<p>We know Pogson formula, which relates difference in magnitude to light fluxes:</p>
<p><span class="math-container">$$10^{0.4 (m_1-m_2)}=\frac{j_2}{j_1}$$</span></p>
<p>Let's assume that Moon reflects light isotropically. Moon is scattering the light that falls onto it, so we can say that the light flux at full moon is double of that at first quarter: <span class="math-container">$\frac{j_2}{j_1}=\frac{1}{2}$</span>. Also, we are given <span class="math-container">$m_1=-12^m5$</span>:</p>
<p><span class="math-container">$$10^{0.4 (m_1-m_2)}=\frac{j_2}{j_1}$$</span>
<span class="math-container">$$m_2=m_1-2.5\log{(\frac{j_2}{j_1})}$$</span>
<span class="math-container">$$m_2=-12^m5-2.5\log{(\frac{1}{2})}=-11^m74$$</span></p>
<p>Thus, the magnitude of the first quarter of moon is equal to <span class="math-container">$-11^m74$</span>.</p>
<hr />
<p>We made an assumption that Moon reflects light isotropically (same in every direction), but this is not quite true. We must also include angle of scattering. If we look in Stellarium for dates around May 2021, we see that the magnitude of full moon is <span class="math-container">$-12^m35$</span> and of first quarter is <span class="math-container">$-11^m$</span>. Thus, the <em>real</em> ratio between light fluxes is around:</p>
<p><span class="math-container">$$\frac{j_2}{j_1}=10^{0.4(m_1-m_2)}=3.47$$</span></p>
<p>And if we insert such ratio for your given magnitude (<span class="math-container">$-12^m5$</span>) into Pogson equation, we get:
<span class="math-container">$$m_2=m_1+2.5\cdot\log{\frac{j_1}{j_2}}=-12^m5-2.5\cdot\log{3.47}=-11^m15$$</span>
So that would be the actual answer.</p>

CC BY-SA 4.0
2021-06-12T16:05:28.613Z
The area factor is only one factor. The scattering angle matters. The real answer is considerably less than half as bright. More like a factor of 5-10.
ProfRob
CC BY-SA 4.0
2021-06-12T18:59:06.417Z
@ProfRob Not quite 5–10.
User123
CC BY-SA 4.0
2021-06-12T19:09:33.653Z
I see your edit, I - haven't done the calculation myself - but it doesn't agree with all these sources which suggest a factor of 5-10. I don't know the origin of the discrepancy, maybe I am misunderstanding the "phase curve".
 https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170425081806866-0333:9781316443545:13174fig1_5.png?pub-status=live ; https://en.wikipedia.org/wiki/Phase_curve_(astronomy)#The_Moon ;  https://skyandtelescope.org/astronomy-news/how-bright-moon-exactly/   ; https://www.chicagotribune.com/weather/ct-wea-0404-asktom-20150403-column.html
ProfRob
CC BY-SA 4.0
2021-06-02T18:41:08.637Z
2021-06-12T18:58:41.360Z
2021-06-12T18:58:41.360Z
User123
<p>Given
Apparent magnitude of moon is -12.5.
Distance of moon and earth is 384400 km.
Diameter of Moon is 31 arcmin.
Phase of moon is 0.5.</p>
<p>Find the apparent magnitude of first quarter moon.</p>

CC BY-SA 4.0
2021-06-01T19:25:39.987Z
guys this is more a project please help me
Rutherford paul
CC BY-SA 4.0
2021-06-01T19:29:25.130Z
Also, please add what you have tried to solve your problem.
WarpPrime
CC BY-SA 4.0
2021-06-01T22:23:27.900Z
Maybe take a look at Sir Cumference's post here:  https://astronomy.stackexchange.com/questions/18073/how-do-i-know-mathematically-rather-than-from-observation-if-a-moon-is-full
Connor Garcia
CC BY-SA 4.0
2021-06-02T12:32:54.220Z
@uhoh Yes, I understand the counterproductivity of my actions and will work to improve them.
WarpPrime
CC BY-SA 4.0
2021-06-02T20:04:58.170Z
@uhoh I've responded to your meta post concerning the issue: https://astronomy.meta.stackexchange.com/a/763/31410
WarpPrime
CC BY-SA 4.0
2021-06-03T01:20:53.267Z
@uhoh Done. We can continue this in the meta discussion.
WarpPrime
CC BY-SA 4.0
2021-06-03T01:47:31.050Z
thanks! comment cleanup! :-)
uhoh
<the-moon><apparent-magnitude><moon-phases><angular-diameter>
Apparent Magnitude of First quarter Moon
Rutherford paul
# Apparent Magnitude of First quarter Moon

(Asked by: username_0 on 2021-06-01)

Given
Apparent magnitude of moon is -12.5.
Distance of moon and earth is 384400 km.
Diameter of Moon is 31 arcmin.
Phase of moon is 0.5.

Find the apparent magnitude of first quarter moon.

**Tags:** the-moon, apparent-magnitude, moon-phases, angular-diameter
> **Comments:**
> **username_0**: guys this is more a project please help me  
> **username_1**: Also, please add what you have tried to solve your problem.  
> **username_2**: Maybe take a look at Sir Cumference's post here:  https://astronomy.stackexchange.com/questions/18073/how-do-i-know-mathematically-rather-than-from-observation-if-a-moon-is-full  
> **username_1**: @uhoh Yes, I understand the counterproductivity of my actions and will work to improve them.  
> **username_1**: @uhoh I've responded to your meta post concerning the issue: https://astronomy.meta.stackexchange.com/a/763/31410  
> **username_1**: @uhoh Done. We can continue this in the meta discussion.  
> **username_3**: thanks! comment cleanup! :-)  

---

## 1 answers

### Answer by username_4 on 2021-06-02. Score: 1

We know Pogson formula, which relates difference in magnitude to light fluxes:

$$10^{0.4 (m\_1-m\_2)}=\frac{j\_2}{j\_1}$$

Let's assume that Moon reflects light isotropically. Moon is scattering the light that falls onto it, so we can say that the light flux at full moon is double of that at first quarter: $\frac{j\_2}{j\_1}=\frac{1}{2}$. Also, we are given $m\_1=-12^m5$:

$$10^{0.4 (m\_1-m\_2)}=\frac{j\_2}{j\_1}$$
$$m\_2=m\_1-2.5\log{(\frac{j\_2}{j\_1})}$$
$$m\_2=-12^m5-2.5\log{(\frac{1}{2})}=-11^m74$$

Thus, the magnitude of the first quarter of moon is equal to $-11^m74$.

---

We made an assumption that Moon reflects light isotropically (same in every direction), but this is not quite true. We must also include angle of scattering. If we look in Stellarium for dates around May 2021, we see that the magnitude of full moon is $-12^m35$ and of first quarter is $-11^m$. Thus, the *real* ratio between light fluxes is around:

$$\frac{j\_2}{j\_1}=10^{0.4(m\_1-m\_2)}=3.47$$

And if we insert such ratio for your given magnitude ($-12^m5$) into Pogson equation, we get:
$$m\_2=m\_1+2.5\cdot\log{\frac{j\_1}{j\_2}}=-12^m5-2.5\cdot\log{3.47}=-11^m15$$
So that would be the actual answer.

> **Comments:**
> **username_5**: The area factor is only one factor. The scattering angle matters. The real answer is considerably less than half as bright. More like a factor of 5-10.  
> **username_4**: @ProfRob Not quite 5–10.  
> **username_5**: I see your edit, I - haven't done the calculation myself - but it doesn't agree with all these sources which suggest a factor of 5-10. I don't know the origin of the discrepancy, maybe I am misunderstanding the "phase curve".  https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170425081806866-0333:9781316443545:13174fig1_5.png?pub-status=live ; https://en.wikipedia.org/wiki/Phase_curve_(astronomy)#The_Moon ;  https://skyandtelescope.org/astronomy-news/how-bright-moon-exactly/   ; https://www.chicagotribune.com/weather/ct-wea-0404-asktom-20150403-column.html  

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