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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.

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    $\begingroup$ guys this is more a project please help me $\endgroup$ Jun 1 at 19:25
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    $\begingroup$ Also, please add what you have tried to solve your problem. $\endgroup$ Jun 1 at 19:29
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    $\begingroup$ @uhoh Yes, I understand the counterproductivity of my actions and will work to improve them. $\endgroup$ Jun 2 at 12:32
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    $\begingroup$ @uhoh I've responded to your meta post concerning the issue: astronomy.meta.stackexchange.com/a/763/31410 $\endgroup$ Jun 2 at 20:04
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    $\begingroup$ @uhoh Done. We can continue this in the meta discussion. $\endgroup$ Jun 3 at 1:20
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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.

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