Question

The best estimate of the orbital period of the Solar System around the center of the Milky Way is between 220 and 250 million years. How much mass (in terms of solar masses) is enclosed by the 26,000 light-years (1.7 \(\cdot 10^{9} \mathrm{AU}\) ) radius of the Solar System's orbit? (An orbital period of \(1 \mathrm{yr}\) for an orbit of radius 1 AU corresponds to 1 solar mass.) a) 90 billion solar masses b) 7.2 billion solar masses c) 52 million solar masses d) 3.7 million solar masses e) 432,000 solar masses

Short answer

Answer: b) 7.2 billion solar masses

Step-by-step solution

Write down given information

The orbital period of the Solar System is between 220 and 250 million years, and its orbit has a radius of 26,000 light-years or 1.7 x 10^9 AU. We will assume a circular orbit for simplicity.

Calculate the average orbital period

To get the average orbital period, add the lower and upper estimates and divide by 2: \(\text{Average orbital period} = \frac{220 + 250}{2} \text{million years} = 235 \text{million years}\)

Express the orbital period in years

The average orbital period is given in million years. To convert it to years, we multiply by a million: \(\text{Orbital period} = 235 \times 10^6 \ \mathrm{years}\)

Use Kepler's Third Law to determine the mass enclosed by the orbit

According to Kepler's Third Law, in a simplified form for our particular case, we have the relation: \(\frac{(235 \times 10^6)^2}{(1.7 \times 10^9)^3} = \frac{M}{1 \ \mathrm{solar \ mass}}\) Rearrange to solve for the enclosed mass \(M\): \(M = \frac{(235 \times 10^6)^2}{(1.7 \times 10^9)^3}\)

Calculate the enclosed mass in solar masses

Plug the numbers into the expression to get the result: \(M = \frac{(235 \times 10^6)^2}{(1.7 \times 10^9)^3} = \frac{(235 \times 10^6)^2}{(1.7 \times 10^9)^3} = 7.2 \times 10^9\) solar masses

Match the result with the given options

The calculated enclosed mass is approximately 7.2 billion solar masses. Comparing with the options given, we find that the closest match is: b) 7.2 billion solar masses

Key concepts

Kepler's Third Law

Understanding Kepler's Third Law is essential for mastering the basics of orbital mechanics. This law, also known as the Law of Periods, provides a relationship between the time it takes for an object to orbit another (orbital period) and the size of its orbit (semi-major axis).

The Third Law can be expressed simply as: the square of the orbital period is directly proportional to the cube of the semi-major axis of the orbit. In a formula,\[\begin{equation}T^2 \text{ is proportional to } a^3\text{or}\frac{T^2}{a^3} = \text{constant}.\end{equation}\]where \(T\) is the orbital period and \(a\) is the semi-major axis.

In the context of our solar system, if we know the orbital period of the Solar System around the Milky Way, Kepler's Third Law allows us to estimate the mass inside the orbit by assuming the orbit is circular. The 'constant' in this law, when applied to objects orbiting our Sun, is often related to the Sun's mass, which is why Kepler's Third Law is particularly handy for estimating solar masses in exercises such as the given textbook problem.

Solar Masses

The term 'solar mass' (often denoted by \(M_\text{☉}\)) is a standard unit of mass in astronomy used to describe the masses of other stars and galaxies. It is equivalent to the mass of our Sun, approximately \(1.989 \times 10^{30} \text{ kilograms}\).

Solar masses are helpful for two reasons; they provide a common scale for comparing celestial objects' masses, and they play a crucial role in calculations involving gravitational forces and orbital mechanics. For instance, in our textbook problem, the solar mass serves as a baseline to express how much mass the Solar System's orbit encloses when we're using Kepler's Third Law.

By comparing orbital periods and distances in units where our Sun's mass and orbit size equals '1', we can make direct and meaningful comparisons to determine the influence of other massive objects, like the collective mass within the Milky Way that governs the orbit of our whole Solar System.

Orbital Mechanics

Orbital mechanics, or celestial mechanics, is the field of study that involves the motion of objects in space under the influence of gravitational forces. It relies on both Newtonian mechanics and Kepler's laws to predict the motion of celestial bodies, including planets, moons, and artificial satellites.

This domain of physics plays a crucial role when we're trying to understand phenomena like the orbits of planets around a star or the relaxation of galaxies over eons. In our textbook example, we applied orbital mechanics principles to determine the Galaxy's mass within our Solar System's orbit. We assumed a circular orbit for simplicity, which is often done in basic calculations, but real orbits can be more elliptical.

Learning to calculate orbital periods and the resulting forces can seem daunting. Still, as with our textbook solution, breaking down complex problems into smaller, understandable steps can make the subject far more approachable, especially when taking into account the tremendously vast scale of space.