An extrasolar planet can be detected by observing the wobble it produces on the star around which it revolves. Suppose an extrasolar planet of mass m B revolves around its star of mass m A . If no external force acts on this simple two-object system, then its cm is stationary. Assume m A and m B are in circular orbits with radii r A and r B about the system’s CM. ( a ) Show that r A = m B m A r B . ( b ) Now consider a Sun-like star and a single planet with the same characteristics as Jupiter. That is, m B = 1.0 × 10 –3 m a and the planet has an orbital radius of 8.0 × 10 11 m. Determine the radius r A of the star’s orbit about the system’s cm. ( c ) When viewed from Earth, the distant system appears to wobble over a distance of 2 r A . If astronomers are able to detect angular displacements q of about 1 milliarcsec ( 1 arcsec = 1 3600 of a degree ) , from what distance d (in light-years) can the star’s wobble be detected (l ly = 9.46 × 10 15 m)? ( d ) The star nearest to our Sun is about 4 ly away. Assuming stars are uniformly distributed throughout our region of the Milky Way Galaxy, about how many stars can this technique be applied to in the search for extrasolar planetary systems?
An extrasolar planet can be detected by observing the wobble it produces on the star around which it revolves. Suppose an extrasolar planet of mass m B revolves around its star of mass m A . If no external force acts on this simple two-object system, then its cm is stationary. Assume m A and m B are in circular orbits with radii r A and r B about the system’s CM. ( a ) Show that r A = m B m A r B . ( b ) Now consider a Sun-like star and a single planet with the same characteristics as Jupiter. That is, m B = 1.0 × 10 –3 m a and the planet has an orbital radius of 8.0 × 10 11 m. Determine the radius r A of the star’s orbit about the system’s cm. ( c ) When viewed from Earth, the distant system appears to wobble over a distance of 2 r A . If astronomers are able to detect angular displacements q of about 1 milliarcsec ( 1 arcsec = 1 3600 of a degree ) , from what distance d (in light-years) can the star’s wobble be detected (l ly = 9.46 × 10 15 m)? ( d ) The star nearest to our Sun is about 4 ly away. Assuming stars are uniformly distributed throughout our region of the Milky Way Galaxy, about how many stars can this technique be applied to in the search for extrasolar planetary systems?
An extrasolar planet can be detected by observing the wobble it produces on the star around which it revolves. Suppose an extrasolar planet of mass mB revolves around its star of mass mA. If no external force acts on this simple two-object system, then its cm is stationary. Assume mA and mB are in circular orbits with radii rA and rB about the system’s CM. (a) Show that
r
A
=
m
B
m
A
r
B
.
(b) Now consider a Sun-like star and a single planet with the same characteristics as Jupiter. That is, mB= 1.0 × 10–3ma and the planet has an orbital radius of 8.0 × 1011 m. Determine the radius rA of the star’s orbit about the system’s cm. (c) When viewed from Earth, the distant system appears to wobble over a distance of 2rA. If astronomers are able to detect angular displacementsq of about 1 milliarcsec
(
1
arcsec
=
1
3600
of a degree
)
, from what distance d (in light-years) can the star’s wobble be detected (l ly = 9.46 × 1015m)? (d) The star nearest to our Sun is about 4 ly away. Assuming stars are uniformly distributed throughout our region of the Milky Way Galaxy, about how many stars can this technique be applied to in the search for extrasolar planetary systems?
Definition Definition Angle at which a point rotates around a specific axis or center in a given direction. Angular displacement is a vector quantity and has both magnitude and direction. The angle built by an object from its rest point to endpoint created by rotational motion is known as angular displacement. Angular displacement is denoted by θ, and the S.I. unit of angular displacement is radian or rad.
A binary-star system contains a visible star and a black hole moving around their center of mass in circular orbits with radii r1 and r2 , respectively. The visible star has an orbital speed of v=5.36x105 ms-1 and a mass of m1 =5Ms ,where Ms= 1.98x1030kg is the mass of our Sun. Moreover, the orbital period of the visible star is T = 30 hours.(a) What is the radius r1 of the orbit of the visible star?(b) Calculate the mass m2 of the black hole in terms of MS . [Hint: One root of the equation x3 = 20a(5a+x)2 , where a is a constant, is x = 28a .]
Stars and black holes in a binary system orbit each other in circular orbits of radius r1 and r2 around their center of mass. Its mass is equal to 1.98x1030 kg, and its speed is 5.36 times faster than our Sun's. Furthermore, the visible star has an orbital period of 30 hours.(a) What is the apparent star's orbital radius, r1, in units of radii?In terms of MS, determine the black hole's mass m2. In the equation x3 = x(5a+5a)2, where an is the constant, x = 28a is a root.
Often in designing orbits for satellites, people use what is termed a "gravitational slingshot effect." The idea is as follows: A satellite of mass ms and speed vsi circles around a planet of mass mp that is moving with speed vpi in the opposite direction. See the diagram below:
Although the satellite never touches the planet, this interaction can still be treated as a collision because of the gravitational interaction between the planet and satellite during the slingshot. Since gravity is a conservative force, the collision is elastic.Use an x-axis with positive pointing to the right.Solve for the unknowns below algebraically first, then use the following values for the parameters.
mp = 4.60E+24 kgms = 1440 kgvsix = 3.740E+3 m/svpix = -2.20E+3 m/s
Solve for the final velocity of the satellite after the collision.
Find the final velocity of the planet.
Chapter 9 Solutions
Physics for Scientists and Engineers with Modern Physics
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