Question

The Parker Solar Probe, launched in 2018, was the first spacecraft to explore the solar corona, the hot gases and flares that extend outward from the solar surface. The probe is in a highly elliptical orbit that, using the gravity of Venus, will be nudged ever closer to the sun until, in 2025, it reaches a closest approach of 6.9 million kilometers from the center of the sun. Its maximum speed as it whips through the corona will be 192 km/s. The probe's highly elliptical orbit carries it out to a maximum distance of 160 Rs with a period of 88 days. What is its slowest speed, in km/s?

Accepted Answer

Step 1: Recognize that the problem involves an elliptical orbit, and the relationship between the speeds at the closest and farthest points of the orbit can be determined using the conservation of angular momentum. Angular momentum is conserved because no external torques act on the system. Step 2: Write the expression for angular momentum conservation: $$ m v_{min} r_{max} = m v_{max} r_{min} $$, where $$ m  is $$the mass of the probe, $$ v_{min}  is $$the slowest speed, $$ v_{max}  is $$the maximum speed, $$ r_{max}  is $$the farthest distance from the center of the Sun, and $$ r_{min}  is $$the closest distance from the center of the Sun. Step 3: Simplify the equation by canceling the mass $$ m $$, as it appears on both sides. This gives $$ v_{min} r_{max} = v_{max} r_{min} . $$Rearrange to solve for $$ v_{min} $$: $$ v_{min} = \frac{v_{max} r_{min}}{r_{max}} . $$Step 4: Substitute the given values into the equation. The maximum speed $$ v_{max}  is 192 km/s$$, the closest distance $$ r_{min}  is 6.9 $$million kilometers ($$ 6.9 	imes 10^6  km$$), and the farthest distance $$ r_{max}  is 160 $$times the Sun's radius $$ R_\odot $$, where $$ R_\odot = 6.96 	imes 10^5  km. $$Thus, $$ r_{max} = 160 	imes 6.96 	imes 10^5  km. $$Step 5: Perform the substitution: $$ v_{min} = \frac{192 	imes (6.9 	imes 10^6)}{160 	imes (6.96 	imes 10^5)} . $$Simplify the expression to find $$ v_{min} $$, the slowest speed of the probe in km/s.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Elliptical Orbits

Kepler's Laws of Planetary Motion

Conservation of Energy in Orbits