Question

Mass of the Milky Way's Halo. The Large Magellanic Cloud orbits the Milky Way at a distance of roughly 160,000 light-years from the galactic center and a velocity of about \(300 \mathrm{km} / \mathrm{s}\). Use these values in the orbital velocity formula to estimate the Milky Way's mass within 160,000 light-years from the center.

Short answer

The Milky Way's mass within 160,000 light-years is approximately \(10^{12}\) solar masses.

Step-by-step solution

Identify the Orbital Velocity Formula

The orbital velocity formula relates the orbital velocity (\(v\)), the mass of the central object (\(M\)), the gravitational constant (\(G\)), and the radius of the orbit (\(r\)). The formula is \(v = \sqrt{\frac{G \cdot M}{r}}\). We will use this formula to find the mass of the Milky Way's halo within 160,000 light-years.

Convert Units and Assign Values

We need to convert the distance from light-years to meters and use the appropriate values for calculations. \(r = 160,000\) light-years, which equals \(160,000 \times 9.461 \times 10^{15} \) meters. The velocity (\(v\)) is \(300,000\) meters/second, and the gravitational constant (\(G\)) is \(6.674 \times 10^{-11} \, \text{m}^3 \cdot \text{kg}^{-1} \cdot \text{s}^{-2}\).

Re-arrange Formula to Solve for Mass

We rearrange the formula to solve for mass (\(M\)).\[M = \frac{v^2 \cdot r}{G}\] This formula gives us the mass of the central object (Milky Way's halo, in this case) that causes the Large Magellanic Cloud to orbit at such velocity and distance.

Substitute Values and Calculate

Substitute the known values into the equation:\[M = \frac{(300,000)^2 \cdot (160,000 \times 9.461 \times 10^{15})}{6.674 \times 10^{-11}}\]Calculate the result to find the mass of the Milky Way's halo within 160,000 light-years.

Compute the Mass

Perform the calculation.1. Compute the square of the velocity: \( (300,000)^2 = 9 \times 10^{10} \).2. Calculate the radius in meters: \( 160,000 \times 9.461 \times 10^{15} = 1.51376 \times 10^{21} \) m.3. Substitute these into the mass formula: \[M = \frac{9 \times 10^{10} \cdot 1.51376 \times 10^{21}}{6.674 \times 10^{-11}}\]4. Compute: \[ M \approx 2.0 \times 10^{42} \, \text{kg} \].

Convert Mass to Solar Masses

To represent the mass in a more intuitive astrophysical unit, convert the mass into solar masses (\(M_\odot\)). One solar mass is approximately \(2.0 \times 10^{30} \) kg. divide the computed mass by one solar mass: \[ \frac{2.0 \times 10^{42}}{2.0 \times 10^{30}} = 10^{12} \]Therefore, the mass within 160,000 light-years is approximately \(10^{12} \) solar masses.

Key concepts

Orbital Velocity Formula

To understand the orbital dynamics around celestial objects, the orbital velocity formula is crucial. It helps us calculate the mass of a central object based on the motion of a smaller orbiting body. The formula itself is \( v = \sqrt{\frac{G \cdot M}{r}} \), where \( v \) is the orbital velocity, \( G \) is the gravitational constant, \( M \) is the mass of the central object, and \( r \) is the radius of the orbit. By rearranging the formula as \( M = \frac{v^2 \cdot r}{G} \), we can solve for the mass of a galaxy or massive object given the speed and distance of an orbiting body. This is useful in astrophysics to study galaxies like the Milky Way.

Gravitational Constant

The gravitational constant, denoted as \( G \), is a fundamental constant in physics that quantifies the strength of gravity. It appears in Newton's law of universal gravitation and is crucial for calculating gravitational forces. Its value is approximately \( 6.674 \times 10^{-11} \, \text{m}^3 \cdot \text{kg}^{-1} \cdot \text{s}^{-2} \). This constant allows us to measure how massive objects like planets and stars attract other masses. In calculations involving the Milky Way, \( G \) helps determine the mass of the galaxy, given the velocities and distances of objects like the Large Magellanic Cloud.

Large Magellanic Cloud

The Large Magellanic Cloud (LMC) is a satellite galaxy that orbits the Milky Way. It plays a crucial role in understanding the mass distribution in the Milky Way's halo. The LMC is located about 160,000 light-years from the Milky Way’s central region and travels at approximately 300 km/s. By using its known orbital velocity and distance, we can estimate the mass of the Milky Way's halo. The LMC's movement offers insights into the gravitational forces at play, contributing to the understanding of both local and broader cosmic dynamics.

Light-year Conversion

Light-years are a common astronomical distance measurement unit. One light-year is the distance light travels in one year, which equates to roughly 9.461 trillion kilometers or \( 9.461 \times 10^{15} \) meters. Converting light-years into meters is essential for calculations involving astronomical distances, such as estimating the Milky Way’s mass. In the example given, 160,000 light-years is the orbital radius converted into \( 1.51376 \times 10^{21} \) meters. This conversion facilitates the use of standard equations, enabling a deeper understanding of massive-scale cosmic structures.

Solar Mass Conversion

A solar mass is the standard unit of mass in astronomy, equivalent to the mass of our Sun. It is approximately \( 2.0 \times 10^{30} \) kilograms. When astronomical calculations give results in kilograms, like \( 2.0 \times 10^{42} \) kg as in our example, converting this figure into solar masses helps contextualize the mass in terms that are familiar and intuitive to astronomers. This conversion involves dividing the mass in kilograms by the solar mass, leading us to express the Milky Way's mass within 160,000 light-years as about \( 10^{12} \) solar masses. This represents a meaningful way to grasp the vast scales involved in galactic measurements.