If the RR Lyrae stars in a globular cluster have average apparent magnitudes of +19, how far away (in pc) is the cluster? (Hints: See the following figure, and use the magnitude- distance formula: d = 10(my- Myt5)/5,) -6 Туре (classical) Cepheido 10 10 Туре I Cepheids 10 AR Lyrae stars 0.3 1 3 10 30 100 Pulsation period (days) pc Luminosity, L/L 4, apnyubew anpsqy
If the RR Lyrae stars in a globular cluster have average apparent magnitudes of +19, how far away (in pc) is the cluster? (Hints: See the following figure, and use the magnitude- distance formula: d = 10(my- Myt5)/5,) -6 Туре (classical) Cepheido 10 10 Туре I Cepheids 10 AR Lyrae stars 0.3 1 3 10 30 100 Pulsation period (days) pc Luminosity, L/L 4, apnyubew anpsqy
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If the RR Lyrae stars in a globular cluster have average apparent magnitudes of +19, how far away (in pc) is the cluster? (Hints: See the following figure, and use the magnitude–distance formula:
d = 10(mV − MV+5)/5.)
![**Transcription and Explanation for Educational Website**
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**Problem Statement:**
If the RR Lyrae stars in a globular cluster have average apparent magnitudes of +19, how far away (in pc) is the cluster?
*(Hints: See the following figure, and use the magnitude–distance formula: \( d = 10^{(m_v - M_v + 5)/5} \).)*
**Figure Description:**
The graph provided is a logarithmic plot showing the relationship between the "Pulsation period (days)" on the x-axis and "Luminosity \( L / L_{\odot} \)" on the right y-axis as well as "Absolute magnitude" on the left y-axis.
1. **Axes:**
- X-axis (Pulsation period): Ranges from 0.3 to 100 days.
- Left Y-axis (Absolute magnitude): Ranges from -2 to -7.
- Right Y-axis (Luminosity \( L / L_{\odot} \)): Ranges from \( 10^{-1} \) to \( 10^{4} \).
2. **Curves:**
- **RR Lyrae stars:** Show a near-vertical line starting at around 0.5 days in pulsation period and an absolute magnitude of approximately +0.75.
- **Type II Cepheids:** This curve is an upward sloping line located above the RR Lyrae stars curve.
- **Type I (classical) Cepheids:** This curve is the highest and steepest, indicating higher luminosities for equivalent pulsation periods compared to RR Lyrae and Type II Cepheids.
This graph helps in understanding the correlation between the pulsation period and luminosity for different types of variable stars, which is key in calculating distances to astronomical objects like globular clusters.
**Calculation Instructions:**
To find the distance to the globular cluster containing RR Lyrae stars:
1. Use the provided magnitude–distance formula:
\[
d = 10^{(m_v - M_v + 5)/5}
\]
2. Identify the absolute magnitude (\( M_v \)) of RR Lyrae stars from the graph.
3. Plug \( m_v = +19 \) and the identified \( M_v \) from the graph into the formula.
This calculation will yield the distance \( d \) in parsecs (pc).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd0530e16-0f9d-4caf-90d6-684cf5a3ef1c%2F48eba977-9847-491f-9899-f69fd9d16eb4%2Fa3r8c98_processed.png&w=3840&q=75)
Transcribed Image Text:**Transcription and Explanation for Educational Website**
---
**Problem Statement:**
If the RR Lyrae stars in a globular cluster have average apparent magnitudes of +19, how far away (in pc) is the cluster?
*(Hints: See the following figure, and use the magnitude–distance formula: \( d = 10^{(m_v - M_v + 5)/5} \).)*
**Figure Description:**
The graph provided is a logarithmic plot showing the relationship between the "Pulsation period (days)" on the x-axis and "Luminosity \( L / L_{\odot} \)" on the right y-axis as well as "Absolute magnitude" on the left y-axis.
1. **Axes:**
- X-axis (Pulsation period): Ranges from 0.3 to 100 days.
- Left Y-axis (Absolute magnitude): Ranges from -2 to -7.
- Right Y-axis (Luminosity \( L / L_{\odot} \)): Ranges from \( 10^{-1} \) to \( 10^{4} \).
2. **Curves:**
- **RR Lyrae stars:** Show a near-vertical line starting at around 0.5 days in pulsation period and an absolute magnitude of approximately +0.75.
- **Type II Cepheids:** This curve is an upward sloping line located above the RR Lyrae stars curve.
- **Type I (classical) Cepheids:** This curve is the highest and steepest, indicating higher luminosities for equivalent pulsation periods compared to RR Lyrae and Type II Cepheids.
This graph helps in understanding the correlation between the pulsation period and luminosity for different types of variable stars, which is key in calculating distances to astronomical objects like globular clusters.
**Calculation Instructions:**
To find the distance to the globular cluster containing RR Lyrae stars:
1. Use the provided magnitude–distance formula:
\[
d = 10^{(m_v - M_v + 5)/5}
\]
2. Identify the absolute magnitude (\( M_v \)) of RR Lyrae stars from the graph.
3. Plug \( m_v = +19 \) and the identified \( M_v \) from the graph into the formula.
This calculation will yield the distance \( d \) in parsecs (pc).
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