# An Introduction to Celestial Mechanics by Professor Richard Fitzpatrick By Professor Richard Fitzpatrick

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Extra resources for An Introduction to Celestial Mechanics

Example text

9), where a is the orbital major radius and e the eccentricity. Moreover, the distance S G and the angle GS P correspond to the radial distance, r, and the true anomaly, θ, respectively. Let PRA be a circle of radius a centered on C. It follows that AP is a diameter of this circle. Let RGQ be a line, perpendicular to AP, that passes through G and joins the circle to the diameter. It follows that CR = a. Let us denote the angle RCS as E. Simple trigonometry reveals that S Q = r cos θ and CQ = a cos E.

Show that if the variation of gravity with height is allowed for, but the resistance of air is neglected, then the height reached will be greater by h2 /(R − h), where R is the Earth’s radius. ) A particle is projected vertically upward from the Earth’s surface with a velocity just suﬃcient for it to reach infinity (neglecting air resistance). Prove that the time needed to reach a height h is ⎡ ⎤ 1/2 ⎢ 3/2 ⎥⎥ 1 2R h ⎢⎢⎢ − 1⎥⎥⎥⎦ , ⎢⎣ 1 + 3 g R where R is the Earth’s radius, and g its surface gravitational acceleration.

B. F1 and F2 are not parallel (or antiparallel), but their lines of action cross at a point. In this case, the line of action of F passes through the crossing point. Deduce that if an isolated system consists of three extended bodies, A, B, and C, where FA is the resultant force acting on A (due to B and C), FB is the resultant force acting on B, and FC is the resultant force acting on C; then FA +FB +FC = 0, and the forces either all have parallel lines of action or have lines of action that cross at a common point.