<p>Last week, one of my exercise tasks was about determining the an observer's position on Earth, when in their zenith is a certain star with known right ascension and declination. We were supposed to use this relation for the hour angle <span class="math-container">$\tau$</span> to determine the longitude <span class="math-container">$\lambda$</span>: <span class="math-container">$\tau = \text{GST} - \lambda - \alpha$</span>, which would transform to <span class="math-container">$\lambda = \text{GST}-\alpha-\tau$</span>. Everything to here is clear to me, including calculating <span class="math-container">$\lambda$</span>.</p>
<p>My problem is that in the case of our exercise, the calculated longitude is bigger than 180°. For simplicity, let's say <span class="math-container">$\lambda=230°\text{E}$</span>. My naive thought process last week was, that this would mean that the longitude would simply continue going an additional 50° into the Western hemisphere, such one could say the true <span class="math-container">$\lambda$</span> on the Western side is 360° minus the calculated value. In my example: <span class="math-container">$\lambda=130° \text{W}$</span>.<br />
However, this week, I was told this is wrong and that we should still calculated 360° minus the calculated value, but instead on the Western hemisphere, we still would be on the Eastern one. Thus, <span class="math-container">$\lambda=130°\text{E}$</span>. Can someone explain to me, why this is the case? I've asked our tutor, but they couldn't explain this, without confusing me (and my fellow students) even more.</p>
<p>EDIT:
Exercise task<br />
After an emergency landing, you find yourself on an unknown beach. Directly above you, you can locate the star <span class="math-container">$\Psi$</span> Leo. You know that for this star <span class="math-container">$\alpha=9^\text{h} 43^\text{m} 54^\text{s}$</span> and <span class="math-container">$\delta=14^\circ1'15"$</span>. The time is <span class="math-container">$17^\text{h} 17^\text{m} 12^\text{s}$</span> UT. In addition, you know that for the current day at 0 UT, we have <span class="math-container">$8^\text{h} 19^\text{m} 50^\text{s}$</span> GMST.  Where are you located? Determine the longitude and latitude, as well as the name of the location.</p>
<p>EDIT 2: Detailed Calculations<br />
Relation between hour angle and right ascension is given by
<span class="math-container">$$\tau = \theta_\text{G}(t=0) + t\left(\frac{366.24}{365.24}\right) - \lambda - \alpha.$$</span>
Since the star is directly above us, it's in the zenith and thus, <span class="math-container">$\tau=0$</span>. It follows for the longitude that
<span class="math-container">$$\lambda = \theta_\text{G}(t=0) + t\left(\frac{366.24}{365.24}\right) - \alpha.$$</span>
It is given from the exercise that
<span class="math-container">$$\theta_\text{G}(t=0) = 8^\text{h} 19^\text{m} 50^\text{s} = 8.33056\text{h},$$</span>
<span class="math-container">$$t = 17^\text{h} 17^\text{m} 12^\text{s} = 17.28667\text{h},$$</span>
<span class="math-container">$$\alpha = 9^\text{h} 43^\text{m} 54^\text{s} = 9.73167\text{h}.$$</span>
Plugging everything into the equation for <span class="math-container">$\lambda$</span> leads to
<span class="math-container">$$\lambda \approx 15.93289\text{h}=238.99335^\circ E.$$</span>
Until this point everything was marked as correct (and matches with the sample solutions of the lecturer).</p>

CC BY-SA 4.0
2023-11-01T05:21:32.793Z
It is fairly common to use longitudes greater than 180 to represent Western longitudes.  But, as you expected, they are in the Western hemisphere.
Greg Miller
CC BY-SA 4.0
2023-11-01T13:58:45.370Z
Your idea sounds correct to me, 230 E and 130 W are the same place. But maybe there's a detail of the exercise itself that makes it different? Maybe if you provide the question as written it might be easier to see where the issue is.
Darth Pseudonym
CC BY-SA 4.0
2023-11-03T21:22:23.863Z
Hi, thanks for answering me! I've added the exercise as written into my question.
LittleCactus
CC BY-SA 4.0
2023-11-04T00:03:58.247Z
Did you take into account that the GMST is given for 0 UT and not for 17h17m12s UT?
Pierre Paquette
CC BY-SA 4.0
2023-11-04T01:38:23.733Z
I have to say I’m puzzled, because the formulas give ≈121° East, whereas SkySafari puts me at ≈157° West. However, if I do the calculations for another star for my own location, I get good results.
Pierre Paquette
CC BY-SA 4.0
2023-11-04T04:19:42.340Z
I too get about 121deg E, so I'm not sure where the > 180 comes in to play.
Greg Miller
CC BY-SA 4.0
2023-11-04T09:06:53.257Z
@PierrePaquette Yeah, I think, I did take this into account c:

To make more obvious where my value > 180 comes into play, I added my calculations for the value of the longitude to the question.
LittleCactus
<coordinate><coordinate-systems><spherical-astronomy>
Correct longitude from longitude over 180°
LittleCactus
# Correct longitude from longitude over 180°

(Asked by: username_0 on 2023-10-31)

Last week, one of my exercise tasks was about determining the an observer's position on Earth, when in their zenith is a certain star with known right ascension and declination. We were supposed to use this relation for the hour angle $\tau$ to determine the longitude $\lambda$: $\tau = \text{GST} - \lambda - \alpha$, which would transform to $\lambda = \text{GST}-\alpha-\tau$. Everything to here is clear to me, including calculating $\lambda$.

My problem is that in the case of our exercise, the calculated longitude is bigger than 180°. For simplicity, let's say $\lambda=230°\text{E}$. My naive thought process last week was, that this would mean that the longitude would simply continue going an additional 50° into the Western hemisphere, such one could say the true $\lambda$ on the Western side is 360° minus the calculated value. In my example: $\lambda=130° \text{W}$.  
However, this week, I was told this is wrong and that we should still calculated 360° minus the calculated value, but instead on the Western hemisphere, we still would be on the Eastern one. Thus, $\lambda=130°\text{E}$. Can someone explain to me, why this is the case? I've asked our tutor, but they couldn't explain this, without confusing me (and my fellow students) even more.

EDIT:
Exercise task  
After an emergency landing, you find yourself on an unknown beach. Directly above you, you can locate the star $\Psi$ Leo. You know that for this star $\alpha=9^\text{h} 43^\text{m} 54^\text{s}$ and $\delta=14^\circ1'15"$. The time is $17^\text{h} 17^\text{m} 12^\text{s}$ UT. In addition, you know that for the current day at 0 UT, we have $8^\text{h} 19^\text{m} 50^\text{s}$ GMST. Where are you located? Determine the longitude and latitude, as well as the name of the location.

EDIT 2: Detailed Calculations  
Relation between hour angle and right ascension is given by
$$\tau = \theta\_\text{G}(t=0) + t\left(\frac{366.24}{365.24}\right) - \lambda - \alpha.$$
Since the star is directly above us, it's in the zenith and thus, $\tau=0$. It follows for the longitude that
$$\lambda = \theta\_\text{G}(t=0) + t\left(\frac{366.24}{365.24}\right) - \alpha.$$
It is given from the exercise that
$$\theta\_\text{G}(t=0) = 8^\text{h} 19^\text{m} 50^\text{s} = 8.33056\text{h},$$
$$t = 17^\text{h} 17^\text{m} 12^\text{s} = 17.28667\text{h},$$
$$\alpha = 9^\text{h} 43^\text{m} 54^\text{s} = 9.73167\text{h}.$$
Plugging everything into the equation for $\lambda$ leads to
$$\lambda \approx 15.93289\text{h}=238.99335^\circ E.$$
Until this point everything was marked as correct (and matches with the sample solutions of the lecturer).

**Tags:** coordinate, coordinate-systems, spherical-astronomy
> **Comments:**
> **username_1**: It is fairly common to use longitudes greater than 180 to represent Western longitudes.  But, as you expected, they are in the Western hemisphere.  
> **username_2**: Your idea sounds correct to me, 230 E and 130 W are the same place. But maybe there's a detail of the exercise itself that makes it different? Maybe if you provide the question as written it might be easier to see where the issue is.  
> **username_0**: Hi, thanks for answering me! I've added the exercise as written into my question.  
> **username_3**: Did you take into account that the GMST is given for 0 UT and not for 17h17m12s UT?  
> **username_3**: I have to say I’m puzzled, because the formulas give ≈121° East, whereas SkySafari puts me at ≈157° West. However, if I do the calculations for another star for my own location, I get good results.  
> **username_1**: I too get about 121deg E, so I'm not sure where the > 180 comes in to play.  
> **username_0**: @PierrePaquette Yeah, I think, I did take this into account c:  To make more obvious where my value > 180 comes into play, I added my calculations for the value of the longitude to the question.  

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