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Mobile App Chapter 15: Problem 21
From observations of the masers in Orion, we find an average radial velocity of \(24.0 \mathrm{km} / \mathrm{s},\) and proper motions of 0.01 arc sec/yr. How far away are these masers?
Short Answer
Expert verified
The distance to the masers in Orion is 506 parsecs.
Step by step solution
01
Understand the Given Information
We are given the radial velocity of the masers in Orion as \(24.0 \, \text{km/s}\) and the proper motion as \(0.01\) arc seconds per year. We need to determine the distance to these masers.
02
Proper Motion Formula
Proper motion can be used to calculate the distance using the relationship \[ \mu = \frac{v_t}{d} \]where \(\mu\) is proper motion in arc seconds per year, \(v_t\) is the transverse velocity, and \(d\) is the distance in parsecs (pc).
03
Relate Radial Velocity and Transverse Velocity
Use the radial velocity to find transverse velocity if both components are similar. For masers, it's common to assume \(v_t \approx v_r\). Thus, \(v_t \approx 24.0 \, \text{km/s}\).
04
Convert Transverse Velocity to Proper Units
1 parsec per year is roughly equal to \(4.74 \, \text{km/s}\). So we convert transverse velocity: \[v_t = \frac{24.0 \, \text{km/s}}{4.74 \, \text{km/s per pc/year}} \approx 5.06 \, \text{pc/year}\]
05
Solve for Distance
Use the relationship formulated before: \[ \mu = \frac{v_t}{d} \]Rearrange to solve for distance: \[ d = \frac{v_t}{\mu} \]\[ d = \frac{5.06 \, \text{pc/year}}{0.01 \, \text{arc seconds/year}} = 506 \, \text{parsecs}\]
06
Final Answer
The distance to the masers in Orion is 506 parsecs.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Radial Velocity
Radial velocity is the speed at which an object moves directly towards or away from an observer. It's an important component in understanding the motion of celestial objects.
In this exercise, the radial velocity of masers in Orion is given as 24 km/s. This value helps in estimating their distance when combined with other motions.
Radial velocity is often measured using the Doppler Effect, where the frequency of light from an object changes due to its motion relative to the observer.
In this exercise, the radial velocity of masers in Orion is given as 24 km/s. This value helps in estimating their distance when combined with other motions.
Radial velocity is often measured using the Doppler Effect, where the frequency of light from an object changes due to its motion relative to the observer.
Proper Motion
Proper motion refers to the apparent motion of a star across the sky, measured in angles per unit of time, usually arc seconds per year.
In this exercise, the proper motion of masers in Orion is given as 0.01 arc seconds/year. Proper motion is crucial as it's used to calculate distances to celestial objects.
The formula \[ \mu = \frac{v_t}{d} \ \] relates proper motion (\( \mu \ \)) to transverse velocity and distance, where transverse velocity (\( v_t \ \)) is perpendicular to the line of sight, and distance (\( d \ \)) is in parsecs.
In this exercise, the proper motion of masers in Orion is given as 0.01 arc seconds/year. Proper motion is crucial as it's used to calculate distances to celestial objects.
The formula \[ \mu = \frac{v_t}{d} \ \] relates proper motion (\( \mu \ \)) to transverse velocity and distance, where transverse velocity (\( v_t \ \)) is perpendicular to the line of sight, and distance (\( d \ \)) is in parsecs.
Transverse Velocity
Transverse velocity is the component of an object's velocity perpendicular to the line of sight, i.e., how fast it's moving across the sky.
In the case of the masers in Orion, the transverse velocity is assumed to be roughly equal to the radial velocity, giving approximately 24 km/s.
The conversion of transverse velocity to proper units benefits calculations. Here, 1 parsec/year is approximately equal to 4.74 km/s, thus: \[ v_t \approx \frac{24 \ \text{km/s}}{4.74 \ \text{km/s per pc/year}} \approx 5.06 \ \text{pc/year} \ \]
In the case of the masers in Orion, the transverse velocity is assumed to be roughly equal to the radial velocity, giving approximately 24 km/s.
The conversion of transverse velocity to proper units benefits calculations. Here, 1 parsec/year is approximately equal to 4.74 km/s, thus: \[ v_t \approx \frac{24 \ \text{km/s}}{4.74 \ \text{km/s per pc/year}} \approx 5.06 \ \text{pc/year} \ \]
Parsecs
Parsec is a unit of distance used in astronomy, equivalent to approximately 3.26 light-years. It is abbreviated as 'pc'.
This unit is particularly helpful when dealing with the vast distances between celestial objects.
Calculating distance using parsecs in our exercise involves the formula: \[ d = \frac{v_t}{\mu} \ \] where \( d \ \) is the distance in parsecs, \( v_t \ \) is the transverse velocity in parsecs/year, and \( \mu \ \) is the proper motion.
For our masers, we have the transverse velocity as 5.06 pc/year and the proper motion as 0.01 arc seconds/year: \[ d = \frac{5.06 \ \text{pc/year}}{0.01 \ \text{arc seconds/year}} = 506 \ \text{parsecs} \]
This unit is particularly helpful when dealing with the vast distances between celestial objects.
Calculating distance using parsecs in our exercise involves the formula: \[ d = \frac{v_t}{\mu} \ \] where \( d \ \) is the distance in parsecs, \( v_t \ \) is the transverse velocity in parsecs/year, and \( \mu \ \) is the proper motion.
For our masers, we have the transverse velocity as 5.06 pc/year and the proper motion as 0.01 arc seconds/year: \[ d = \frac{5.06 \ \text{pc/year}}{0.01 \ \text{arc seconds/year}} = 506 \ \text{parsecs} \]
Orion
Orion is a prominent constellation located on the celestial equator and visible throughout the world. Named after a hunter in Greek mythology, Orion is among the most recognizable constellations.
The Orion constellation contains many interesting objects, including the Orion Nebula, a region close to the masers discussed in our exercise.
Masers within Orion help astronomers study the motion and properties of interstellar objects and provide valuable information about the structure and dynamics of our galaxy.
The Orion constellation contains many interesting objects, including the Orion Nebula, a region close to the masers discussed in our exercise.
Masers within Orion help astronomers study the motion and properties of interstellar objects and provide valuable information about the structure and dynamics of our galaxy.
