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

CC BY-SA 4.0
2023-02-11T08:20:05.033Z
I would hate to see Astronomy SE become a site for checking/doing homework.
ProfRob
CC BY-SA 4.0
2023-02-11T12:08:13.853Z
Is there a reference to the "accepted" way to render the symbols for the planets on StackExchange sites?  Unicode / MathJax / other?
GrapefruitIsAwesome
CC BY-SA 4.0
2023-02-11T17:13:22.527Z
@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.
misha.physics
CC BY-SA 4.0
2023-02-11T17:23:39.217Z
@misha.physics so a valid, but uninteresting, answer, that you would accept, is ",Yes this is correct"? Why do you think it might not be correct? Perhaps asking about whichever steps in your calculation are troubling you and why, would make a better question.
ProfRob
CC BY-SA 4.0
2023-02-11T17:25:00.347Z
Another way to handle this would be to ask how to calculate the flux deficit and provide an answer yourself. Then you might get feedback on your answer.
ProfRob
CC BY-SA 4.0
2023-02-11T17:42:27.513Z
@ProfRob, thanks. I think I just wanted to make sure my evaluating is correct, I would be glad to know if it's so. I built this solution by myself, but I'm not sure in some steps. Where I calculate the time of Venus transit, subtract orbital speeds, and consider the Earth's area like a circle area. Of course I would be grateful to know another way to calculate the flux deficit.
misha.physics
CC BY-SA 4.0
2023-11-22T11:16:18.157Z
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 ($150,000,000\text{ km}$)
Snack Exchange
CC BY-SA 4.0
2023-11-22T17:42:32.790Z
@SnackExchange, thank you, yes, I agree, it seems I was wrong about this problem.
misha.physics
CC BY-SA 4.0
2023-11-23T13:40:21.707Z
@misha.physics And also, the $2$ should be in the numinator instead of denominator.
Snack Exchange
CC BY-SA 4.0
2023-11-23T20:19:20.793Z
@SnackExchange, yes indeed. Thank you for noticing this.
misha.physics
CC BY-SA 4.0
2023-11-24T11:56:59.457Z
@misha.physics :-)
Snack Exchange
<planetary-transits>
Calculation of solar energy lack during Venus transit
misha.physics
# Calculation of solar energy lack during Venus transit

(Asked by: username_0 on 2023-02-10)

**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:  
$\alpha\_V=\displaystyle\frac{D\_V}{r\_{EV}}$;  
$\alpha\_\odot=\displaystyle\frac{ D\_\odot}{r\_{SE}}$;  
$t=\displaystyle\frac{\alpha\_\odot r\_{EV}}{v\_V-v\_\oplus}$;  
$v\_V=\displaystyle\frac{2\pi r\_{SV}}{T\_V}$;  
$v\_\oplus=\displaystyle\frac{2\pi r\_{SE}}{T\_\oplus}$.  
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)?

**Tags:** planetary-transits

> **Comments:**
> **username_1**: I would hate to see Astronomy SE become a site for checking/doing homework.  
> **username_2**: Is there a reference to the "accepted" way to render the symbols for the planets on StackExchange sites?  Unicode / MathJax / other?  
> **username_0**: @username_1, 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.  
> **username_1**: @username_0 so a valid, but uninteresting, answer, that you would accept, is ",Yes this is correct"? Why do you think it might not be correct? Perhaps asking about whichever steps in your calculation are troubling you and why, would make a better question.  
> **username_1**: Another way to handle this would be to ask how to calculate the flux deficit and provide an answer yourself. Then you might get feedback on your answer.  
> **username_0**: @username_1, thanks. I think I just wanted to make sure my evaluating is correct, I would be glad to know if it's so. I built this solution by myself, but I'm not sure in some steps. Where I calculate the time of Venus transit, subtract orbital speeds, and consider the Earth's area like a circle area. Of course I would be grateful to know another way to calculate the flux deficit.  
> **username_3**: 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 ($150,000,000\text{ km}$)  
> **username_0**: @SnackExchange, thank you, yes, I agree, it seems I was wrong about this problem.  
> **username_3**: @username_0 And also, the $2$ should be in the numinator instead of denominator.  
> **username_0**: @SnackExchange, yes indeed. Thank you for noticing this.  
> **username_3**: @username_0 :-)  

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## 0 answers

