VIPSolutionsA. The hour angle (HA) is the difference between the local sidereal time (LST) and the right ascension (RA) of the star, given by HA = LST - RA. At transit, HA = 0. Since Rigel will transit in 75 minutes (1.25 hours), it is currently east of the meridian, so its HA is negative. Thus, the HA is -1.25. B. LST = RA - 1.25 hours = 05h 14m 32.3s - 1h 15m = 03h 59m 32.3s. C. A star due east on the horizon must have a declination of approximately 0° (on the celestial equator) to rise exactly due east. Such a star has HA = -6 hours, so its RA = LST - HA = LST + 6 hours = 03h 59m 32.3s + 6h = 09h 59m 32.3s. D. Rigel will set when its HA is approximately +6 hours (since its declination is close to 0°, similar to an equatorial star, which sets 6 sidereal hours after transit). Thus, LST at setting ≈ RA + 6 hours = 05h 14m 32.3s + 6h = 11h 14m 32.3s.
Question:
A. The hour angle (HA) is the difference between the local sidereal time (LST) and the right ascension (RA) of the star, given by HA = LST - RA. At transit, HA = 0. Since Rigel will transit in 75 minutes (1.25 hours), it is currently east of the meridian, so its HA is negative. Thus, the HA is -1.25. B. LST = RA - 1.25 hours = 05h 14m 32.3s - 1h 15m = 03h 59m 32.3s. C. A star due east on the horizon must have a declination of approximately 0° (on the celestial equator) to rise exactly due east. Such a star has HA = -6 hours, so its RA = LST - HA = LST + 6 hours = 03h 59m 32.3s + 6h = 09h 59m 32.3s. D. Rigel will set when its HA is approximately +6 hours (since its declination is close to 0°, similar to an equatorial star, which sets 6 sidereal hours after transit). Thus, LST at setting ≈ RA + 6 hours = 05h 14m 32.3s + 6h = 11h 14m 32.3s.
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Asked by: vishal
Created at: 2025-09-11 03:25:25
Asked by: vishal
Created at: 2025-09-11 03:25:25
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# Astronomy Problem Solutions
## A. Hour Angle (HA) of Rigel
- **Formula:**
\[
\text{HA} = \text{LST} - \text{RA}
\]
- **At transit:** HA = 0.
- **Time until transit:** 75 minutes = 1.25 hours.
- **Current position:** Rigel is east of the meridian, so HA is negative.
- **Result:**
\[
\boxed{\text{HA} = -1.25~\text{hours}}
\]
---
## B. Local Sidereal Time (LST) Now
- **Given:**
- \(\text{RA}_\text{Rigel} = 05^{\mathrm{h}}\,14^{\mathrm{m}}\,32.3^{\mathrm{s}}\)
- \(\text{HA} = -1.25~\text{hours}\)
- **Formula:**
\[
\text{LST} = \text{RA} + \text{HA}
\]
- **Calculation:**
\[
\text{LST} = 05^{\mathrm{h}}\,14^{\mathrm{m}}\,32.3^{\mathrm{s}} + (-1^{\mathrm{h}}\,15^{\mathrm{m}})
\]
\[
= 03^{\mathrm{h}}\,59^{\mathrm{m}}\,32.3^{\mathrm{s}}
\]
- **Result:**
\[
\boxed{\text{LST} = 03^{\mathrm{h}}\,59^{\mathrm{m}}\,32.3^{\mathrm{s}}}
\]
---
## C. RA of a Star Rising Due East
- **Condition:** Star due east means it is on the horizon at azimuth \(90^\circ\).
- **For an observer at mid-latitude:**
- A star rises due east if declination \(\delta \approx 0^\circ\) (celestial equator).
- Such a star rises when HA = \(-6~\text{hours}\).
- **Formula:**
\[
\text{RA} = \text{LST} - \text{HA} = \text{LST} + 6~\text{hours}
\]
- **Calculation:**
\[
\text{RA}_\text{east} = 03^{\mathrm{h}}\,59^{\mathrm{m}}\,32.3^{\mathrm{s}} + 06^{\mathrm{h}} = 09^{\mathrm{h}}\,59^{\mathrm{m}}\,32.3^{\mathrm{s}}
\]
- **Result:**
\[
\boxed{\text{RA}_\text{east} = 09^{\mathrm{h}}\,59^{\mathrm{m}}\,32.3^{\mathrm{s}}}
\]
---
## D. LST When Rigel Sets
- **Setting condition:** For a star near the celestial equator, it sets when HA = \(+6~\text{hours}\).
- **Formula:**
\[
\text{LST}_{\text{set}} = \text{RA} + 6~\text{hours}
\]
- **Calculation:**
\[
\text{LST}_{\text{set}} = 05^{\mathrm{h}}\,14^{\mathrm{m}}\,32.3^{\mathrm{s}} + 06^{\mathrm{h}} = 11^{\mathrm{h}}\,14^{\mathrm{m}}\,32.3^{\mathrm{s}}
\]
- **Result:**
\[
\boxed{\text{LST}_{\text{set}} = 11^{\mathrm{h}}\,14^{\mathrm{m}}\,32.3^{\mathrm{s}}}
\]
---
## Summary Table
| Part | Question | Answer |
|------|--------------------------------------------------------|------------------------------|
| A | Current hour angle of Rigel | \(-1.25~\text{hours}\) |
| B | Current LST | \(03^{\mathrm{h}}\,59^{\mathrm{m}}\,32.3^{\mathrm{s}}\) |
| C | RA of a star rising due east | \(09^{\mathrm{h}}\,59^{\mathrm{m}}\,32.3^{\mathrm{s}}\) |
| D | LST when Rigel sets | \(11^{\mathrm{h}}\,14^{\mathrm{m}}\,32.3^{\mathrm{s}}\) |
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