# Calculation of solar energy lack during Venus transit

The solar constant in Earth's orbit is $$k$$ ($$k=1.37\text{ kW/m}^2$$). Evaluate the solar energy lack due to the transit of Venus through the Sun diameter. Radius and orbit radius of Venus are $$R_V$$ and $$r_{SV}$$, respectively. Consider the orbits of Earth and Venus circular.
$$R_\oplus$$ is Earth's radius
$$D_\odot$$ is Sun diameter
$$D_V$$ is Venus diameter
$$r_{SE}$$ is distance from Sun to Earth
$$r_{EV}$$ is distance from Earth to Venus
$$T_\oplus$$ is orbital period of Earth
$$T_V$$ is orbital period of Venus
My calculations
The solar constant is $$k=\displaystyle\frac{E}{St}$$, where $$E$$ is the total solar energy, $$S$$ is sphere area of radius $$1\text{ au}$$, $$t$$ is time of solar energy radiation. Let $$t$$ be the time of Venus transit, then Earth gets energy $$E=4\pi kR_\oplus^2t$$ (without Venus). The Sun unit angular area gives $$\displaystyle\frac{E}{\alpha_\odot^2}=W$$ energy, where $$\alpha_\odot^2$$ is angular area of Sun. During Venus transit Earth gets solar energy $$E'=W(\alpha_\odot^2-\alpha_V^2)$$, where $$\alpha_V^2$$ is angular area of Venus. The energy lack is:
$$\Delta E=E-E'=W\alpha_\odot^2-W(\alpha_\odot^2-\alpha_V^2)=W\alpha_V^2=E\left(\displaystyle\frac{\alpha_V}{\alpha_\odot}\right)^2=4\pi k\left(\displaystyle\frac{\alpha_V}{\alpha_\odot}\right)^2R_\oplus^2t$$.
Then we have:
$$\alpha_V=\displaystyle\frac{D_V}{r_{EV}}$$;
$$\alpha_\odot=\displaystyle\frac{ D_\odot}{r_{SE}}$$;
$$t=\displaystyle\frac{\alpha_\odot r_{EV}}{v_V-v_\oplus}$$;
$$v_V=\displaystyle\frac{2\pi r_{SV}}{T_V}$$;
$$v_\oplus=\displaystyle\frac{2\pi r_{SE}}{T_\oplus}$$.
And substituting it all we obtain:
$$\Delta E=\displaystyle\frac{kD_V^2R_\oplus^2r_{SE}}{2 D_\odot r_{EV}\left(\displaystyle\frac{r_{SV}}{T_V}-\frac{r_{SE}}{T_\oplus}\right)}$$.
I'd like to clarify if this evaluation is correct.

And, I'm wondering how to use (type) astronomical symbols of planet here (if it's possible)?

• I would hate to see Astronomy SE become a site for checking/doing homework. Feb 11 at 8:20
• Is there a reference to the "accepted" way to render the symbols for the planets on StackExchange sites? Unicode / MathJax / other? Feb 11 at 12:08
• @ProfRob, thanks for the answer. I've read "What topics can I ask about here?" and "What types of questions should I avoid asking?" in "Help center", and haven't found any reasons not to ask my question. I haven't written that my question is "homework". It's just a question regarding "Planetary Science and Celestial mechanics". I've described the problem and provided my reasoning with calculations. Maybe someone can help me to clarify this. Feb 11 at 17:13
• While you say $S$ is the area of a sphere with radius $1\text{ au}$, you write $S=4\pi R_\oplus^2$. But $R_\oplus=6378\text{ km}$ which is the Earth's radius. $1\text{ au}$ is the distance from Earth to Sun ($150,000,000\text{ km}$) Nov 22 at 11:16
• @misha.physics And also, the $2$ should be in the numinator instead of denominator. Nov 23 at 13:40