SCI238Assignment5Solutions - SCI 238 Online - Fall 2023

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Physics

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Dec 6, 2023

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© Michel Fich, 2023 SCIENCE 238 Assignment #5 (Requires course material up to and including Module 6e) 1) Abinary star system is observed. (a) The stars are 0.8 arcsec apart and orbit with a period of 12.0 years. If the stars are 10 parsecs from the Sun then what is the mass of these two stars (together)? Assume circular orbits. (b) If one of the stars is three times as bright as the other AND Stellar luminosity is related to stellar mass by L o< M3 then what is the mass of each of the two stars? (Marks: 4) (a) (2 marks) The distance between the stars is the semi-major axis given by angle x distance which gives (with correction from arcseconds to radians %810 x 3.086 x 1016 = 206265 and putting in parsecs in meters) a = 1.2x10?m =8.0A4.U. am? G(my+mg) plus constants: (12.0 X 365.25 X 24 X 60 X 60)? = 2 m(1.20 X 10'2)3 which then gives m; +m, = 7.13 x 10*°kg or 3.59 Mgy, [Note: the mass of the Sunis Mg, = 1.989 x 103%kg]. Rounding this is 3.6Mg,n e Alternative (partial) solution: (my + my)P2(yr) = a3(A.U.) = (my +my)122 =8.0°= m, + m, = 3.56 Mgy, = 3.6Mgyn Note on significant figures. Because the angle is only given to 1 digit accuracy (0.8) the final answer of 3.6 could be rounded to 4.0 here. But keep an extra digit or two in order to do the calculations in (b) e Kepler's Third Law is P2 = and putting in the values above (b) (2 marks) Since brightness depends on luminosity and distance, and the stars are the same distance from us, and we can write L (3)3: 3= (fl)s and Ly my ms. my = 1.44m,. From (a): 1.44m, + m, = 3.59Ms,,, and m, = 1.47Mg,,,,. This then gives my = 2.12 Mg,,,,. After rounding: my = 2.1Mg,,, and m, = 1.5Mg,, Note on significant figures: students may choose to round to one digit and that is fine as long as the unrounded number is shown too so that the two stars have different mass. (Marks: 10 out of 60 Assignment Marks)
© Michel Fich, 2023 2) Investigate the average density of Main-Sequence stars. (a) Make a log-log plot of Mass versus Radius for spheres with constant densities of 102 10° and 10° kg/m? and Masses between 0.05 Msu, and 200 Msu, . Note that this plot will consist of four (changed to three to simplify problem) lines, one for each density. (b) On the plot in (a) add the Mass and Radius of the stars on the Main Sequence stars shown on the table below (This is a standard data table taken from a graduate course textbook). (c) Describe and discuss any trends that you see in the figure produced above. (Marks: 6) (a) (2 marks) Mass is given by M = %nR3p where p is the density. To simplify the calculation of the lines put in units of kg/m3 for density and Solar Radii and Solar Masses: [note that Mgy = M/Mgyy] Mgorar X 1.989 x 1030 = %nR;,m,(e.957 % 108)3p and after calculations: or Mgoiar = 7.09 X 107*R3 0P [Note that for density of water, p = 103, this gives Mg, 1q, = 0.709R3, 4, ... a one Solar Radius sphere has a mass of 0.709 Solar Masses] The table below gives a set of possible endpoints for the lines to be drawn in the same units as above. (Note that mass limits are used here for plotting, but could use radius to set plotting line limits.) p R log (R) M log (M) 10?2 0.89 —0.051 0.05 -1.3 102 141 1.15 200 2.30 103 0.41 —0.38 0.05 -1.3 103 6.56 0.82 200 2.30 10* 0.19 -0.72 0.05 -1.3 10* 3.04 0.48 200 2.30 (Marks: 10 out of 60 Assignment Marks)
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