(II) Our Sun rotates about the center of our Galaxy (mG ≈4 ×10^41 kg) at a distance of about 3 ×

step 1 6 .41. So we want to find how long it takes for the sun to orbit around the center of our galaxy

(II) Our Sun rotates about the center of our Galaxy (mG ≈4...

(II) Our Sun rotates about the center of our Galaxy $\left(m_{\mathrm{G}} \approx 4 \times 10^{41} \mathrm{kg}\right)$ at a distance of about $3 \times 10^{4}$ light-years $\left[11 \mathrm{y}=\left(3.00 \times 10^{8} \mathrm{m} / \mathrm{s}\right) \cdot\left(3.16 \times 10^{7} \mathrm{yr}\right) \cdot(1.00 \mathrm{yr})\right] .$ What is the period of the Sun's orbital motion about the center of the Galaxy?

(II) Our Sun rotates about the center of our Galaxy
$\left(m_{\mathrm{G}} \approx 4 \times 10^{41} \mathrm{kg}\right)$ at a distance of about $3 \times 10^{4}$ light-years $\left[11 \mathrm{y}=\left(3.00 \times 10^{8} \mathrm{m} / \mathrm{s}\right) \cdot\left(3.16 \times 10^{7} \mathrm{yr}\right) \cdot(1.00 \mathrm{yr})\right] .$ What is the
period of the Sun's orbital motion about the center of the Galaxy?

Key Concepts

Kepler's Third Law

Kepler's Third Law establishes a relationship between the orbital period and the radius of the orbit around a central mass, often expressed in its generalized form as T^2 ? r^3. When combined with Newton’s gravitational theory, it allows for the calculation of the orbital period of any object in a circular orbit by linking the period to the distance from the center of mass and the mass causing the gravitational pull.

Astronomical Unit Conversion

Astronomical calculations frequently require converting large-scale distance units, such as light-years, into more standard units like meters. Accurate unit conversion is crucial for applying physical laws—such as those for orbital dynamics—and ensuring consistency across calculations. This step is particularly important when assessing movements on the scale of galaxies or other astronomical systems.

Circular Orbital Motion

This concept refers to the motion of an object in a circular path under the influence of a central force. In the context of orbital dynamics, the gravitational force acting between two bodies provides the necessary centripetal force to maintain the orbit. By equating the gravitational force with the centripetal force, one can derive expressions for the orbital speed and period of the object.

Newton's Law of Universal Gravitation

Newton's Law of Universal Gravitation states that every pair of masses in the universe attracts each other with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. This principle is fundamental in calculating the gravitational force that acts as the centripetal force needed for an orbit, and it is essential when determining the orbital period of a body under gravitational influence.

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