>1</sub> is equal to the time to cover segment ''A''<sub>2</sub>. <br><br> (3) The total orbit times for planet 1 and planet 2 have a ratio ''a''<sub>1</sub><sup>3/2</sup>&nbsp;:&nbsp;''a''<sub>2</sub><sup>3/2</sup>.]]
-==Examples==
+'''Kepler's laws of planetary motion''' are three laws that describe the motion  of [[Planet|planets]] around the [[sun]]:
-*The first example is 945   (''[[3 (number)|3]]<sup>3</sup>× [[5 (number)|5]]× [[7 (number)|7]]''). Its [[prime number|prime]] [[factor|factors]] are [[3 (number)|3]], [[5 (number)|5]], and [[7 (number)|7]]. The next following eleven odd abundant numbers are
+#Planets move around the sun in [[Ellipse|elliptic]] [[Orbit|orbits]]. The sun is in one of the two foci of the orbit.
-, 2205, 2835, 3465, 4095, 4725, 5355, 5775, 5985, 6435, 6615.
+#A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
-*Odd abundant numbers below 500000 are in On-Line Encyclopedia of Integer Sequences [http://oeis.org/A005231/b005231.txt, A005231].
+#The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.
+[[Johannes Kepler]] found these laws, between 1609 and 1619.
-==Formulas==
+==Comparison to Copernicus==
-The following [[formula]]
+Kepler's laws improve the [[Scientific model|model]] of Copernicus. If the eccentricities of the planetary [[orbit]]s are taken as zero, then Kepler basically agrees with Copernicus:
+#The planetary orbit is a [[circle]]
+#The Sun at the center of the orbit
+#The speed of the planet in the orbit is constant
-<math>945+630n</math><ref>{{Cite web|url=http://ms.appliedprobability.org/data/files/Articles%2038/38-1-2.pdf|title=More Odd Abundant  Sequences|first=|last=|date=2005|website=|publisher=JAY. SCHIFFMAN|accessdate=2017-01-2}}</ref>
+The eccentricities of the orbits of those planets known to Copernicus and Kepler are small, so the rules above give good approximations of planetary motion; but Kepler's laws fit the observations better than Copernicus's.
-presents 62 [[abundant number]]s, but it fails if
-<math>n\le62</math>.
+Kepler's corrections are not at all obvious:
+#The planetary orbit is ''not'' a circle, but an ''ellipse''.
+#The Sun is ''not'' at the center but at a ''focal point'' of the elliptical orbit.
+#Neither the linear speed nor the angular speed of the planet in the orbit is constant, but the ''area speed'' is constant.
-The second formula
+The eccentricity of the orbit of the Earth makes the time from the [[March equinox]] to the [[September equinox]], around 186 days, unequal to the time from the September equinox to the March equinox, around 179 days. A diameter would cut the orbit into equal parts, but the plane through the sun parallel to the [[equator]] of the earth cuts the orbit into two parts with areas in a 186 to 179 ratio, so the eccentricity of the orbit of the Earth is approximately
+:<math>\varepsilon\approx\frac \pi 4 \frac {186-179}{186+179}\approx 0.015,</math>
+which is close to the correct value (0.016710219) (see [[Earth's orbit]]).
+The calculation is correct when [[perihelion]], the date the Earth is closest to the Sun, falls on a [[solstice]]. The current perihelion, near January 4, is fairly close to the solstice of December 21 or 22.
-<math>3465+2310n</math><ref>{{Cite web|url=http://ms.appliedprobability.org/data/files/Articles%2038/38-1-2.pdf|title=More Odd Abundant  Sequences|first=|last=|date=2005|website=|publisher=JAY. SCHIFFMAN|accessdate=2017-01-26}}</ref>
+[[Category:Astronomy]]
-presents 192 [[abundant number]]s, but fails if
-<math>n\le192</math>
-The [[third]] formula
-<math>2446903305+1631268870n</math> <ref>{{Cite web|url=http://ms.appliedprobability.org/data/files/Articles%2038/38-1-2.pdf|title=More Odd Abundant  Sequences|first=|last=|date=2005|website=|publisher=JAY. SCHIFFMAN|accessdate=2017-01-26}}</ref>
-fails if <math>n\le135939073</math>.
-==Properties==
-* An calculation was given by Iannucci shows how to find the smallest abundant number not divisible by the first '''''n''''' primes.
-*An abundant number with abundance 1 is called a [[quasiperfect number]], although none have yet been found. A quasiperfect number must be an odd square number having a value above 10<sup>30<sup>.
-==References==
-{{Reflist}}
-{{Math-stub}}
-[[Category:Integer sequences]]

Odd abundant number and Kepler's Laws: Difference between pages

Latest revision as of 22:12, 30 May 2017

Comparison to Copernicus

Navigation menu

Search