<html><head><title>Math Addendum 5 Gravitation</title><style type="text/css"> p { margin: 0 0 0 0; font-family: Arial, sans-serif; font-size:10pt; font-weight: normal; font-style: normal; vertical-align: baseline; color: black; text-decoration: none; } div { border-width:0px; border-style: solid; border-color: red; } body { background-color: #fff; } p img { vertical-align: middle; } p b, li b { font-weight : bold; } p i, li i { font-style : italic; } p u, li u { text-decoration : underline; } p so, li so { text-decoration : line-through; } p sub, p sub { font-size : 70%; vertical-align: sub; } p sup, p sup { font-size : 70%; vertical-align: super; } ul { padding-left : 2em; /* for mozilla list marker box */ margin-left : 0; /* for IE list marker box */ } ol { padding-left : 2em; /* for mozilla list marker box */ margin-right : -0.5em; margin-left : 0; /* for IE list marker box */ } p.Style1, li.Style1 /*Normal*/ { margin-left: 0pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; 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top: 72.75pt; left: 30pt; width: 332pt; "><a name="" /><p class="Style1"><span style="font-size: 12pt; font-family: 'Arial', sans-serif;"><b><span>Addendum 5: Gravitation, orbits, Tidal forces, Roche limit </span></b></span></p></div><div id="mc-region-541" style="position: absolute; top: 105pt; left: 30pt; width: 122.25pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0004_1211411015.PNG" id="generatedImage541" path=".\addendum 5 gravity kepler 3rd law_images/IMG0004_1211411015.PNG" style="width: 122.25pt;height: 18.75pt;"></img></div><div id="mc-region-540" style="position: absolute; top: 105pt; left: 168pt; width: 65.25pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0005_1211411031.PNG" id="generatedImage540" path=".\addendum 5 gravity kepler 3rd law_images/IMG0005_1211411031.PNG" style="width: 65.25pt;height: 22.5pt;"></img></div><div id="mc-region-539" style="position: absolute; top: 111pt; left: 258pt; width: 56.25pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0006_1211411031.PNG" id="generatedImage539" path=".\addendum 5 gravity kepler 3rd law_images/IMG0006_1211411031.PNG" style="width: 56.25pt;height: 16.5pt;"></img></div><div id="mc-region-605" style="position: absolute; top: 135pt; left: 18pt; width: 94.5pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0007_1211411031.PNG" id="generatedImage605" path=".\addendum 5 gravity kepler 3rd law_images/IMG0007_1211411031.PNG" style="width: 94.5pt;height: 22.5pt;"></img></div><div id="mc-region-606" style="position: absolute; top: 141pt; left: 126pt; width: 74.25pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0008_1211411031.PNG" id="generatedImage606" path=".\addendum 5 gravity kepler 3rd law_images/IMG0008_1211411031.PNG" style="width: 74.25pt;height: 16.5pt;"></img></div><div id="mc-region-607" style="position: absolute; top: 141pt; left: 222pt; width: 78pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0009_1211411031.PNG" id="generatedImage607" path=".\addendum 5 gravity kepler 3rd law_images/IMG0009_1211411031.PNG" style="width: 78pt;height: 16.5pt;"></img></div><div id="mc-region-608" style="position: absolute; top: 141pt; left: 330pt; width: 70.5pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0010_1211411031.PNG" id="generatedImage608" path=".\addendum 5 gravity kepler 3rd law_images/IMG0010_1211411031.PNG" style="width: 70.5pt;height: 16.5pt;"></img></div><div id="mc-region-631" style="position: absolute; top: 165pt; left: 30pt; width: 69.75pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0011_1211411046.PNG" id="generatedImage631" path=".\addendum 5 gravity kepler 3rd law_images/IMG0011_1211411046.PNG" style="width: 69.75pt;height: 18.75pt;"></img></div><div id="mc-region-638" style="position: absolute; top: 165pt; left: 126pt; width: 75.75pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0012_1211411046.PNG" id="generatedImage638" path=".\addendum 5 gravity kepler 3rd law_images/IMG0012_1211411046.PNG" style="width: 75.75pt;height: 22.5pt;"></img></div><div id="mc-region-646" style="position: absolute; top: 171pt; left: 228pt; width: 72pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0013_1211411046.PNG" id="generatedImage646" path=".\addendum 5 gravity kepler 3rd law_images/IMG0013_1211411046.PNG" style="width: 72pt;height: 16.5pt;"></img></div><div id="mc-region-19" style="position: absolute; top: 206.25pt; left: 24pt; width: 408.5pt; "><a name="" /><p class="Style1">1. <b><i>Law of Gravity</i></b><span style="font-family: 'Arial', sans-serif;"><b><i><span>. </span></i></b></span><span style="font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">Newton's law of gravity state that the gravitatinoal force between two bodies of mass m1 and m2 separated by distance d is</span></span><span style="font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">. </span></span></p></div><div id="mc-region-572" style="position: absolute; top: 238.5pt; left: 180pt; width: 54.75pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0014_1211411046.PNG" id="generatedImage572" path=".\addendum 5 gravity kepler 3rd law_images/IMG0014_1211411046.PNG" style="width: 54.75pt;height: 37.5pt;"></img></div><div id="mc-region-563" style="position: absolute; top: 284.25pt; left: 18pt; width: 402.5pt; "><a name="" /><p class="Style1">Ex: What is the gravitational force between two people separated by 10cm? Compare with the force between each person and the Earth (the person's weight)</p></div><div id="mc-region-564" style="position: absolute; top: 327pt; left: 36pt; width: 46.5pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0015_1211411046.PNG" id="generatedImage564" path=".\addendum 5 gravity kepler 3rd law_images/IMG0015_1211411046.PNG" style="width: 46.5pt;height: 16.5pt;"></img></div><div id="mc-region-565" style="position: absolute; top: 327pt; left: 96pt; width: 46.5pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0016_1211411062.PNG" id="generatedImage565" path=".\addendum 5 gravity kepler 3rd law_images/IMG0016_1211411062.PNG" style="width: 46.5pt;height: 16.5pt;"></img></div><div id="mc-region-566" style="position: absolute; top: 327pt; left: 162pt; width: 39.75pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0017_1211411062.PNG" id="generatedImage566" path=".\addendum 5 gravity kepler 3rd law_images/IMG0017_1211411062.PNG" style="width: 39.75pt;height: 12.75pt;"></img></div><div id="mc-region-567" style="position: absolute; top: 316.5pt; left: 222pt; width: 65.25pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0018_1211411062.PNG" id="generatedImage567" path=".\addendum 5 gravity kepler 3rd law_images/IMG0018_1211411062.PNG" style="width: 65.25pt;height: 37.5pt;"></img></div><div id="mc-region-568" style="position: absolute; top: 321pt; left: 318pt; width: 84.75pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0019_1211411062.PNG" id="generatedImage568" path=".\addendum 5 gravity kepler 3rd law_images/IMG0019_1211411062.PNG" style="width: 84.75pt;height: 22.5pt;"></img></div><div id="mc-region-569" style="position: absolute; top: 346.5pt; left: 42pt; width: 65.25pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0020_1211411062.PNG" id="generatedImage569" path=".\addendum 5 gravity kepler 3rd law_images/IMG0020_1211411062.PNG" style="width: 65.25pt;height: 41.25pt;"></img></div><div id="mc-region-570" style="position: absolute; top: 357pt; left: 150pt; width: 63.75pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0021_1211411062.PNG" id="generatedImage570" path=".\addendum 5 gravity kepler 3rd law_images/IMG0021_1211411062.PNG" style="width: 63.75pt;height: 16.5pt;"></img></div><div id="mc-region-571" style="position: absolute; top: 346.5pt; left: 240pt; width: 78.75pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0022_1211411078.PNG" id="generatedImage571" path=".\addendum 5 gravity kepler 3rd law_images/IMG0022_1211411078.PNG" style="width: 78.75pt;height: 35.25pt;"></img></div><div id="mc-region-658" style="position: absolute; top: 398.25pt; left: 12pt; width: 408.5pt; "><a name="" /><p class="Style1">2. <b>Orbital Speed</b>. The speed of any objectorbiting a much larger object [e.h. moon, satellite] has a speed given by equating the gravitational force with the centripedal force</p></div><div id="mc-region-576" style="position: absolute; top: 440.25pt; left: 60pt; width: 66pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0023_1211411078.PNG" id="generatedImage576" path=".\addendum 5 gravity kepler 3rd law_images/IMG0023_1211411078.PNG" style="width: 66pt;height: 39.75pt;"></img></div><div id="mc-region-577" style="position: absolute; top: 444.75pt; left: 162pt; width: 45.75pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0024_1211411078.PNG" id="generatedImage577" path=".\addendum 5 gravity kepler 3rd law_images/IMG0024_1211411078.PNG" style="width: 45.75pt;height: 29.25pt;"></img></div><div id="mc-region-659" style="position: absolute; top: 494.25pt; left: 24pt; width: 249.5pt; "><a name="" /><p class="Style1">Ex: The average orbital speed of the Moon in its orbit is </p></div><div id="mc-region-660" style="position: absolute; top: 531pt; left: 36pt; width: 59.25pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0025_1211411078.PNG" id="generatedImage660" path=".\addendum 5 gravity kepler 3rd law_images/IMG0025_1211411078.PNG" style="width: 59.25pt;height: 12.75pt;"></img></div><div id="mc-region-661" style="position: absolute; top: 519pt; left: 144pt; width: 74.25pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0026_1211411078.PNG" id="generatedImage661" path=".\addendum 5 gravity kepler 3rd law_images/IMG0026_1211411078.PNG" style="width: 74.25pt;height: 33pt;"></img></div><div id="mc-region-662" style="position: absolute; top: 524.25pt; left: 264pt; width: 72pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0027_1211411078.PNG" id="generatedImage662" path=".\addendum 5 gravity kepler 3rd law_images/IMG0027_1211411078.PNG" style="width: 72pt;height: 27.75pt;"></img></div><div id="mc-region-663" style="position: absolute; top: 638.25pt; left: 12pt; width: 408.5pt; "><a name="" /><p class="Style1" style="margin-left: 0pt; margin-right: 0pt; text-align: left;">3.<b> Escape Speed</b><span style="font-family: 'Arial', sans-serif;"><b><i><span>.</span></i></b></span><span style="font-family: 'Arial', sans-serif;"><i><span style="font-weight: normal;"> </span></i></span><span style="font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">The escape speed of an object from the surface of a spherical body xcan by computed by summing the gravitational potential energy with the inital kinetic ene</span></span><span style="font-family: 'Arial', sans-serif;"><i><span style="font-weight: normal;">rgy =0</span></i></span></p></div><div id="mc-region-588" style="position: absolute; top: 680.25pt; left: 60pt; width: 95.25pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0028_1211411093.PNG" id="generatedImage588" path=".\addendum 5 gravity kepler 3rd law_images/IMG0028_1211411093.PNG" style="width: 95.25pt;height: 27.75pt;"></img></div><div id="mc-region-590" style="position: absolute; top: 686.25pt; left: 186pt; width: 18.5pt; "><a name="" /><p class="Style1">so </p></div><div id="mc-region-591" style="position: absolute; top: 678.75pt; left: 234pt; width: 66.75pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0029_1211411093.PNG" id="generatedImage591" path=".\addendum 5 gravity kepler 3rd law_images/IMG0029_1211411093.PNG" style="width: 66.75pt;height: 29.25pt;"></img></div><div id="mc-region-595" style="position: absolute; top: 722.25pt; left: 72pt; width: 207.5pt; "><a name="" /><p class="Style1">Ex: The escape speed from Martian surface is:</p></div><div id="mc-region-611" style="position: absolute; top: 741pt; left: 96pt; width: 87pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0030_1211411093.PNG" id="generatedImage611" path=".\addendum 5 gravity kepler 3rd law_images/IMG0030_1211411093.PNG" style="width: 87pt;height: 36.75pt;"></img></div><div id="mc-region-612" style="position: absolute; top: 752.25pt; left: 216pt; width: 63pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0031_1211411093.PNG" id="generatedImage612" path=".\addendum 5 gravity kepler 3rd law_images/IMG0031_1211411093.PNG" style="width: 63pt;height: 27.75pt;"></img></div><div id="mc-region-621" style="position: absolute; top: 794.25pt; left: 18pt; width: 414.5pt; "><a name="" /><p class="Style1">4. <b>Kepler's Third Law. </b>Kepler's third states that the orbital period is proportional to the semi-major axis of the orbit to the 3/2 power. With proper constants we get </p></div><div id="mc-region-623" style="position: absolute; top: 830.25pt; left: 180pt; width: 54.75pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0032_1211411093.PNG" id="generatedImage623" path=".\addendum 5 gravity kepler 3rd law_images/IMG0032_1211411093.PNG" style="width: 54.75pt;height: 34.5pt;"></img></div><div id="mc-region-625" style="position: absolute; top: 872.25pt; left: 36pt; width: 365pt; "><a name="" /><p class="Style1">If the period is expressed in (Earth) years and the semi-major axis in AU, then the constants cancel to give the simplified form (valid only for objects orbiting the Sun!):</p></div><div id="mc-region-627" style="position: absolute; top: 909pt; left: 198pt; width: 30.75pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0033_1211411109.PNG" id="generatedImage627" path=".\addendum 5 gravity kepler 3rd law_images/IMG0033_1211411109.PNG" style="width: 30.75pt;height: 18.75pt;"></img></div><div id="mc-region-628" style="position: absolute; top: 938.25pt; left: 48pt; width: 357.5pt; "><a name="" /><p class="Style1">Ex 1: What is the orbital period of an asteroid with a semi-major axis a = 3.8 AU?</p></div><div id="mc-region-636" style="position: absolute; top: 962.25pt; left: 66pt; width: 89.75pt; "><a name="" /><p class="Style1">Simplified equation:</p></div><div id="mc-region-633" style="position: absolute; top: 993pt; left: 66pt; width: 46.5pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0034_1211411109.PNG" id="generatedImage633" path=".\addendum 5 gravity kepler 3rd law_images/IMG0034_1211411109.PNG" style="width: 46.5pt;height: 12.75pt;"></img></div><div id="mc-region-629" style="position: absolute; top: 971.25pt; left: 174pt; width: 66.75pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0035_1211411125.PNG" id="generatedImage629" path=".\addendum 5 gravity kepler 3rd law_images/IMG0035_1211411125.PNG" style="width: 66.75pt;height: 48.75pt;"></img></div><div id="mc-region-632" style="position: absolute; top: 999pt; left: 282pt; width: 42.75pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0036_1211411125.PNG" id="generatedImage632" path=".\addendum 5 gravity kepler 3rd law_images/IMG0036_1211411125.PNG" style="width: 42.75pt;height: 12.75pt;"></img></div><div id="mc-region-634" style="position: absolute; top: 1034.25pt; left: 66pt; width: 66.5pt; "><a name="" /><p class="Style1">Full equation: </p></div><div id="mc-region-640" style="position: absolute; top: 1024.5pt; left: 168pt; width: 86.25pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0037_1211411125.PNG" id="generatedImage640" path=".\addendum 5 gravity kepler 3rd law_images/IMG0037_1211411125.PNG" style="width: 86.25pt;height: 59.25pt;"></img></div><div id="mc-region-639" style="position: absolute; top: 1059pt; left: 276pt; width: 42.75pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0038_1211411125.PNG" id="generatedImage639" path=".\addendum 5 gravity kepler 3rd law_images/IMG0038_1211411125.PNG" style="width: 42.75pt;height: 12.75pt;"></img></div><div id="mc-region-645" style="position: absolute; top: 1100.25pt; left: 48pt; width: 390.5pt; "><a name="" /><p class="Style1">Ex 2 Find the semi-major axis of the Jovian satellite Io's orbit if the orbital period is 1.77 days.</p></div><div id="mc-region-652" style="position: absolute; top: 1130.25pt; left: 48pt; width: 372.5pt; "><a name="" /><p class="Style1">Note: This <i>cannot</i> be solved using the simplified equation since the central body is not the Sun.</p></div><div id="mc-region-653" style="position: absolute; top: 1185pt; left: 60pt; width: 52.5pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0039_1211411125.PNG" id="generatedImage653" path=".\addendum 5 gravity kepler 3rd law_images/IMG0039_1211411125.PNG" style="width: 52.5pt;height: 12.75pt;"></img></div><div id="mc-region-654" style="position: absolute; top: 1153.5pt; left: 174pt; width: 84pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0040_1211411140.PNG" id="generatedImage654" path=".\addendum 5 gravity kepler 3rd law_images/IMG0040_1211411140.PNG" style="width: 84pt;height: 59.25pt;"></img></div><div id="mc-region-655" style="position: absolute; top: 1179pt; left: 282pt; width: 69.75pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0041_1211411140.PNG" id="generatedImage655" path=".\addendum 5 gravity kepler 3rd law_images/IMG0041_1211411140.PNG" style="width: 69.75pt;height: 18.75pt;"></img></div><div id="mc-region-664" style="position: absolute; top: 1256.25pt; left: 18pt; width: 426.5pt; "><a name="" /><p class="Style1">5. Roche limit. Since the gravitational force depends on the distance, a small bosy (satellite near a larger bosy (Planet) will feel a stronger force on the near side than the far side. This is referrrede to as the <b>tidal force</b>. In the frame of the center of the small body, this has the effect of stretching the body apart. This effect is counted by the self-gravity of the small body. As the distance to the larger bosdy decreases, the tidal force increases rapdly (as r<sup>-3</sup> as shown below), but the slef-gravity doesn't change. There is a crirical distance, called the Roche limit, for which the tidal force is equal to the self-gravity force. For distnaces closed than the Roch limit, the small body is pulled apart.</p></div><div id="mc-region-667" style="position: absolute; top: 1376.25pt; left: 30pt; width: 335.75pt; "><a name="" /><p class="Style1">Consider two identical small spheres at a distance d from a large mass M: </p></div><div id="mc-region-672" style="position: absolute; top: 1410pt; left: 66pt; width: 314.25pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0043_1211411140.PNG" id="generatedImage672" path=".\addendum 5 gravity kepler 3rd law_images/IMG0043_1211411140.PNG" style="width: 314.25pt;height: 142.5pt;"></img></div><div id="mc-region-741" style="position: absolute; top: 1586.25pt; left: 120pt; width: 52.5pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0044_1211411140.PNG" id="generatedImage741" path=".\addendum 5 gravity kepler 3rd law_images/IMG0044_1211411140.PNG" style="width: 52.5pt;height: 33.75pt;"></img></div><div id="mc-region-740" style="position: absolute; top: 1586.25pt; left: 198pt; width: 60.75pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0045_1211411140.PNG" id="generatedImage740" path=".\addendum 5 gravity kepler 3rd law_images/IMG0045_1211411140.PNG" style="width: 60.75pt;height: 34.5pt;"></img></div><div id="mc-region-673" style="position: absolute; top: 1598.25pt; left: 318pt; width: 83.75pt; "><a name="" /><p class="Style1">Newtonian gravity.</p></div><div id="mc-region-735" style="position: absolute; top: 1640.25pt; left: 30pt; width: 113pt; "><a name="" /><p class="Style1">The difference in force is:</p></div><div id="mc-region-736" style="position: absolute; top: 1676.25pt; left: 30pt; width: 258pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0046_1211411156.PNG" id="generatedImage736" path=".\addendum 5 gravity kepler 3rd law_images/IMG0046_1211411156.PNG" style="width: 258pt;height: 49.5pt;"></img></div><div id="mc-region-737" style="position: absolute; top: 1760.25pt; left: 30pt; width: 341pt; "><a name="" /><p class="Style1">Now from elementary calculus (Taylor series expansion of function) we know: </p></div><div id="mc-region-687" style="position: absolute; top: 1790.25pt; left: 168pt; width: 91.5pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0047_1211411156.PNG" id="generatedImage687" path=".\addendum 5 gravity kepler 3rd law_images/IMG0047_1211411156.PNG" style="width: 91.5pt;height: 19.5pt;"></img></div><div id="mc-region-738" style="position: absolute; top: 1823.25pt; left: 30pt; width: 335pt; "><a name="" /><p class="Style1">where <span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;">e</span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">&lt;&lt;1 and the ellipses (...) indicate the next term is small (of order </span></span><span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;">e</span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;"><sup>2</sup></span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">)</span></span></p></div><div id="mc-region-725" style="position: absolute; top: 1868.25pt; left: 24pt; width: 197pt; "><a name="" /><p class="Style1">Applying this to the force difference equation</p></div><div id="mc-region-691" style="position: absolute; top: 1903.5pt; left: 114pt; width: 237.75pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0048_1211411156.PNG" id="generatedImage691" path=".\addendum 5 gravity kepler 3rd law_images/IMG0048_1211411156.PNG" style="width: 237.75pt;height: 34.5pt;"></img></div><div id="mc-region-726" style="position: absolute; top: 1946.25pt; left: 24pt; width: 372.5pt; "><a name="" /><p class="Style1">Now imagine the 2 small spheres just touching. Assume the radius of each sphere is a, so the distance between centers is 2a. The self gravity holding them together is:</p></div><div id="mc-region-720" style="position: absolute; top: 1988.25pt; left: 186pt; width: 96.75pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0049_1211411156.PNG" id="generatedImage720" path=".\addendum 5 gravity kepler 3rd law_images/IMG0049_1211411156.PNG" style="width: 96.75pt;height: 39.75pt;"></img></div><div id="mc-region-727" style="position: absolute; top: 2042.25pt; left: 24pt; width: 396.5pt; "><a name="" /><p class="Style1">The Roche limit is the critical distance r<sub>Roche</sub> at which the tidal force and the self gravity terms are equal (note that <span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;">D</span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">r = 2a)</span></span>:</p></div><div id="mc-region-728" style="position: absolute; top: 2078.25pt; left: 24pt; width: 88.5pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0050_1211411156.PNG" id="generatedImage728" path=".\addendum 5 gravity kepler 3rd law_images/IMG0050_1211411156.PNG" style="width: 88.5pt;height: 43.5pt;"></img></div><div id="mc-region-718" style="position: absolute; top: 2096.25pt; left: 186pt; width: 78.5pt; "><a name="" /><p class="Style1">solving for r<sub>Roche:</sub></p></div><div id="mc-region-719" style="position: absolute; top: 2081.25pt; left: 288pt; width: 91.5pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0051_1211411171.PNG" id="generatedImage719" path=".\addendum 5 gravity kepler 3rd law_images/IMG0051_1211411171.PNG" style="width: 91.5pt;height: 48.75pt;"></img></div><div id="mc-region-742" style="position: absolute; top: 2150.25pt; left: 18pt; width: 380pt; "><a name="" /><p class="Style1">We can write this expression more conveniently using the mass densities of the planet (<span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;">r</span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;"><sub>P</sub></span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">) </span></span>and satellite (<span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;">r</span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;"><sub>s</sub></span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">)</span></span></p></div><div id="mc-region-704" style="position: absolute; top: 2191.5pt; left: 132pt; width: 70.5pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0052_1211411171.PNG" id="generatedImage704" path=".\addendum 5 gravity kepler 3rd law_images/IMG0052_1211411171.PNG" style="width: 70.5pt;height: 28.5pt;"></img></div><div id="mc-region-705" style="position: absolute; top: 2191.5pt; left: 234pt; width: 60.75pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0053_1211411171.PNG" id="generatedImage705" path=".\addendum 5 gravity kepler 3rd law_images/IMG0053_1211411171.PNG" style="width: 60.75pt;height: 28.5pt;"></img></div><div id="mc-region-743" style="position: absolute; top: 2228.25pt; left: 30pt; width: 18.5pt; "><a name="" /><p class="Style1">so </p></div><div id="mc-region-710" style="position: absolute; top: 2230.5pt; left: 180pt; width: 74.25pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0054_1211411171.PNG" id="generatedImage710" path=".\addendum 5 gravity kepler 3rd law_images/IMG0054_1211411171.PNG" style="width: 74.25pt;height: 42pt;"></img></div><div id="mc-region-744" style="position: absolute; top: 2276.25pt; left: 30pt; width: 180.5pt; "><a name="" /><p class="Style1">substituting into the Roche formula gives</p></div><div id="mc-region-714" style="position: absolute; top: 2298.75pt; left: 78pt; width: 181.5pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0055_1211411171.PNG" id="generatedImage714" path=".\addendum 5 gravity kepler 3rd law_images/IMG0055_1211411171.PNG" style="width: 181.5pt;height: 57.75pt;"></img></div><div id="mc-region-745" style="position: absolute; top: 2372.25pt; left: 30pt; width: 372.5pt; "><a name="" /><p class="Style1">This is a reasonable estimate, but is idealized iusing two identical spheres. A more accurate numerical calculation for a single sphere gives:</p></div><div id="mc-region-747" style="position: absolute; top: 2400.75pt; left: 138pt; width: 111pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0056_1211411171.PNG" id="generatedImage747" path=".\addendum 5 gravity kepler 3rd law_images/IMG0056_1211411171.PNG" style="width: 111pt;height: 68.25pt;"></img></div><div id="mc-region-748" style="position: absolute; top: 2492.25pt; left: 48pt; width: 368.75pt; "><a name="" /><p class="Style1">Example: Saturn system. Assume the mean densities of the planet and satellite are equal:</p></div><div id="mc-region-754" style="position: absolute; top: 2541pt; left: 54pt; width: 81.75pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0057_1211411187.PNG" id="generatedImage754" path=".\addendum 5 gravity kepler 3rd law_images/IMG0057_1211411187.PNG" style="width: 81.75pt;height: 16.5pt;"></img></div><div id="mc-region-753" style="position: absolute; top: 2541pt; left: 186pt; width: 92.25pt; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0058_1211411187.PNG" id="generatedImage753" path=".\addendum 5 gravity kepler 3rd law_images/IMG0058_1211411187.PNG" style="width: 92.25pt;height: 16.5pt;"></img></div><div id="mc-region-759" style="position: absolute; top: 2577pt; left: 72pt; width: 87pt; background-color: inherit; "><a name="" /><img border="0" src=".\addendum 5 gravity kepler 3rd law_images/IMG0059_1211411187.PNG" id="generatedImage759" path=".\addendum 5 gravity kepler 3rd law_images/IMG0059_1211411187.PNG" style="width: 87pt;height: 16.5pt;"></img></div><div id="mc-region-760" style="position: absolute; top: 2576.25pt; left: 198pt; width: 159.5pt; "><a name="" /><p class="Style1">This is the (very close to) the radius of the outermost rings (137,000 km)</p></div><div></div></body></html>