The sun moves at a speed vS=220 km / s in a circular orbit of radius rS=30,000 light years arou

step 1 Question number 18 wants us to determine the circumference of the Sun's orbit around the galacti

The sun moves at a speed vS=220 km / s in a circular orbit...

The sun moves at a speed $v_S=220 \mathrm{~km} / \mathrm{s}$ in a circular orbit of radius $r_S=30,000$ light years around the center of the Milky Way galaxy. The earth requires $T_E=1$ year to orbit the sun, at a radius of $1.50 \times 10^{11} \mathrm{~m}$. (a) Using this information, find a formula for the total mass responsible for keeping the sun in its orbit, as a multiple of the sun's mass $M_0$, in terms also of the parameters $v_S, r_S, T_E$, and $r_E$. Note that $G$ is not needed here! (b) Find this mass numerically.

Key Concepts

Orbital Mass Determination

This concept involves determining the total mass causing an orbit by combining the measurements of orbital speed and radius. For an object in circular motion, the mass within the orbit can be expressed as a function of the orbital speed and radius. When paired with an independent reference (like Earth's orbital characteristics around the Sun), one can scale this derived mass in terms of a known mass, thereby quantifying the mass of systems such as galaxies without an explicit dependence on the gravitational constant.

Circular Orbital Motion

This concept describes the motion of an object moving along a circular path where the required inward (centripetal) acceleration is provided by a force—in many astrophysical contexts, the gravitational force. In orbital mechanics, equating the centripetal acceleration (v²/r) to the gravitational acceleration allows one to derive expressions that relate orbital speeds, radii, and the mass of the central object or system.

Gravitational Parameter Elimination

By using the known parameters from one orbit—such as Earth's orbit around the Sun—to express the product of the gravitational constant and the mass (GM) of the central body, one can eliminate the explicit appearance of the gravitational constant in further calculations. This technique leverages well-known orbital relationships in systems where direct measurements of G are not necessary, instead using comparative orbital data to infer masses.

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