Celestial Mechanics
Celestial Mechanics
Kepler’s Laws:I. A planet orbits the Sun in an ellipse, with the Sun at one focus of
the ellipseII. A line connecting a planet to the Sun sweeps out equal areas in
equal time intervals
Kepler’s Laws:I. A planet orbits the Sun in an ellipse, with the Sun at one focus of
the ellipseII. A line connecting a planet to the Sun sweeps out equal areas in
equal time intervalsIII. P[yr]2=a[AU]3, with P orbital period of the planet, a average
distance of the planet from the Sun
log(a) = (2/3) log(P)
BELLIPSE:defined by a set of points satisfying the equationr+r’=2a
Eccentricity:e = FF’/2a0<e<1
If F=F’, then r=r’, then r=aeq. of a CIRCLE (e=0)
A = ⇡ab
<latexit sha1_base64="fr6EuGRq/kdw9STSBPxZM1NjXyI=">AAAB8HicbVBNSwMxEJ2tX7V+rXr0EiyCp7IrBfUgVL14rGA/pF1KNs22oUl2SbJCWforvHhQxKs/x5v/xrTdg7Y+GHi8N8PMvDDhTBvP+3YKK6tr6xvFzdLW9s7unrt/0NRxqghtkJjHqh1iTTmTtGGY4bSdKIpFyGkrHN1O/dYTVZrF8sGMExoIPJAsYgQbKz1eX3UThjAKe27Zq3gzoGXi56QMOeo996vbj0kqqDSEY607vpeYIMPKMMLppNRNNU0wGeEB7VgqsaA6yGYHT9CJVfooipUtadBM/T2RYaH1WIS2U2Az1IveVPzP66QmuggyJpPUUEnmi6KUIxOj6feozxQlho8twUQxeysiQ6wwMTajkg3BX3x5mTTPKn61cnlfLddu8jiKcATHcAo+nEMN7qAODSAg4Ble4c1Rzovz7nzMWwtOPnMIf+B8/gCCpY+W</latexit>
BELLIPSE:defined by a set of points satisfying the equationr+r’=2a
Eccentricity:e = FF’/2a0<e<1
If F=F’, then r=r’, then r=aeq. of a CIRCLE (e=0)
A = ⇡ab
<latexit sha1_base64="fr6EuGRq/kdw9STSBPxZM1NjXyI=">AAAB8HicbVBNSwMxEJ2tX7V+rXr0EiyCp7IrBfUgVL14rGA/pF1KNs22oUl2SbJCWforvHhQxKs/x5v/xrTdg7Y+GHi8N8PMvDDhTBvP+3YKK6tr6xvFzdLW9s7unrt/0NRxqghtkJjHqh1iTTmTtGGY4bSdKIpFyGkrHN1O/dYTVZrF8sGMExoIPJAsYgQbKz1eX3UThjAKe27Zq3gzoGXi56QMOeo996vbj0kqqDSEY607vpeYIMPKMMLppNRNNU0wGeEB7VgqsaA6yGYHT9CJVfooipUtadBM/T2RYaH1WIS2U2Az1IveVPzP66QmuggyJpPUUEnmi6KUIxOj6feozxQlho8twUQxeysiQ6wwMTajkg3BX3x5mTTPKn61cnlfLddu8jiKcATHcAo+nEMN7qAODSAg4Ble4c1Rzovz7nzMWwtOPnMIf+B8/gCCpY+W</latexit>
BELLIPSE:defined by a set of points satisfying the equationr+r’=2a
Eccentricity:e = FF’/2a0<e<1
If F=F’, then r=r’, then r=aeq. of a CIRCLE (e=0)
b2 = a2 (1-e2)Relation between semi-major, semi-minor, and ellipticity A = ⇡ab
<latexit sha1_base64="fr6EuGRq/kdw9STSBPxZM1NjXyI=">AAAB8HicbVBNSwMxEJ2tX7V+rXr0EiyCp7IrBfUgVL14rGA/pF1KNs22oUl2SbJCWforvHhQxKs/x5v/xrTdg7Y+GHi8N8PMvDDhTBvP+3YKK6tr6xvFzdLW9s7unrt/0NRxqghtkJjHqh1iTTmTtGGY4bSdKIpFyGkrHN1O/dYTVZrF8sGMExoIPJAsYgQbKz1eX3UThjAKe27Zq3gzoGXi56QMOeo996vbj0kqqDSEY607vpeYIMPKMMLppNRNNU0wGeEB7VgqsaA6yGYHT9CJVfooipUtadBM/T2RYaH1WIS2U2Az1IveVPzP66QmuggyJpPUUEnmi6KUIxOj6feozxQlho8twUQxeysiQ6wwMTajkg3BX3x5mTTPKn61cnlfLddu8jiKcATHcAo+nEMN7qAODSAg4Ble4c1Rzovz7nzMWwtOPnMIf+B8/gCCpY+W</latexit>
BELLIPSE:defined by a set of points satisfying the equationr+r’=2a
Eccentricity:e = FF’/2a0<e<1
If F=F’, then r=r’, then r=aeq. of a CIRCLE (e=0)
b2 = a2 (1-e2)Relation between semi-major, semi-minor, and ellipticity A = ⇡ab
<latexit sha1_base64="fr6EuGRq/kdw9STSBPxZM1NjXyI=">AAAB8HicbVBNSwMxEJ2tX7V+rXr0EiyCp7IrBfUgVL14rGA/pF1KNs22oUl2SbJCWforvHhQxKs/x5v/xrTdg7Y+GHi8N8PMvDDhTBvP+3YKK6tr6xvFzdLW9s7unrt/0NRxqghtkJjHqh1iTTmTtGGY4bSdKIpFyGkrHN1O/dYTVZrF8sGMExoIPJAsYgQbKz1eX3UThjAKe27Zq3gzoGXi56QMOeo996vbj0kqqDSEY607vpeYIMPKMMLppNRNNU0wGeEB7VgqsaA6yGYHT9CJVfooipUtadBM/T2RYaH1WIS2U2Az1IveVPzP66QmuggyJpPUUEnmi6KUIxOj6feozxQlho8twUQxeysiQ6wwMTajkg3BX3x5mTTPKn61cnlfLddu8jiKcATHcAo+nEMN7qAODSAg4Ble4c1Rzovz7nzMWwtOPnMIf+B8/gCCpY+W</latexit>
BELLIPSE:defined by a set of points satisfying the equationr+r’=2a
Eccentricity:e = FF’/2a0<e<1
If F=F’, then r=r’, then r=aeq. of a CIRCLE (e=0)
b2 = a2 (1-e2)Relation between semi-major, semi-minor, and ellipticity
r =a(1� e2)
1 + e cos ✓, 0 e < 1
<latexit sha1_base64="DfFjjCi1rmjD1iv50RLY41IcqjU=">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</latexit>
If a, e, and theta are known, the position from the focus can be derived
A = ⇡ab
<latexit sha1_base64="fr6EuGRq/kdw9STSBPxZM1NjXyI=">AAAB8HicbVBNSwMxEJ2tX7V+rXr0EiyCp7IrBfUgVL14rGA/pF1KNs22oUl2SbJCWforvHhQxKs/x5v/xrTdg7Y+GHi8N8PMvDDhTBvP+3YKK6tr6xvFzdLW9s7unrt/0NRxqghtkJjHqh1iTTmTtGGY4bSdKIpFyGkrHN1O/dYTVZrF8sGMExoIPJAsYgQbKz1eX3UThjAKe27Zq3gzoGXi56QMOeo996vbj0kqqDSEY607vpeYIMPKMMLppNRNNU0wGeEB7VgqsaA6yGYHT9CJVfooipUtadBM/T2RYaH1WIS2U2Az1IveVPzP66QmuggyJpPUUEnmi6KUIxOj6feozxQlho8twUQxeysiQ6wwMTajkg3BX3x5mTTPKn61cnlfLddu8jiKcATHcAo+nEMN7qAODSAg4Ble4c1Rzovz7nzMWwtOPnMIf+B8/gCCpY+W</latexit>
BELLIPSE:defined by a set of points satisfying the equationr+r’=2a
Eccentricity:e = FF’/2a0<e<1
If F=F’, then r=r’, then r=aeq. of a CIRCLE (e=0)
b2 = a2 (1-e2)Relation between semi-major, semi-minor, and ellipticity
r =a(1� e2)
1 + e cos ✓, 0 e < 1
<latexit sha1_base64="DfFjjCi1rmjD1iv50RLY41IcqjU=">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</latexit>
If a, e, and theta are known, the position from the focus can be derived
PerihelionAphelion
Area of ellipse: A = ⇡ab
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Conic curves
E<0E<0
E total energyE<0—> bound systemE>0 —> unbound system
E=0 E>0
0 e < 1
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r =2p
1 + cos ✓
<latexit sha1_base64="x8BFW1+WyxuM+Zxy2VFVgnFeqw8=">AAACBnicbVBNS8NAEN34WetX1KMIi0UQhJKUgnoQil48VrAf0JSy2W7apZtN2J0IJeTkxb/ixYMiXv0N3vw3btsctPXBwOO9GWbm+bHgGhzn21paXlldWy9sFDe3tnd27b39po4SRVmDRiJSbZ9oJrhkDeAgWDtWjIS+YC1/dDPxWw9MaR7JexjHrBuSgeQBpwSM1LOPFL7CXqAITStxlrpnHo106sGQAcmynl1yys4UeJG4OSmhHPWe/eX1I5qETAIVROuO68TQTYkCTgXLil6iWUzoiAxYx1BJQqa76fSNDJ8YpY+DSJmSgKfq74mUhFqPQ990hgSGet6biP95nQSCi27KZZwAk3S2KEgEhghPMsF9rhgFMTaEUMXNrZgOickETHJFE4I7//IiaVbKbrV8eVct1a7zOAroEB2jU+Sic1RDt6iOGoiiR/SMXtGb9WS9WO/Wx6x1ycpnDtAfWJ8/wuSYuQ==</latexit>
r =a(e2 � 1)
1 + e cos ✓
<latexit sha1_base64="gfb778pS6KqElA2Fsjqti8fb0aE=">AAACD3icbVA9SwNBEN2L3/ErammzGJSIGO6CoBZC0MYygomBXAx7mzmzZO+D3TkhHPcPbPwrNhaK2Nra+W/cxBSa+GDg8d4MM/O8WAqNtv1l5WZm5+YXFpfyyyura+uFjc2GjhLFoc4jGammxzRIEUIdBUpoxgpY4Em48foXQ//mHpQWUXiNgxjaAbsLhS84QyN1CnuKnlHXV4ynrAS3FXro7GepcwDU5ZFOXewBsizrFIp22R6BThNnTIpkjFqn8Ol2I54EECKXTOuWY8fYTplCwSVkeTfREDPeZ3fQMjRkAeh2Ovono7tG6VI/UqZCpCP190TKAq0HgWc6A4Y9PekNxf+8VoL+STsVYZwghPxnkZ9IihEdhkO7QgFHOTCEcSXMrZT3mAkHTYR5E4Iz+fI0aVTKzlH59OqoWD0fx7FItskOKRGHHJMquSQ1UiecPJAn8kJerUfr2Xqz3n9ac9Z4Zov8gfXxDXa8mxs=</latexit>
e > 1
<latexit sha1_base64="TlqOVUrn78U50330Rm29rnbCBsc=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoF6k6MVjBdMW2lA220m7dLMJuxuhlP4GLx4U8eoP8ua/cdvmoK0PBh7vzTAzL0wF18Z1v53C2vrG5lZxu7Szu7d/UD48auokUwx9lohEtUOqUXCJvuFGYDtVSONQYCsc3c381hMqzRP5aMYpBjEdSB5xRo2VfCQ3xOuVK27VnYOsEi8nFcjR6JW/uv2EZTFKwwTVuuO5qQkmVBnOBE5L3UxjStmIDrBjqaQx6mAyP3ZKzqzSJ1GibElD5urviQmNtR7Hoe2MqRnqZW8m/ud1MhNdBRMu08ygZItFUSaIScjsc9LnCpkRY0soU9zeStiQKsqMzadkQ/CWX14lzYuqV6teP9Qq9ds8jiKcwCmcgweXUId7aIAPDDg8wyu8OdJ5cd6dj0VrwclnjuEPnM8fbNONzA==</latexit>
Each type of conic section is related to a specific form of celestial motion
Newton’s Laws:I. An object at rest will remain at rest and an object in motion will
remain in motion in a straight line at a constant speed unless acted upon by an external force (law of inertia)
= the momentum of an object remains constant unless it experiences an external force
p = mv
<latexit sha1_base64="2+ehx4S79iWORJsd4XP8Kx1+LXo=">AAAB/HicbZDLSsNAFIZP6q3WW7RLN4NFcFUSEdSFUHTjsoK9QBvKZDpph84kYWZSKCG+ihsXirj1Qdz5Nk7TLLT1h4GP/5zDOfP7MWdKO863VVpb39jcKm9Xdnb39g/sw6O2ihJJaItEPJJdHyvKWUhbmmlOu7GkWPicdvzJ3bzemVKpWBQ+6llMPYFHIQsYwdpYA7ua9v0AxRm6QQLlPM0Gds2pO7nQKrgF1KBQc2B/9YcRSQQNNeFYqZ7rxNpLsdSMcJpV+omiMSYTPKI9gyEWVHlpfnyGTo0zREEkzQs1yt3fEykWSs2EbzoF1mO1XJub/9V6iQ6uvJSFcaJpSBaLgoQjHaF5EmjIJCWazwxgIpm5FZExlphok1fFhOAuf3kV2ud196J+/XBRa9wWcZThGE7gDFy4hAbcQxNaQGAGz/AKb9aT9WK9Wx+L1pJVzFThj6zPHzV3k90=</latexit>
Newton’s Laws:I. An object at rest will remain at rest and an object in motion will
remain in motion in a straight line at a constant speed unless acted upon by an external force (law of inertia)
= the momentum of an object remains constant unless it experiences an external force
II. The net force (the sum of all forces) acting on an object is proportional to the object’s mass and its resultant acceleration
Fnet =nX
i=1
Fi = ma
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m = constant
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a =dv
dt
<latexit sha1_base64="/15KA5E3HKSgPYA3Pcd3XL8IOI4=">AAACBnicbVDLSsNAFJ3UV62vqEsRBovgqiRSUBdC0Y3LCvYBTSmTyaQdOnkwc1MoISs3/oobF4q49Rvc+TdO0yy09cCFwzn3cu89biy4Asv6Nkorq2vrG+XNytb2zu6euX/QVlEiKWvRSESy6xLFBA9ZCzgI1o0lI4ErWMcd3878zoRJxaPwAaYx6wdkGHKfUwJaGpjHqeP6mGT4Gju+JDT1cmGSZakH2cCsWjUrB14mdkGqqEBzYH45XkSTgIVABVGqZ1sx9FMigVPBsoqTKBYTOiZD1tM0JAFT/TR/I8OnWvGwH0ldIeBc/T2RkkCpaeDqzoDASC16M/E/r5eAf9lPeRgnwEI6X+QnAkOEZ5lgj0tGQUw1IVRyfSumI6LTAJ1cRYdgL768TNrnNbteu7qvVxs3RRxldIRO0Bmy0QVqoDvURC1E0SN6Rq/ozXgyXox342PeWjKKmUP0B8bnD95pmMk=</latexit>
Fnet =dp
dt
<latexit sha1_base64="Y9PeB+CwbkLqDvMqQKNGtByxodo=">AAACEHicbVBNS8NAEN3Ur1q/qh69LBbRU0mkoB6EoiAeK9gPaErZbDbt0s0m7E6EEvITvPhXvHhQxKtHb/4bt2kP2vpg4PHeDDPzvFhwDbb9bRWWlldW14rrpY3Nre2d8u5eS0eJoqxJIxGpjkc0E1yyJnAQrBMrRkJPsLY3up747QemNI/kPYxj1gvJQPKAUwJG6pePU9cL8E0/dVWIJYMsw5fYDRShqZ9bcZalPmT9csWu2jnwInFmpIJmaPTLX64f0SRkEqggWncdO4ZeShRwKlhWchPNYkJHZMC6hkoSMt1L84cyfGQUHweRMiUB5+rviZSEWo9Dz3SGBIZ63puI/3ndBILzXsplnACTdLooSASGCE/SwT5XjIIYG0Ko4uZWTIfEpAEmw5IJwZl/eZG0TqtOrXpxV6vUr2ZxFNEBOkQnyEFnqI5uUQM1EUWP6Bm9ojfryXqx3q2PaWvBms3soz+wPn8ApXGdBQ==</latexit>
The net force on an object is equal to the time rate of change of its momentum
Newton’s Laws:I. An object at rest will remain at rest and an object in motion will
remain in motion in a straight line at a constant speed unless acted upon by an external force (law of inertia)
= the momentum of an object remains constant unless it experiences an external force
II. The net force (the sum of all forces) acting on an object is proportional to the object’s mass and its resultant acceleration
III. For every action there is an equal and opposite reaction
F12 = �F21
<latexit sha1_base64="yEvAYxHY4QOORrq/NXpN5Hwx0jE=">AAACDXicbVDLSsNAFJ34rPUVdelmsApuLEkpqAuhKIjLCvYBTSiT6aQdOjMJMxOhhPyAG3/FjQtF3Lp35984bbOorQcuHM65l3vvCWJGlXacH2tpeWV1bb2wUdzc2t7Ztff2mypKJCYNHLFItgOkCKOCNDTVjLRjSRAPGGkFw5ux33okUtFIPOhRTHyO+oKGFCNtpK59nHpBCG+7qSc5dCtZBq/g2axWcbOsa5ecsjMBXCRuTkogR71rf3u9CCecCI0ZUqrjOrH2UyQ1xYxkRS9RJEZ4iPqkY6hAnCg/nXyTwROj9GAYSVNCw4k6O5EirtSIB6aTIz1Q895Y/M/rJDq88FMq4kQTgaeLwoRBHcFxNLBHJcGajQxBWFJzK8QDJBHWJsCiCcGdf3mRNCtlt1q+vK+Watd5HAVwCI7AKXDBOaiBO1AHDYDBE3gBb+DderZerQ/rc9q6ZOUzB+APrK9fg5iZ9w==</latexit>
Action and reaction are forces acting on DIFFERENT objects
m1
M2
Newton’s Laws:I. An object at rest will remain at rest and an object in motion will
remain in motion in a straight line at a constant speed unless acted upon by an external force (law of inertia)
= the momentum of an object remains constant unless it experiences an external force
II. The net force (the sum of all forces) acting on an object is proportional to the object’s mass and its resultant acceleration
III. For every action there is an equal and opposite reaction
Newton’s Law of Universal Gravitation:
F = GMm
r2, G = 6.673⇥ 10�11 N m2
kg2
<latexit sha1_base64="D7G+ClwIqz0pPvF55CKv8Gg+FUA=">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</latexit>
Work and EnergyThe amount of energy (the work) needed to raise an object of mass m to a height h against a gravitational force is equal to the change in the potential energy of the system:
Work Integral:
For the gravitational force on m being due to a mass M located at the orgin:
Uf � Ui = �U ⌘ �Z rf
ri
F · dr
<latexit sha1_base64="/d2pQ2W1avn7wHppdc2PhnyBVpM=">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</latexit>
F · dr = �Fdr
<latexit sha1_base64="7MK/qTtQmARkPdXq+mJW/oT25ik=">AAACB3icbVDLSsNAFJ3UV62vqEtBBovgxpJIQV0IRaG4rGAf0IQymUzaoZNJmJkIJWTnxl9x40IRt/6CO//GaZqFth64cOace5l7jxczKpVlfRulpeWV1bXyemVjc2t7x9zd68goEZi0ccQi0fOQJIxy0lZUMdKLBUGhx0jXG99M/e4DEZJG/F5NYuKGaMhpQDFSWhqYh6njBbCZQQf7kYJ+/hQZvIKnTeiLgVm1alYOuEjsglRBgdbA/HL8CCch4QozJGXftmLlpkgoihnJKk4iSYzwGA1JX1OOQiLdNL8jg8da8WEQCV1cwVz9PZGiUMpJ6OnOEKmRnPem4n9eP1HBhZtSHieKcDz7KEgYVBGchgJ9KghWbKIJwoLqXSEeIYGw0tFVdAj2/MmLpHNWs+u1y7t6tXFdxFEGB+AInAAbnIMGuAUt0AYYPIJn8ArejCfjxXg3PmatJaOY2Qd/YHz+ADv7l6M=</latexit>
�U =
Z rf
ri
GmM
r2dr
<latexit sha1_base64="t4ekb6e6V4I+XLMwP10yooDbE+Q=">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</latexit>
U = �GMm
r(where Uf = 0 at rf = 1)
<latexit sha1_base64="SEENi6AshcFG0UP95tZb+9cnalQ=">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</latexit>
Work and EnergyWork must be performed on a massive object if its speed is to be changed:
Kinetic energy of an object
The work done on the particle results in an equivalent change in the particle’s kinetic energy (conservation of energy)
W ⌘ ��U = . . . =1
2mv2f �
1
2mv2i
<latexit sha1_base64="vQJ1vf5dmPT4qLMmIZWLVD7GiW4=">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</latexit>
K ⌘ 1
2mv2
<latexit sha1_base64="WqcgCJkb7s0tYdvyRcZ5WLwd/74=">AAACA3icdVDLSsNAFJ3UV62vqDvdDBbBVUli6GNXdCO4qWAf0MQymU7aoZOHM5NCCQU3/oobF4q49Sfc+TdO2goqeuDC4Zx7ufceL2ZUSMP40HJLyyura/n1wsbm1vaOvrvXElHCMWniiEW84yFBGA1JU1LJSCfmBAUeI21vdJ757THhgkbhtZzExA3QIKQ+xUgqqacfXEKH3CZ0DB2fI5ya09SawgCOb6yeXjRKtWrZssvQKBlGxbTMjFgV+9SGplIyFMECjZ7+7vQjnAQklJghIbqmEUs3RVxSzMi04CSCxAiP0IB0FQ1RQISbzn6YwmOl9KEfcVWhhDP1+0SKAiEmgac6AySH4reXiX953UT6VTelYZxIEuL5Ij9hUEYwCwT2KSdYsokiCHOqboV4iFQWUsVWUCF8fQr/Jy2rZNql2pVdrJ8t4siDQ3AEToAJKqAOLkADNAEGd+ABPIFn7V571F6013lrTlvM7IMf0N4+Adjelww=</latexit>
W = �K =1
2m(v2f � v2i )
<latexit sha1_base64="ZK8ogzfLRU62PVIv/IqSku4lnSM=">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</latexit>
E = K + U =1
2mv2 �G
Mm
r
<latexit sha1_base64="a8sWr1WFZRqaESqs0cZMlncAn90=">AAACFnicbVDLSgMxFM3UV62vUZdugkUQpGWmFNSFUBRREKGC0xbaWjJppg1NZoYkUyjDfIUbf8WNC0Xcijv/xrSdhbYeSO7hnHtJ7nFDRqWyrG8js7C4tLySXc2trW9sbpnbOzUZRAITBwcsEA0XScKoTxxFFSONUBDEXUbq7uBi7NeHREga+PdqFJI2Rz2fehQjpaWOWbiEZ/DmyNF3yxMIx3YSlxLI4fBhXAvwairf8iQWScfMW0VrAjhP7JTkQYpqx/xqdQMcceIrzJCUTdsKVTtGQlHMSJJrRZKECA9QjzQ19REnsh1P1krggVa60AuEPr6CE/X3RIy4lCPu6k6OVF/OemPxP68ZKe+kHVM/jBTx8fQhL2JQBXCcEexSQbBiI00QFlT/FeI+0jEonWROh2DPrjxPaqWiXS6e3pXzlfM0jizYA/vgENjgGFTANagCB2DwCJ7BK3gznowX4934mLZmjHRmF/yB8fkDmyudQQ==</latexit>
Energy of a particle of mass m with velocity v at a distance r from the center of a larger mass M
At r = 1, v = 0
<latexit sha1_base64="QY62kjAXZuvnGHDc5hVqRGkxN9k=">AAAB/3icbVDLSgNBEJz1GeNrVfDiZTAIHiTsSkA9BKJePEYwD8iGMDuZTYbMzi4zvYGwJuCvePGgiFd/w5t/4+Rx0MSChqKqm+4uPxZcg+N8W0vLK6tr65mN7ObW9s6uvbdf1VGiKKvQSESq7hPNBJesAhwEq8eKkdAXrOb3bsd+rc+U5pF8gEHMmiHpSB5wSsBILfsw9VSIr2Gkih6XAQzORv2iM2zZOSfvTIAXiTsjOTRDuWV/ee2IJiGTQAXRuuE6MTRTooBTwYZZL9EsJrRHOqxhqCQh0810cv8QnxiljYNImZKAJ+rviZSEWg9C33SGBLp63huL/3mNBILLZsplnACTdLooSASGCI/DwG2uGAUxMIRQxc2tmHaJIhRMZFkTgjv/8iKpnufdQv7qvpAr3cziyKAjdIxOkYsuUAndoTKqIIoe0TN6RW/Wk/VivVsf09YlazZzgP7A+vwBmxqV2w==</latexit>
E = 0
<latexit sha1_base64="CT6cClPX2pe8f54ocXZuft1KuNk=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKexKQD0IQRE8RjQPSJYwO5kkQ2Znl5leISz5BC8eFPHqF3nzb5wke9DEgoaiqpvuriCWwqDrfju5ldW19Y38ZmFre2d3r7h/0DBRohmvs0hGuhVQw6VQvI4CJW/FmtMwkLwZjG6mfvOJayMi9YjjmPshHSjRF4yilR5ur9xuseSW3RnIMvEyUoIMtW7xq9OLWBJyhUxSY9qeG6OfUo2CST4pdBLDY8pGdMDblioacuOns1Mn5MQqPdKPtC2FZKb+nkhpaMw4DGxnSHFoFr2p+J/XTrB/4adCxQlyxeaL+okkGJHp36QnNGcox5ZQpoW9lbAh1ZShTadgQ/AWX14mjbOyVylf3ldK1essjjwcwTGcggfnUIU7qEEdGAzgGV7hzZHOi/PufMxbc042cwh/4Hz+AIyhjVY=</latexit>
1
2mv2 = G
Mm
r
<latexit sha1_base64="wP8WQ3wE0RtYG04OBgYctry9ET0=">AAACDHicbVDLSgMxFL3js9ZX1aWbYBFclZlSUBdC0YVuhAr2AW0tmTTThiYzQ5IplGE+wI2/4saFIm79AHf+jZl2Ftp6IORwzrkk97ghZ0rb9re1tLyyurae28hvbm3v7Bb29hsqiCShdRLwQLZcrChnPq1rpjlthZJi4XLadEdXqd8cU6lY4N/rSUi7Ag985jGCtZF6hWLHk5jEThKXEyTQ+CG9L9D1TL4VSSwTk7JL9hRokTgZKUKGWq/w1ekHJBLU14RjpdqOHepujKVmhNMk34kUDTEZ4QFtG+pjQVU3ni6ToGOj9JEXSHN8jabq74kYC6UmwjVJgfVQzXup+J/XjrR31o2ZH0aa+mT2kBdxpAOUNoP6TFKi+cQQTCQzf0VkiE0N2vSXNyU48ysvkka55FRK53eVYvUyqyMHh3AEJ+DAKVThBmpQBwKP8Ayv8GY9WS/Wu/Uxiy5Z2cwB/IH1+QPW45rj</latexit>
vesc =p2GM/r
<latexit sha1_base64="Qz0zbh1iJBkIYdPOIsu2wOmz5GY=">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</latexit>
The escape velocity is independent on m
Center of mass
position vector R as the weighted average of the position vectors of the individual massesR ⌘ m1r1 +m2r2 + . . .
m1 +m2 + . . .
<latexit sha1_base64="fMSt6o8AXStkQ+4cw7noGE2qKCw=">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</latexit>
MR =nX
i=1
miri
<latexit sha1_base64="+a5n9KPdoewtV8CoRJuHucpB+6U=">AAACEnicbVDLSsNAFJ34rPUVdelmsAi6KYkU1EWh6MaNUMU+oIlhMp20Q2cmYWYilJBvcOOvuHGhiFtX7vwbp4+Fth64cDjnXu69J0wYVdpxvq2FxaXlldXCWnF9Y3Nr297Zbao4lZg0cMxi2Q6RIowK0tBUM9JOJEE8ZKQVDi5HfuuBSEVjcaeHCfE56gkaUYy0kQL7+BpmXhjB2xxWoadSHmS06ub3mcgzHtCJKQOa54FdcsrOGHCeuFNSAlPUA/vL68Y45URozJBSHddJtJ8hqSlmJC96qSIJwgPUIx1DBeJE+dn4pRweGqULo1iaEhqO1d8TGeJKDXloOjnSfTXrjcT/vE6qozM/oyJJNRF4sihKGdQxHOUDu1QSrNnQEIQlNbdC3EcSYW1SLJoQ3NmX50nzpOxWyuc3lVLtYhpHAeyDA3AEXHAKauAK1EEDYPAInsEreLOerBfr3fqYtC5Y05k98AfW5w8cqp0v</latexit>
r = r02 � r01
<latexit sha1_base64="rc5i/ethk5ardnrxjxKSA+EL9Ho=">AAACHnicbZDLSgMxFIYz9VbrbdSlm2AR3FhmSkVdCEU3LivYC3TGkkkzbWiSGZKMUIZ5Eje+ihsXigiu9G1M21lo6w+Bj/+cw8n5g5hRpR3n2yosLa+srhXXSxubW9s79u5eS0WJxKSJIxbJToAUYVSQpqaakU4sCeIBI+1gdD2ptx+IVDQSd3ocE5+jgaAhxUgbq2efpl4QQpnBSzij+9SLJeUk66XVLIMni7abZT277FScqeAiuDmUQa5Gz/70+hFOOBEaM6RU13Vi7adIaooZyUpeokiM8AgNSNegQJwoP52el8Ej4/RhGEnzhIZT9/dEirhSYx6YTo70UM3XJuZ/tW6iw3M/pSJONBF4tihMGNQRnGQF+1QSrNnYAMKSmr9CPEQSYW0SLZkQ3PmTF6FVrbi1ysVtrVy/yuMoggNwCI6BC85AHdyABmgCDB7BM3gFb9aT9WK9Wx+z1oKVz+yDP7K+fgDtiqJs</latexit>
displacement vector
MV =nX
i=1
mivi
<latexit sha1_base64="QypGTMFXI0tF3MiEKQ+p5BZGfIQ=">AAACEnicbVDLSsNAFJ34rPUVdelmsAi6KYkU1EWh6MaNUME+oIlhMp20Q2cmYWZSKCHf4MZfceNCEbeu3Pk3Th8LbT1w4XDOvdx7T5gwqrTjfFtLyyura+uFjeLm1vbOrr2331RxKjFp4JjFsh0iRRgVpKGpZqSdSIJ4yEgrHFyP/daQSEVjca9HCfE56gkaUYy0kQL79BZmXhjBZg6r0FMpDzJadfOHTOQZD+jUHAY0zwO75JSdCeAicWekBGaoB/aX141xyonQmCGlOq6TaD9DUlPMSF70UkUShAeoRzqGCsSJ8rPJSzk8NkoXRrE0JTScqL8nMsSVGvHQdHKk+2reG4v/eZ1URxd+RkWSaiLwdFGUMqhjOM4HdqkkWLORIQhLam6FuI8kwtqkWDQhuPMvL5LmWdmtlC/vKqXa1SyOAjgER+AEuOAc1MANqIMGwOARPINX8GY9WS/Wu/UxbV2yZjMH4A+szx8pXp03</latexit>
R is the position of the center of mass of the system, and V is the velocity of the center of mass. P=MV is the momentum of the center of mass.
Center of massr = r02 � r01
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displacement vector
If all forces acting on individual particles in the system are due to other particles contained within the system (Newton’s 3rd law):
F =dP
dt= 0
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i.e., the center of mass does not accelerate if no external force exists, i.e., the reference frame associated to the center of mass is inertial
Center of massr = r02 � r01
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displacement vector
It is therefore convenient to place the coordinate center at the center of mass, choosing R=0. Substituting r2=r1+r, and defining the reduced mass (for a binary system) as: µ ⌘ m1m2
m1 +m2
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r1 = � µ
m1r
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r2 =µ
m2r
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E =1
2µv2 �G
Mµ
r
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i.e., the total energy of the system is equal to the kinetic energy of the reduced mass plus the potential energy of the reduced mass moving about a mass M located and fixed at the origin.
Center of massr = r02 � r01
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displacement vector
Total angular momentum:
L = m1r1 ⇥ v1 +m2r2 ⇥ v2 = µr⇥ v = r⇥ p
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i.e., the total angular momentum equals the angular momentum of the reduced mass onlyThe two-body problem can be treated as an equivalent one-body problem with the reduced mass moving about a fixed mass M at a distance r.µ
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dL
dt=
d
dt(r⇥ p) =
dr
dt⇥ p+ r⇥ dp
dt= v ⇥ p+ r⇥ F = 0
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L of a system is a constant for a central law force
Revisited Kepler’s 1st Law
r =L2/µ2
GM(1 + e cos ✓)General equation of a conic section
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i.e., the path of the reduced mass about the center of mass under the influence of gravity is a conic section.
Elliptical orbits result from an attractive r-2 central-force law (i.e., gravity) when the total energy of the system is less than 0 (E<0, bound system); parabolic trajectories when E=0; and hyperbolic trajectories when E>0 (unbound systems).
NOTE: both objects in a binary system move about the center of mass in ellipses, with the center of mass occupying one focus of each ellipse.
For closed planetary orbits, comparing the above with the previous equation of an ellipse, I obtain the total angular momentum of the system (max for e=0, i.e., circular motions):
L = µp
GMa(1� e2)
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Derivation of Kepler’s 2nd Law
d�
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Derivation of Kepler’s 2nd Law
Kepler’s revised1st law
Derivation of Kepler’s 2nd Law
Kepler’s revised1st law
Derivation of Kepler’s 2nd Law
Kepler’s revised1st law
Derivation of Kepler’s 3rd Law