In 2004 astronomers reported the discovery of a large Jupiter-sized planet orbiting very close to the star HD 179949 (hence the term 'hot Jupiter'). The orbit was just 1 9 the distance of Mercury from our sun, and it takes the planet only 3.09 days to make one orbit (assumed to be circular). (b) How fast (in km/s) is this planet moving?

Verified step by step guidance

1. Step 1: Calculate the circumference of the planet's orbit using the formula for the circumference of a circle, C = 2\pi r, where r is the radius of the orbit.
2. Step 2: Convert the orbital radius from astronomical units (AU) to kilometers. Note that 1 AU is approximately 149.6 million kilometers.
3. Step 3: Determine the orbital period in seconds, knowing that there are 86,400 seconds in a day.
4. Step 4: Calculate the orbital speed using the formula v = \frac{C}{T}, where C is the circumference of the orbit and T is the orbital period in seconds.
5. Step 5: Express the orbital speed in kilometers per second (km/s) to find how fast the planet is moving.

Key Concepts

Orbital Speed

Orbital speed refers to the velocity at which an object travels along its orbit around a celestial body. For a circular orbit, this speed can be calculated using the formula v = 2πr/T, where r is the radius of the orbit and T is the orbital period. Understanding this concept is crucial for determining how fast the 'hot Jupiter' is moving around its star.

Kepler's Laws of Planetary Motion

Kepler's Laws describe the motion of planets around the sun, with the first law stating that planets move in elliptical orbits with the sun at one focus. The second law indicates that a line segment joining a planet and the sun sweeps out equal areas during equal intervals of time, which implies that planets move faster when closer to the sun. These laws help in understanding the dynamics of the 'hot Jupiter' and its rapid orbit.

Centripetal Acceleration

Centripetal acceleration is the acceleration directed towards the center of a circular path, necessary for an object to maintain its circular motion. It is calculated using the formula a_c = v²/r, where v is the orbital speed and r is the radius of the orbit. This concept is important for analyzing the forces acting on the 'hot Jupiter' as it orbits its star at high speeds.