Question

Our Sun is \(2.3 \times 10^{4}\) ly (light-years) from the center of our Milky Way galaxy and is moving in a circle around this center at a speed of \(250 \mathrm{~km} / \mathrm{s}\). (a) How long does it take the Sun to make one revolution about the galactic center? \((b)\) How many revolutions has the Sun completed since it was formed about \(4.5 \times 10^{9}\) years ago?

Short answer

The Sun takes approximately 561 million years to complete one revolution around the Milky Way. Since its formation 4.5 billion years ago, it has made approximately 8 full revolutions.

Step-by-step solution

(a) Calculate Time for One Revolution

First, calculate the time it takes for the Sun to make one complete revolution around the Milky Way. The Milky Way's radius is \(2.3 \times 10^{4}\) light-years so its circumference, \(c\), can be calculated using the formula \(C = 2\pi r\), where \(r\) is the radius. So, \(C = 2 \pi \times 2.3 \times 10^{4}\) light-years. Then calculate how many kilometers there are in a light-year to convert the distance to kilometers. Given that 1 light-year ≈ \(9.461 \times 10^{12}\) km, the circumference \(C\) in kilometers is \(C = 2 \pi \times 2.3 \times 10^{4} \times 9.461 \times 10^{12}\) km. Then use the formula for time, \(t = \frac{d}{v}\), where \(d\) is distance and \(v\) is velocity, to convert this into years: \(t = \frac{2 \pi \times 2.3 \times 10^{4} \times 9.461 \times 10^{12}}{250} = 5.612 \times 10^{8}\) km/yr, which is 561 million years.

(b) Calculate Total Sun Revolutions

Next, calculate how many revolutions the Sun has completed since it was formed about 4.5 billion years ago. Use the formula for number of revolutions, \(n = \frac{T}{t}\), where \(T\) is the total time and \(t\) is the time for one revolution. So, \(n = \frac{4.5 \times 10^{9}}{5.612 \times 10^{8}} \approx 8\). Thus, the Sun has made approximately 8 revolutions around the Milky Way since it formed.

Key concepts

Astronomical Calculations

Astronomical calculations are essential in understanding the vast cosmos we live in. These mathematical procedures allow us to calculate distances, speeds, and periods of celestial objects. In the context of our solar system and its journey through the Milky Way, we need several key pieces of information to determine the path and travel time of our Sun.

For instance, knowing the speed of the Sun as it orbits the galactic center is a starting point. With a speed of 250 kilometers per second, we can begin to unravel the journey of our Sun. Astronomical calculations help us translate these raw figures into more conceptual understandings, such as the period of the Sun's galactic orbit. To make such a calculation, we also need the concept of a light-year, which is the distance light travels in one year. It's a vital conversion tool for handling the massive scales in astronomy.

Light-year Conversion

Understanding the light-year conversion is crucial when dealing with astronomical distances. A light-year is approximately 9.461 trillion kilometers (or about 5.879 trillion miles). It's a measurement that allows astronomers to convey the vast distances in the universe in a more comprehensible way.

To illustrate, calculating the circumference of our galaxy for the Sun's orbit requires conversion from light-years to kilometers. Such conversion puts the incomprehensible distances into practical numbers we can work with. For example, the Sun's distance to the center of the Milky Way is 23,000 light-years. By converting light-years to kilometers, we can then calculate the solar orbital path and further derive the time it takes for the Sun to complete one circuit around the galactic center.

Galactic Revolution Period

The galactic revolution period is the time it takes for an object, like our Sun, to orbit once around the center of the galaxy. This period depends on both the speed of the object and the distance of its orbit. In astronomical calculations, once we know the orbit's circumference (after a light-year conversion) and the object's speed, we can calculate the revolution period using the simple formula for time, which is distance divided by speed.

In the case of our Sun, its revolution period around the Milky Way's center is approximately 561 million years. This astounding figure allows astrophysicists to estimate the number of orbits the Sun has completed since its formation. These calculations are critical for developing models of galactic evolution and understanding the lifecycle of stars within the grand architecture of the cosmos.