NATS 1880 Project 3 Due: Monday, April 8, 2024 at 11:59 pm Please submit electronically via eClass Student name: Yusra Rostom Student number: 218647735 1 Project Description For nearly 20 years, astronomers have been monitoring the brightness of distant stars in an attempt to determine whether or not any of them host a planetary system. A measure of a star's brightness as a function of time is known as a light curve. Periodic fluctuations within a star's light curve are an indication that planetary transits may be occurring, where a planet is periodically passing between Earth and the distant star causing a temporary decrease in its brightness (although fluctuations can be caused by other astrophysical phenomena as well!). Over 3000 confirmed exoplanets have been detected via this Transit Method, mostly due to the Kepler Space Satellite and the Transit Exoplanet Survey Satellite. In Project 3, you will be provided with a simulated light curve that is comparable to those studied by astronomers today. Through an analysis of this light curve, you will determine whether or not your target star has any planets orbiting around it and what some of their properties are. You will also be asked to infer what other properties the planet(s) may have and evaluate its habitability. Both course materials and independent sources will have to be consulted in order to complete this project. Be sure to cite any sources used in your analysis. 2 Data The target star that was studied by your fictitious telescope has a mass, luminosity, radius, and temperature of 0.30 Mo, 0.008 Lo, 0.31 AU and 3157.18K. To access the observational data, go to link posted on eClass and find the .csv file named after your student number. A small portion of the data is illustrated in Figure 1. If you don't have experience with comma separated values (csv) files, note that they can be opened with Microsoft Excel. However you may use any software program that you wish (e.g. Google Sheets, TOPCAT). Each file contains measurements of the star's light curve, with the first two columns marking the time of a brightness measurement (in years) and the relative brightness. For this project, we will ignore measurement uncertainties that typically come with this type of 1 Relative Brigthness 1.000 0.999 0.998 0.997 0.996 0.00 0.02 0.04 0.06 0.10 0.12 0.14 0.08 Time (Years) Figure 1: Portion of your simulated light curve, which is a plot of relative brightness versus time (in years). 2 data. I highly recommend first trying to reproduce Figure 1 to make sure you are handling the data correctly, however a reproduction of Figure 1 is not necessary for your report. 3 Analysis Your report should begin with an Abstract (maximum 250 words) that summarizes your findings and an Introduction (maximum 500 words) that outlines our motivation for finding exoplanets and how the transit method works. The rest of your report should be followed by sections titled Results, Analysis, and Bibliography, which have no word limits. For your analysis, make a plot of the entire light curve and then create a zoomed-in graph of one transit for each confirmed exoplanet. Note that, just like real observations, there are some times when it was not possible to observe the target star. So be sure to choose a well resolved transit to plot as your zoomed-in image. Also be sure the axis are properly labelled and the axis range is appropriate to view the transit event such that the detailed shape of the transit can be seen. Your submitted report should contain a section with all the necessary figures in the Results section. From these figures, answer the following questions and include them in the Analysis section of your report. a) How many exoplanets are you able to confirm are orbiting your target star? How do you know? [2 marks, approx. 25-100 words] b) How many unconfirmed exoplanets do you have orbiting your target star? How do you know and why are they unconfirmed? [2 marks, approx. 25-100 words] c) For the confirmed exoplanets that you have discovered, what are their orbital periods and size (e.g. radius)? Put the data in a table and explicitly show your calculations below the table for one exoplanet. [4 marks] d) What properties of the confirmed exoplanets can you derive from the information given, other than their orbital periods and radii? Put the data in a table and explicitly show your calculations below the table for one exoplanet. [4 marks] e) What can you say about the habitability of this planet to life-like-us based on the evidence alone? [2 marks, approx. 25-100 words] f) Would your analysis of any exoplanets that you deemed habitable in the previous question change if, through transit spectroscopy, you learn the exoplanet has a 90.00atm atmosphere that is 96.00% carbon dioxide? Explain your reasoning [2 marks, approx. 25-100 words] 3 g) How confident are you in your conclusion about each planet's habitability? What informa- tion would you need to increase your confidence. Be specific! [2 marks, approx. 25-100 words] Finally, end your report with a Bibliography section if you used any additional sources. You should also have in-line citations throughout the report wherever the information was used. + 4 Additional Notes • Due to the uniqueness of each dataset, it is entirely possible that you have been provided a simulated light curve is difficult to analyse. So please note that all light curves are meant to have a minimum of 1 exoplanet (if not more). If you feel like your data contains an error and has no confirmed exoplanets, please contact nats1880@yorku.ca or stop by my virtual office hours. • You are all encouraged to discuss the project amongst yourselves, but the submitted work must be done by you and you alone. Directly collaborating on another student's analysis is an academic offense. • In addition to the posting questions on eClass and providing help after class or during office hours, there will be a dedicated Project 3 help session on March 21 instead of a lecture. 5

The University of Queensland, School of Mechanical and Mining Engineering MECH2210 Dynamics and Orbital Mechanics Section B1- Tutorial Problems 2022 Assignment B1, Submit: 0.3 & 17 due Mon wk 11 Online Dynamics Revision Quiz The following problems cover orbital mechanics material - please attempt all: Module/Question - 1/7, 2/12, 2/21, 3/5-12, 4/5, 6/3. (Answers online) F ᏛᎾ . B m Problem 1 Explorer No.1 launched in January 1958 had perigee and apogee heights above the Earth's surface of 360 km and 2550 km and an inclination of 33.2°. Calculate the: a) orbit period, b) its eccentricity, c) the total energy (per unit mass of the orbit) d) the angular momentum (per unit mass of the orbit), e) the maximum and minimum speeds and orbit positions f) the minimum impulse to change the inclination to 0° assuming the satellite passes the equator at its perigee, g) the minimum impulse Av required to escape Earth. (Answers: 1.92hr,0.14, -25.4 MJ/kg, 5.53x1010 m²/s, 8.2p & 6.2a km/s, 3.54 km/s, 2.67 km/s) 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) Problem 3 An early warning defence system detects an UFO travelling at 800km above earth's surface with a horizontal (perpendicular) speed of 8 km/s and a vertical (radial) speed of 1.3 km/s. Determine: a) the total energy (per unit mass of the orbit) b) the angular momentum (per unit mass of the orbit), c) the eccentricity of the orbit e d) the minimum and maximum radii of the orbit e) the true anomaly 9 at the point of observation. f) whether the UFO is possibly a missile, Earth satellite or comet and explain your answer. g) the minimum measured speed v of the UFO for it to be a comet and explain why. (Tot. 21 mks = 3 x 7) Problem 4

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)

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?

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.

MATINS BETERE 1) If an object is in Earth's orbit (a=1.000 A.U., e=0.0167) and suddenly increases its energy by 10 per cent without increasing its angular momentum [perhaps it fires a rocket engine towards the Sun, accelerating it directly outwards]: what is the final semi-major axis and eccentricity of the object's orbit? Note: E<0. Therefore increasing E by 10% means less negative or 90% of the original value. (Marks: 4) 2) The Earth was last at perihelion on January 3, 2022. The period of the Earth's orbit is 365.25 days. (a) If the Earth's orbit was a perfect circle when (give the actual calendar date in your answer!) would it reach a True Anomaly (angle with respect to perihelion) of 160 degrees? (b) But given that the Earth's orbit is slightly elliptical (e-0.0167), when (calendar date) will the Earth actually be at True Anomaly of 160 degrees? (c) How far will the Earth be from the Sun at that time? (d) How fast will the Earth be moving in its orbit on that day? (e) (for comparison to (c) and (d)) what is the distance (from the Sun) and speed of the Earth when it is at perihelion? (Marks 10) 3) How bright will Pluto appear in reflected or scattered sun-light (total Flux received in W/m²) at the Earth when Pluto is at Opposition and a distance from the Sun of 40.0 A.U.? Assume that Pluto has an albedo of 0.60 and a radius of 1.2 x 10³ km. (Marks: 6) DELL

1. (10 pts) Manually draw the layout and patterns of motion of the solar system. Please include the Sun and all planets. Draw the orbits of all planets around the Sun (2 pts) Label the orbit direction and spin direction of all planets. Label the spin direction of the Sun. (2 pts) Label the planets' average orbital distances from the Sun in AU and their orbital periods in Earth years (2 pts) Label the size of the Sun and each planet as compared to Earth's radius (e.g. 0.5 Earth radius, 2 Earth radii,...) (2 pts) Point out the exceptions to the patterns of motion (2 pts). (The sizes and distances do not need to be drawn to scale, which is impossible to do on letter sized paper anyway.) • •

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.

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.

33 A satellite is orbiting at a distance of 4.2 x 106 m from the surface of the Earth. The radius of the Earth is 6.4 x 106 m. What is the ratio of gravitational force on the satellite in orbit/gravitational force on the satellite on the surface of the Earth? A 0.36 B 0.42 C 0.51 D 0.64 Answer give D

Problem 1. (15 points) Ptolemy vs Copernicus (1) In this problem and the next we will compare how Ptolemy and Copernicus handled the inner (or inferior) and outer (or superior) planets in their respective models of the Solar System. To keep things simple, we will neglect the planets' eccentricities for the purposes of these two problems. (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, 0 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 r = 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?