Physics: Wok & Enegy Beyond Eath Guided Inquiy Elliptical Obits Keple s Fist Law states that all planets move in an elliptical path aound the Sun. This concept can be extended to celestial bodies beyond ou sola system. Fundamentals of an Ellipse An ellipse has two foci, a majo axis and a mino axis. The Sun o moe massive body is located at one of the foci and the planet o satellite obits along the elliptical path. An impotant note, the moe massive body is located at one foci if M >> m. If this is not the case, then the two bodies will still follow an elliptical obit, but the cente of gavity of the system will be at one of the foci instead of the moe massive body. The sketch to the ight shows how to daw an ellipse. Loop a sting aound thumbtacks at each focus and stetch the sting tight with a pencil while moving the pencil aound the tacks. The sum of the two distances fom the ellipse to each of the foci is constant, + = 2a. In this case, is the obital position of the planet. The sketch to the ight shows the majo paametes that descibe an ellipse: Peigee o peihelion the point along the elliptical path that is closest to the focus the Sun o massive body occupies. Apogee o aphelion the point along the elliptical path that is fathest fom the Sun o massive body. R p the distance between the planet s and the Sun s centes of gavity at the peihelion. R a the distance between the planet s and the Sun s centes of gavity at the aphelion. Focus, F Focus, F a the length of the semi- majo axis, it is one- half the total width of the ellipse. b (not shown) the length of the semi- majo axis, it is one- half the total height of the ellipse. e eccenticity of the ellipse, values ange fom 0 < e < 1, as e 0 the ellipse becomes moe cicula. e = a! b! a An impotant paamete is the obital position (distance between the centes of gavity). It is calculated using pola coodinates and the following equation: = a(1 e! ) 1 + e(cos θ)
Use the link, http://nssdc.gsfc.nasa.gov/planetay/factsheet/mecuyfact.html, and the infomation above to find the elliptical paametes fo Mecuy. Show all you wok fo the calculations. 1. Complete the data table below. Note the distances fo a, R p, and R a in the fact sheet ae fom Mecuy s cente of gavity to the Sun s cente of gavity. Planet a, metes R p, metes R a, metes eccenticity V max, m/s V min, m/s Mecuy 2. Calculate the semi- mino axis, b. 3. Calculate the obital position,, whee the mino- axis intesects the ellipse. 4. Sketch the elliptical obit to scale. Label all the paametes in questions #1 - #3.
Consevation of Enegy The total mechanical enegy, E mechanical, is conseved in both cicula and elliptical obits. In cicula obits the gavitational potential enegy, E g, and the kinetic enegy, E k, emain constant at all points of the obit. Howeve, in elliptical obits E g and E k ae diffeent as the planet moves along its obital path. Cicula Obit (clockwise) E mechanical is conseved because no extenal wok is being done on the system. F g is always pependicula to the diection of motion so no intenal wok is pefomed on the obiting body. The foce only changes the diection of motion. E g is constant because all points ae the same distance fom the cente of the massive body. E k must be constant because E g and E mechanical ae constant. Obital velocity is constant because E k is constant. Calculating Enegy Values Elliptical Obit (clockwise) E mechanical is conseved because no extenal wok is being done on the system. F g is pependicula to the diection of motion at A and C. No intenal wok is pefomed on the obiting body. The foce only changes the diection of the motion. A component of F g is opposite the diection of motion at B. This intenal wok educes obital velocity. A component of F g is in the same diection of the motion at D and E. This intenal wok inceases obital velocity. E g is at its minimum at A, and at its maximum at C. In deceasing ode, C > D > B > E > A. E k is at its maximum at A, and at its minimum at C. In deceasing ode, A > E > B > D > C. Obital velocity is at its maximum at A, and is at its minimum at C. In deceasing ode, A > E > B > D > C. The equation, E mechanical = E g + E k, applies to both cicula and elliptical obits. The equation gavitational potential enegy is the same fo both types of obits. Howeve, the velocity equations ae diffeent. The velocity equation fo a cicula obit is the same as the one we calculated in Unit 7 Univesal Gavitation. The velocity equation fo elliptical obits is moe complex because F g is not pependicula to the diection of motion. Note on Potential Enegy - When woking in oute space, we set E g = 0 at an infinite distance fom the massive body s cente of gavity. As an object moves close to the massive body we show the deceasing E g as an inceasing negative numbe. Obit Velocity Kinetic Enegy Gavitational Potential Enegy Total Mechanical Enegy Cicula v = GM E! =!! mv! =!! m GM E! = GMm E!"#! = GMm 2 Elliptical v = GM(!!!! ) E! =!! mv! =!! mgm(!!!! ) E! = GMm E!"#! = GMm 2a
5. Calculate the total mechanical enegy of Mecuy. 6. Calculate Mecuy s gavitational potential enegy, kinetic enegy and velocity at the peigee. 7. Calculate Mecuy s gavitational potential enegy, kinetic enegy and velocity at the apogee. 8. Calculate Mecuy s gavitational potential enegy, kinetic enegy and velocity whee the semi- mino axis intesects the ellipse. 9. How do you velocity values fo #8 and #9 compae to the minimum and maximum velocity values fom the NASA fact sheet?
Escape Velocity If you thow a ball up into the ai, its velocity will decease as the enegy is tansfeed fom the kinetic stoage mechanism to the gavitational potential mechanism. Once the kinetic enegy is depleted the ball has eached its maximum height and will etun to the Eath. The hade you thow the ball the geate its initial velocity. The geate its initial velocity the geate its kinetic enegy. The geate its kinetic enegy the fathe it will tavel vetically befoe it uns out of kinetic enegy and etuns to the Eath. You can eplay this scenaio ove and ove thowing the ball hade and hade each time. Eventually, the initial velocity and kinetic enegy ae lage enough that the ball uns out of kinetic enegy at an infinite distance fom the Eath. At this point the ball has escaped the Eath s gavity. It will not fall back to Eath. It is fee! The initial velocity in this last scenaio is known as the escape velocity. Note: If you ae concened about ai esistance you can conduct this thought expeiment on the moon o any planet without an atmosphee. We can calculate the escape velocity by setting the initial total mechanical enegy equal to the final total mechanical enegy. Since the final conditions ae the ball at est an infinite distance fom the planet, Final E!"#!!"#$!% = E!" + E!" =!! mv!! + GMm This means the initial mechanical enegy must also be zeo, Initial E!"#!!"#$!% = E!" + E!" =!! mv!! + GMm =!! m(0)! + GMm = 0 + 0 = 0 In the above equations, is the distance between the planet s cente of gavity and the object. On the suface of the planet this is the adius of the planet. If the object is in obit, then the distance,, is the planet s adius plus its obital distance. Remembe, M = mass of lage body and m = mass of smalle body tying to escape. 10. Reaange the initial mechanical enegy equation above to solve fo the escape velocity (v esc = v i ). = 0 11. Calculate the escape velocity fom the suface of Mecuy.
12. Calculate the escape velocity fom the apogee of Mecuy s obit. How does this compae to its obital velocity at the apogee? 13. Calculate the escape velocity fom the peigee of Mecuy s obit. How does this compae to its obital velocity at the peigee? Black Holes: A Special Case of Escape Velocity In the eighteenth centuy, scientists John Mitchell and Piee- Simon Leplace hypothesized that if a sta had enough mass its gavity could be so stong that not even light could escape its suface. In 1915, Einstein pedicted the existence of black holes in his Theoy of Geneal Relativity. While we have yet to find diect evidence of a black hole, thee is enough indiect evidence that scientists can infe the existence of black holes. The definition of a black hole is an object that is so massive and compact that its gavitational field is so stong that nothing, including light, can escape fom it. Since a photon of light is the fastest moving paticle in the univese and it is massless, gavity will affect it less than any othe object in the univese. Theefoe, if light can t escape a black hole, nothing can escape a black hole. Cuent theoy is that all of a black hole s mass is contained in a singulaity. Fo a non- otating, unchaged black hole (Schwazschild black hole) this singulaity is a point with no dimension. Fo a otating, unchaged black hole (Ke black hole) this singulaity is a ing with no dimension. In eithe case, the density of a black hole is infinite. All black holes have an event hoizon. The event hoizon is the point of no etun. Objects can coss the event hoizon heading towad the singulaity. Howeve, nothing, not even light, can pass back though the event hoizon. If a celestial body s adius,, is small enough and its mass, M, is lage enough the escape velocity will be equal to the speed of light. This adius is known as the event hoizon. We can use the escape velocity equation to detemine the elationship between the mass of a Schwazschild black hole and the adius of its event hoizon. 14. Calculate the adius of the event hoizon fo a Schwazschild black hole that has the mass of Mecuy, M. [Hint: Reaange the equation to solve fo the adius,, and let v esc = 3.00 x 10 8 m/s (speed of light).]
15. How does this adius compae to the adii of Mecuy and a golf ball? 16. Calculate the mass of a Schwazschild black hole that has an event hoizon that is the adius of Mecuy,. [Hint: Reaange the equation to solve fo the mass, M, and let v esc = 3.00 x 10 8 m/s (speed of light).] 17. How does this mass compae to the masses of Mecuy and the Sun?