homework5

ASTR 3880, Spring 2024 Homework #5 Due Date: 2024-Feb-29 #1 (10 pts) . Consider measuring the area of a triangle when you measure the base to be b = 5 . 0 ± 0 . 1 cm and the height to be h = 10 . 0 ± 0 . 3 cm. What is the area and uncertainty on your measurement of the area? Requirement : On your way to the answer, please derive an expression for σ A using the error propagation formula. #2 (10 pts) . You have observed a star in the night sky and collected 1003 photons in 10 sec- onds. Compute the rate at which you received photons (in photons second − 1 ) and its associated uncertainty. #3a (7 pts) . Kepler’s Third Law is give by, P 2 = 4 π 2 G ( m 1 + m 2 ) a 3 , (1) where all variables are measured in MKS units. Show that if instead we measure P in years, a in au, and m 1 and m 2 in solar mass ( M ⊙ ), that this equation can be written as, 0 . 9998 P ( yr ) 2 = 1 [ m 1 ( M ⊙ ) + m 2 ( M ⊙ )] a ( AU ) 3 , (2) such that the constant is so close to unity that we simply write it as, P 2 = 1 ( m 1 + m 2 ) a 3 . (3) Note: The exact value of the constant will be different depending on how precise you define various values. But it should be very close to unity. #3b (3 pts) . Show that in the case where m 1 is the mass is the Sun and the m 2 is the mass in any body in our solar system, Equation 3 reduces to, 1

P 2 = a 3 (4) #4a (10 pts); a well-labelled figure is required [5 pt]) . The Hubble Space Telescope is in a nearly circular orbit, approximately 610 km above the surface of the Earth. Estimate its orbital period (in minutes) using Newton’s version of Kepler’s Third Law. #4b (10 pts) . What is the orbital speed of the Hubble Space Telescope (in km s − 1 ) and what speed would it have to move at in order to escape Earth’s gravity? #5a (10 pts); a well-labelled figure is required [5 pt]) . Communication and weather satellites are often placed in so-called geostationary orbits that keep them over a fixed point on the Earth’s equator. At what altitude (in km) must these satellites be placed in order to be in geostationary orbit. #5b (5 pts) . How does this altitude compare (by compare I mean compute a multiplicative factor) to the Moon’s distance? (Although not required, a figure showing a scale model of the Earth, geostationary orbit, and the moon’s orbit would probably help you get a sense of it. Don’t worry about being too precise.) #6a (10 pts) . Compute the orbital speed (in km s − 1 ) for each planet in our solar system (assuming they move in circular orbits) and then plot the orbital speed as a function of its distance from the Sun as measured in au. Please include a printout of the figure in your solution and use Python to make the figure. Remember, figures are useless without labels! #6b (10 pts) . Add the Trappist-1 system in a different color from your previous homework to your plot. #7 (5 pts) . Halley’s comet’s orbit has a semimajor axis of 17.8 au and an eccentricity of 0.967. Compute the velocity (in km s − 1 ) of the comet at aphelion and perihelion using the vis viva equa- tion. 2