Astrophysics

Problem 2 An earth satellite is tracked from ground stations and is observed to have an altitude of 2200 km, a velocity magnitude 7 km/s and a radial velocity of 2.7 km/s. Determine: a) the total orbital energy per unit mass b) the orbital eccentricity e c) the minimum altitude and the maximum speed and d) the true anomalies at the observed position and at the point of maximum speed. e) Identify any potential difficulties with the orbit. (Answers: -22 MJ/kg, 0.389, -835km & 10 km/s, 105.3deg & 0 deg, crash into Earth)

1. You have a glass with mass m bouncing off the floor. Its duration of contact with the floor the time difference between first touching the floor and last touching the floor- is T. Its velocity when it first touches the floor is Viy, and its velocity as it launches off the floor is ufy. If, at any time during its contact with the floor, the force on the glass ever exceeds Fmax, even for the tiniest fraction of a second, the glass will break. Note that the force on the glass during its collision with the floor will not be constant, and you have no idea what the exact force graph looks like. (a) Write down an inequality describing the condition under which it is certain that the glass will break. Hint: Sketching an F vs. t graph might help you think about this. (b) Let's say the inequality you wrote down does not hold. Does this mean that the glass is certain to remain unbroken? Explain.

2. You have a horizontal, disc-shaped platform that rotates around its center. Its mass is M and radius is R. There is no friction at the axis of rotation. On the platform, at radius r = R/2, there is a kid with mass m = M/40. They both are initially at rest. (a) The kid steps forward in such a fashion that her r remains constant, propelled by a force F due to the friction between the platform and her shoes. As she moves, what is the ratio of the angular acceleration of the kid to the angular acceleration of the platform? (b) The kid keeps moving in a manner such that her r is constant. What is the ratio of the angular velocity of the kid to the angular velocity of the platform at any instant? (c) What is the ratio of the total angle covered by the kid since she started moving to the angle rotated through by the platform?

3. Astronomers observe a new comet approaching the sun. They obtain the location of the comet relative to the sun, and the velocity of the comet. But they don't have long-term data to directly tell whether the trajectory of the comet is an ellipse or a hyperbola. Still, astronomers can figure it out. After all, an elliptical orbit means that the comet is gravitationally bound to the sun: it can never escape to an infinite distance. But a hyperbolic trajectory extends to infinity: the comet is unbound and must escape the Sun's gravity. Chose, from the following options, the test that astronomers can apply, using their position and velocity observations, that distinguishes between a bound and an unbound comet. If an option is incorrect, briefly explain why. If it is correct, give the exact inequality they will apply, using the velocity and distance to the sun of the comet, and if needed, data about other objects in the solar system. State whether the inequality indicates a bound or unbound comet. (a) Lcomet > LEarth. (Compare angular momenta.) (b) Ecomet > 0. (The total orbital energy of the comet is positive.) (c) Icomet > ISun (Compare moments of inertia.) (d) Apcomet > Fcomet. (Compare the momentum change to the gravity on the comet.) (e) Tcomet > 0. (The total torque on the comet is positive.)

4. You do a collision experiment with carts, with expensive equipment that reduces friction with the track to a negligible level. Cart 1 has mass 1.00 kg with initial velocity v₁ = 3.00 m/s, and it heads toward cart 2 with mass 0.500 kg that starts at rest. (a) Obtain an equation that expresses v2f in terms of vif. (b) Obtain an equation for Eloss, the energy lost during the collision, where v₁f is the one remaining unknown. (c) Using an online graphing calculator or similar software (I suggest desmos.com; it's easy to learn), plot both v2f and Eloss as functions of v₁f (which will be the horizontal axis). Focus on the part of the graph where 0 <₁ < 3.5 m/s, and you can see the maximum of Eloss clearly. Save it as a PDF or screen capture; turn in that image. (d) Some of the v1₁ values on your image are impossible to observe in the lab. Why? Give the inequality that defines these impossible velocities. Hint: this is really simple and has nothing to do with Eloss. (e) At what v₁y and v2, values does that maximum Eloss occur? What would you see hap- pening in your experiment? (f) At some possible values of v₁f, you will see that Eloss < 0. How might this happen?

Using Galilean relativity: Mort is coming toward you at a speed of 95 km/hr. You throw a baseball in his direction at 75 km/hr. How fast does he see the ball moving? B.) Using Einstein's relativity: Mort is coming toward you at a speed of 95,000 km/s. You shine a light in his direction at 300,000 km/s. How fast does he see the light moving?

Verna is moving by you at 75% of the speed of light. You get out a clock and measure 10 seconds going by. If you also watch a clock in Al's ship, how much time will you see it record during your 10 seconds? (Hint: you can use the table shown from about min 32-34 in the lecture 4A)

Suppose a supersonic airplane flies at a speed of 1670 km/hr from Nairobi, Kenya to Quito, Ecuador; note that this is the same speed that Earth rotates, but in the opposite direction. Describe how this flight would look to an observer on the Moon.

Imagine we measure the length of a spaceship to be 30 meters when it is at rest relative to us. What will happen to the length if we somehow measure it as it moves past us? If the speed is 60% of the speed of light (180,000 km/s) what length would we measure? (Hint: you can use the graph shown from about minute 39-41 in the lecture 4A)

Suppose you are sitting in a closed room that is magically transported off Earth so that, as shown in the diagram, you are accelerating through space at 9.8 m/s2. According to the equivalence principle, how will you know that you've left Earth?

For the following line elements and vector V", write down the metric, the inverse metric, and

Consider a metric in a D-dimensional space. Evaluate the following expressions.

3.Consider the line element of the sphere of radius a:ds2 =a2(d02+sin20do2).The only non-vanishing Christoffel symbols arer=-sin0 cos0,=tana)Write down the metric and the inverse metric,and use the definitionTP (ngva+9uo-09m)=TPvuto reproduce the results written above for and[You can also check that the otherChristoffel symbols vanish,for practice,but this will not be marked.b)Write down the two components of the geodesic equation.b)The geodesics of the sphere are great circles.Thinking of 0 =0 as the North pole and 0=as the South pole,find a set a solutions to the geodesic equation corresponding to meridians,andalso the solution corresponding to the equator.

Consider Euclidean space in D = 2. Show by direct computation that the Riemann curvature tensor,

(a) As we discussed in class, in the geocentric model of the Solar System the epicycles of the three superior planets are not independent, but instead are arranged so that they rotate in phase with one another and with the Sun's motion about the Earth. Explain why this was necessary, in terms of the observations available to Ptolemy. (b) How did Copernicus explain these same observations within his heliocentric model? (c) Both Ptolemy and Copernicus knew well the synodic periods of the planets (i.e., the intervals between successive oppositions of each planet, when it appeared directly opposite the Sun in the sky.) Show that in Ptolemy's model the period of a superior planet's motion around its deferent, PD and the period of its motion around its epicycle, PE, both measured relative to a fixed direction, must be given by PD = P₂ and PE = Pe where Pp is the true orbital period of the planet around the Sun and Pe is the true orbital period of the Earth around the Sun. [HINT: You might want to think about the planet's motion relative to the distant stars, and about your answer to part (a) above. You can also look at this in terms of the planet's synodic period, using the equation given in the Class Notes.] (d) For an inferior planet, on the other hand, show that PD Pe and PE = Pp. [Here again, it may be useful to consider the planet's synodic period, which in this case is twice the period between conjunctions with the Sun.] So both the Ptolemaic and Copernican models involve the same underlying periods for the planet's motions, but interpret them very differently.

(a) The Greek astronomer Aristarchus, in the 3rd century BCE, is thought to have been the first person to have deduced the distance from the Earth to the Moon, using observations of lunar eclipses. Aristarchus first observed that the diameter of the Moon was approximately one-third of the diameter of the Earth's shadow, at the distance of the Moon. He also knew that the angular diameters of the Moon and Sun, as seen from the Earth, are almost exactly equal, based on observations of solar eclipses. With the aid of a sketch of the geometry of a lunar eclipse, show that these two observations imply that the physical diameter of the Moon is approximately one-quarter of the diameter of the Earth. (b) Now using the fact that the angular diameter of the Moon is close to 0.5°, or a little under 0.01 radians, estimate the distance of the Moon in units of the Earth's diameter ᎠᏯ . (c) About 19 centuries later, the Italian-French astronomer Giovanni Domenico Cassini attempted to estimate the distance to the Sun by combining observations of the planet Mars when it was closest to the Earth, against the background of distant stars, from two widely-separated places on Earth. (He used data from Paris and French Guyana in South America, having sent out an expedition for this purpose.) From these data he calculated that the difference in the direction to Mars would have been approximately 40 arcseconds, if it had been possible to observe it from the Earth's North and South poles simultaneously. [Recall that there are 3600" in 1°, or 206, 265" in one radian.] Assuming that the distance to Mars at the time was 0.5 AU, use this result to estimate the distance to the Sun (i.e., 1 AU) in terms of the Earth's diameter De. In fact, both of these estimates were quite accurate, for their times, and each became the standard value used by later astronomers for a century or more. Nowadays, we know the distance to the Moon accurate to a few centimeters, thanks to laser retro-reflectors left on its surface by the Apollo missions. The AU was not measured accurately until the advent of radar astronomy and interplanetary spacecraft in the 1960s, though many more attempts were made in the interim, mostly using transits of Venus to gauge the diameter of the Sun.

(a) As we discussed in class, in Ptolemy's geocentric model of the Solar System the centers of the epicycles for the inferior planets Mercury and Venus are tied to the motion of the Sun, in order to keep these two planets from 'wandering' too far away from the Sun in the sky. The maximum elongations (i.e., angular distance from the Sun) observed for Mercury and Venus are max = 22.8° and 46.3°, respectively. Demonstrate that, in the Ptolemaic model, max can be used to estimate the ratio of the radius of the epicycle, E for each planet to that of its deferent, D, but not their absolute values nor their ratios to the Earth-Sun distance. (b) In the Copernican model, on the other hand, show that the orbital radius of an inferior planet is given by T = R sin max, where R is the Earth's orbital radius, or 1 Astronomical Unit (AU). Use the values of max given above to evaluate r for Mercury and Venus. Give your answers to 3 decimal places. (c) With the help of a diagram, calculate the minimum and maximum distances from the Earth for an inferior planet in the Ptolemaic model in terms of D and the parameter max. (d) Do the same for the Copernican model, giving your results in terms of R and max. Show that the ratio of minimum to maximum distance is the same for both models. (Do this analytically, rather than numerically.) In this sense, the two models are equivalent, but Copernicus considered the inability to relate the orbital radii of Mercury & Venus to that of the Earth to be a major failing of the old model. Do you agree?

2. Describe by means of a clearly drawn Minkowski diagram, the pole (or ladder) -barn paradox. If you wish you can use the following values for the set-up. Length of the barn: 10 m, length of the pole: 12 m, relative speed of the pole with respect to the barn: 0.6c.

5. Current observations tell us that the Hubble constant Ho= 72 km s¹ Mpc-¹. When Hubble first attempted to measure this value, he arrived at a much larger value,H₂= 500 km s¹ Mpc-¹. Suppose for a moment that our universe had such a large Hubble constant. a. What would the distance be (in Mpc) of a galaxy with a red shift of z= 0.023?Compare this with the "correct" value. b. What would be the Hubble time? Compare this with the age of some constituents of the universe based on other considerations and comment. c. Discuss the hot topic of "Hubble tension" and what kind of explanations have been offered thus far to address this.

4. From the relativistic energy-momentum equation E² = p²c² + m^2oc^4, find the low-velocity Newtonian approximation. Comment, please, on its significance.

3. The energy density of photons in the frequency range (v, v+ dv), is given by the blackbody or Planck function: \varepsilon(\nu) d \nu=\frac{8 \pi h}{c^{3}} \frac{v^{3} d \nu}{\exp (h v / k T)-1} a.Derive that the peak of this function occurs at an energy hv=2.82 kT. What isthis relation or law usually called in the literature? b. Derive that, integrated over all frequencies, the energy density is equal to: \begin{array}{l} \varepsilon_{\gamma}=\alpha T^{4} \quad \text { where } \\ \alpha=\frac{\pi^{2}}{15} \frac{k^{4}}{\hbar^{3} c^{3}}=7.56 \times 10^{-16} \mathrm{~J} \mathrm{~m}^{-3} \mathrm{~K}^{-4} \end{array} What is this relation generally known as? c. Hence, calculate the total energy density of the Cosmic MicrowaveBackground (assume T = 2.725 K) and its photon density.