Planets Beyond the Solar System. On October 15, 2001, a...

Planets Beyond the Solar System. On October 15, 2001, a planet was discovered orbiting around the star $\mathrm{HD} 68988$ . Its orbital distance was measured to be 10.5 million kilometers from the center of the star, and its orbital period was estimated at 6.3 days. What is the mass of HD 68988$?$ Express your answer in kilograms and in terms of our sun's mass. (Consult Appendix F.)

Key Concepts

Newton's version of Kepler's Third Law

This law relates the orbital period of a planet to the semi-major axis of its orbit and the mass of the central object it orbits. By setting the gravitational force equal to the centripetal force, it provides the formula P² = (4?²/G(M+m))a³, which for cases where the planet's mass is negligible compared to the star simplifies to P² = 4?²a³/GM. This relationship is fundamental in astrophysics for determining the mass of stars and other central bodies from the observed orbital parameters of nearby orbiting objects.

Orbital Mechanics

Orbital mechanics is the study of the motions of objects in space under the influence of forces, primarily gravity. It encompasses the analysis of how bodies move in orbits, based on their velocities, distances, and the gravitational forces they experience. Understanding orbital mechanics is essential for determining the trajectories and periods of celestial bodies, which in turn allows scientists to infer properties such as the mass of the central object around which they orbit.

Unit Conversion in Physics

Accurate computation in astrophysics often requires converting measurements from one set of units to another to maintain consistency, particularly when using universal constants like the gravitational constant (G). For instance, distances may need to be converted from kilometers to meters and time from days to seconds before applying equations derived from Newtonian mechanics. This ensures that all physical quantities are expressed in compatible units, which is crucial for correct calculations.

Mass Comparison in Astronomy

Expressing stellar masses in terms of the Sun's mass provides a convenient scale for comparing the masses of different stars. Since the mass of the Sun is a well-known constant in astrophysics, it serves as a standard reference. This comparison not only aids in understanding the relative scale of celestial bodies but also simplifies the interpretation of results in the broader context of stellar evolution and dynamics.

Transcript

00:01 A planet is orbiting everyone's favorite star, hd 68988, and given the distance from the planet to the star and the orbital period of the planet, we're asked to find the mass of the star.

00:15 So, first off, i want to convert a lot of these quantities they give us.

00:19 So they give us the distance as 10 .5 million kilometers.

00:25 So 10 .5 times 10 to the 6 kilometers, but let's just make this meters right away, so plus three more, so times 10 to the ninth meters.

00:39 And we're given that the period is 6 .3 days, and we'll want that in seconds.

00:46 And so doing that, we should find that it's 5 .44 times 10 to the 5th seconds.

00:55 So now we just don't have to worry about those later on.

00:58 So we want to find the mass of the star.

01:00 So this is circular orbit, so one of our go -to equations for that is that the velocity.

01:05 Of the planet orbiting will be the squared of g times the mass of the star divided by the orbital distance.

01:14 So let's solve for the mass of the star here, and then we'll see what else we need.

01:18 So we square both sides of the equation, v squared equals gm over r, and then we get m by itself as v squared r over g.

01:34 So we know r, but we don't know v, but given that they've given us r and the period, we should be able to find it.

01:42 So we'll come over here, and we know that the period is just the circumference or the total distance of the orbit, divided by the velocity of orbits...

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