The task
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.
My additional notation
$R_\oplus$ 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:
$\alpha_\odot=\displaystyle\frac{ D_\odot}{r_{SE}}$;
$t=\displaystyle\frac{\alpha_\odot r_{EV}}{v_V-v_\oplus}$;
$v_V=\displaystyle\frac{2\pi r_{SV}}{T_V}$;
$v_\oplus=\displaystyle\frac{2\pi r_{SE}}{T_\oplus}$.
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)?

  • 1
    $\begingroup$ I would hate to see Astronomy SE become a site for checking/doing homework. $\endgroup$
    – ProfRob
    Feb 11 at 8:20
  • $\begingroup$ Is there a reference to the "accepted" way to render the symbols for the planets on StackExchange sites? Unicode / MathJax / other? $\endgroup$ Feb 11 at 12:08
  • $\begingroup$ @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. $\endgroup$ Feb 11 at 17:13
  • 1
    $\begingroup$ 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 ($150,000,000\text{ km}$) $\endgroup$ Nov 22 at 11:16
  • 1
    $\begingroup$ @misha.physics And also, the $2$ should be in the numinator instead of denominator. $\endgroup$ Nov 23 at 13:40


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