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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1$\begingroup$ guys this is more a project please help me $\endgroup$– Rutherford paulCommented Jun 1, 2021 at 19:25
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4$\begingroup$ Also, please add what you have tried to solve your problem. $\endgroup$– WarpPrimeCommented Jun 1, 2021 at 19:29
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2$\begingroup$ @uhoh Yes, I understand the counterproductivity of my actions and will work to improve them. $\endgroup$– WarpPrimeCommented Jun 2, 2021 at 12:32
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1$\begingroup$ @uhoh I've responded to your meta post concerning the issue: astronomy.meta.stackexchange.com/a/763/31410 $\endgroup$– WarpPrimeCommented Jun 2, 2021 at 20:04
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1$\begingroup$ @uhoh Done. We can continue this in the meta discussion. $\endgroup$– WarpPrimeCommented Jun 3, 2021 at 1:20
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1 Answer
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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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1$\begingroup$ 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. $\endgroup$– ProfRobCommented Jun 12, 2021 at 16:05
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1$\begingroup$ 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". static.cambridge.org/binary/version/id/… ; en.wikipedia.org/wiki/Phase_curve_(astronomy)#The_Moon ; skyandtelescope.org/astronomy-news/how-bright-moon-exactly ; chicagotribune.com/weather/… $\endgroup$– ProfRobCommented Jun 12, 2021 at 19:09
