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Chapter 13: Problem 65

Studies of the relationship of the Sun to its galaxy-the Milky Way- -have revealed that the Sun is located near the outer edge of the galactic disk, about 30000 lightyears from the center. The Sun has an orbital speed of approximately \(250 \mathrm{km} / \mathrm{s}\) around the galactic center. (a) What is the period of the Sun's galactic motion? (b) What is the order of magnitude of the mass of the Milky Way galaxy? Suppose that the galaxy is made mostly of stars of which the Sun is typical. What is the order of magnitude of the number of stars in the Milky Way?

Short Answer

Expert verified
The period of the Sun's galactic motion is on the order of \(10^8\) years. The mass of the Milky Way is on the order of \(10^{42}\) kg, which corresponds to an estimated number of stars on the order of \(10^{11}\).

Step by step solution

01

Calculate the Orbital Radius of the Sun

The Sun is located about 30000 lightyears from the center of the Milky Way. This distance needs to be converted into a more usable unit, such as kilometers. 1 lightyear \(\approx 9.461 \times 10^{12}\) kilometers. So, the orbital radius r of the Sun around the Milky Way in km is given by \(r = 30000 \times 9.461 \times 10^{12}\).
02

Find the Period of the Sun's Galactic Motion

The period T of the Sun's galactic motion can be found using the equation for the time it takes an object to travel around a circle (the period of rotation), which is \(T = \frac{2 \pi r}{v}\), where r is the orbital radius found in step 1, and v is the orbital speed of the Sun (250 km/s). Substituting the value from step 1 and the given speed into this formula will yield the period in seconds. This result should be converted to years for practicality.
03

Calculate the Mass of the Milky Way Galaxy

To determine the mass of the Milky Way, we can use Newton's version of Kepler's third law, which can be written as \(M = \frac{rv^2}{G}\), where r is the orbital radius, v is the orbital speed, and G is the gravitational constant (\(G \approx 6.674 \times 10^{-11} N(m/kg)^2\)). We can substitute the known values into the formula to find the mass of the Milky Way (M) in kilograms.
04

Estimate the Number of Stars in the Milky Way

To estimate the number of stars in the Milky Way, we would assume the galaxy is composed mostly of stars similar to the Sun. If we divide the total mass of the Milky Way by the mass of the Sun (\(1.989 \times 10^{30}\) kg), we can get an estimated number of stars in our galaxy.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Orbital Period
The concept of an 'orbital period' refers to the time it takes for an object, like a planet or a star, to complete one full orbit around another object. This can be the time it takes for a planet to go around its star, or in our case, the time our Sun takes to complete one full orbit around the center of the Milky Way Galaxy.

To determine this period for the Sun's galactic orbit, we consider the Sun's speed and the distance it travels on its path, which approximates a circle. Using the equation provided in the solution, we see that an object's orbital period increases as the radius of its orbit grows. Thus, the vast distance of 30,000 lightyears leads to a long galactic year 鈥� it's a stark contrast to our Earth year of 365 days!
Mass of the Milky Way
The Milky Way's mass is a measure of the total amount of matter it contains, which includes the stars, planets, dust, gases, dark matter, and more. Determining this figure helps us understand the gravitational forces at play, which govern the orbits of galactic objects and can even affect light's trajectory through space.

The method presented uses the principles articulated by Kepler and refined by Newton. By relating the orbital velocity and radius of an object like the Sun to the gravitational constant, we can infer the mass of the central body, in this case, the Milky Way. It's fascinating that through observing the Sun's motion, we can deduce the scale of our entire galaxy, a mass so immense that it's expressed in terms of solar masses or kilograms.
Number of Stars in the Milky Way
Estimating the number of stars in our galaxy is akin to counting the grains of sand on a beach from afar; it's a daunting task due to the vastness and complexity of the Milky Way. However, with some intelligent assumptions and calculations, it becomes tractable. By dividing the Milky Way鈥檚 mass by the Sun's mass - our average star - we obtain a rough count.

This figure gives us a sense of the scale of our galactic home, filled with possibly hundreds of billions of stars. Additionally, understanding the number of stars helps researchers infer the number of potential solar systems and, by extension, the potential for life-sustaining planets like Earth. Students should appreciate that these numbers, while approximate, provide a window into the sheer scale and wonder of our galaxy.

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Most popular questions from this chapter

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