Solve the problem.
-Given that the polar equation $\mathrm { r } = \frac { \mathrm { a } \left( 1 - \mathrm { e } ^ { 2 } \right) } { 1 + \mathrm { e } \cos \theta }$ models the orbits of the planets, compare Earth's orbit relative to the sun with that of Pluto and describe the difference. For Earth, $\mathrm { a } = 1.00$ and $\mathrm { e } = 0.017$. For Pluto, $\mathrm { a } = 39.4$ and $\mathrm { e } = 0.249$.

A) Earth's orbit closely approximates a circle with the sun positioned at its center. Pluto's orbit is more elliptical with the sun positioned on its long axis about 10 astronomical units away from the center.
B) Pluto's orbit closely approximates a circle with the sun positioned at its center. Earth's orbit is more elliptical with the sun positioned on its long axis about 10 astronomical units away from the center.
C) Earth circles the sun, but Pluto does not.
D) The orbits of Earth and Pluto are both circular, with the sun positioned at their centers, but the diameter of Pluto's orbit is about 40 times that of Earth.

Question

Solve the problem.
-Given that the polar equation $\mathrm { r } = \frac { \mathrm { a } \left( 1 - \mathrm { e } ^ { 2 } \right) } { 1 + \mathrm { e } \cos \theta }$ models the orbits of the planets, compare Earth's orbit relative to the sun with that of Pluto and describe the difference. For Earth, $\mathrm { a } = 1.00$ and $\mathrm { e } = 0.017$. For Pluto, $\mathrm { a } = 39.4$ and $\mathrm { e } = 0.249$.

A) Earth's orbit closely approximates a circle with the sun positioned at its center. Pluto's orbit is more elliptical with the sun positioned on its long axis about 10 astronomical units away from the center.
B) Pluto's orbit closely approximates a circle with the sun positioned at its center. Earth's orbit is more elliptical with the sun positioned on its long axis about 10 astronomical units away from the center.
C) Earth circles the sun, but Pluto does not.
D) The orbits of Earth and Pluto are both circular, with the sun positioned at their centers, but the diameter of Pluto's orbit is about 40 times that of Earth.

Quizplus · Accepted Answer

The eccentricity ($e$) of an orbit determines its shape. Earth's orbit, with $e = 0.017$, is nearly circular, while Pluto's orbit, with $e = 0.249$, is more elliptical. The parameter $a$ represents the semi-major axis, indicating the size of the orbit but not affecting its shape as directly as $e$ does.

Solve the Problem r=a(1−e2)1+ecos⁡θ\mathrm { r } = \frac { \mathrm { a } \left( 1 - \mathrm { e } ^ { 2 } \right) } { 1 + \mathrm { e } \cos \theta }r=1+ecosθa(1−e2)​

Solve the Problem $\mathrm { r } = \frac { \mathrm { a } \left( 1 - \mathrm { e } ^ { 2 } \right) } { 1 + \mathrm { e } \cos \theta }$