<p>Although this question would probably be more suited for the worldbuilding.se, you're asking for scientific references, so I guess this is ok to ask here.</p>

<p>What you describe is a hierarchical system, akin to Sun-Jupiter-Anymoon. I assume the planet orbits star <span class="math-container">$S_1$</span> directly, and <span class="math-container">$S_2$</span> is the larger star, and the planet will be abbreviated with <span class="math-container">$P$</span>.  </p>

<p>I don't know any references, but just from geometric considerations I'll state a formula here that should work in a quasi-2D setting (what I mean by this, later).
The surface illumination function <span class="math-container">$I(t)$</span> should be a simple sum of the two-star contributions <span class="math-container">$$I(t)=I_1\,\sin(\omega_0 t) + I_2(t)\, \sin(\omega_1\,t), $$</span> where <span class="math-container">$\omega_0$</span> is the spin rotation frequency of P around itself. The amplitudes of the sinii are <span class="math-container">$I_1$</span>, the constant amplitude related to the constant distance of <span class="math-container">$S_1$</span> to <span class="math-container">$P$</span>, and <span class="math-container">$I_2(t)$</span> being the irradiation amplitude coming <span class="math-container">$S_2$</span>. As example, <span class="math-container">$I_1$</span> could be a number like the familiar <span class="math-container">$1360 \, kW/m^2$</span> for the sun.<br>
I wrote <span class="math-container">$I_2(t)$</span> as a function of time, because distance from the planet to <span class="math-container">$S_2$</span> varies over time, as it orbits <span class="math-container">$S_1$</span>. Therefore, as simplest approximation, I would assume that this is also a sinus, hence <span class="math-container">$$I_2(t)=I_{2,flct}\; \sin(\omega_1\,t)+I_{2,avg},$$</span> with <span class="math-container">$I_{2,avg}$</span> being the average irradiation at <span class="math-container">$S_1$</span>'s orbit around <span class="math-container">$S_2$</span>, and <span class="math-container">$I_{2,flct}$</span> is given by the minimum and maximum distance of the planet towards <span class="math-container">$S_2$</span>, and finally <span class="math-container">$\omega_1$</span> being the orbital frequency of P around <span class="math-container">$S_1$</span>.</p>

<p>Once you have some numbers for the constants, you can use python, or any other routine, in order to investigate the behaviour of your day/night cycles. Here it is important that your routine can replace the negative contributions in <span class="math-container">$I(t)$</span> with zero values, otherwise the negative values of <span class="math-container">$I_1$</span> could 'eat' the positive values of <span class="math-container">$I_2(t)$</span> and instead of perpetual day you'd get a perpetual night from the formula, when P would be directly between <span class="math-container">$S_1$</span> and <span class="math-container">$S_2$</span>. In python you would do this via some command like "I_updated = [0 if i &lt; 0 else i for i in I_raw]", where "I_raw" would be the values from the individual <span class="math-container">$S_1$</span> or <span class="math-container">$S_2$</span> contributions, before summing them up to <span class="math-container">$I(t)$</span>.</p>

<p>If you want to know 'which sun is up' at any given moment, you can plot the sinii separately. </p>

<p>This quickly scribbled formula ignores possible 3D-effects on the illumination, such as <a href="https://en.wikipedia.org/wiki/Kozai_mechanism" rel="nofollow noreferrer">a possible Kozai resonance in your system</a>, but also misrepresents 'shadowing' of <span class="math-container">$S_2$</span> by <span class="math-container">$S_1$</span>. Therefore the formula represents a very slightly inclined orbital plane w.r.t <span class="math-container">$S_1$</span>.</p>

<p>Please observe that your system, as you want to build it, might not be stable for too long:<br>
For the hierarchical structure of your orbits to exist, the stars must have a mass ratio of at least <span class="math-container">$m_2/m_1\approx 23$</span>. If <span class="math-container">$m_1$</span> is a sun-like G0 star, then <span class="math-container">$m_2$</span> will be an O9, which has a lifetime of only 4 million years on the main sequence.<br>
However, if you assign <span class="math-container">$m_2$</span> to be the G0 star, then <span class="math-container">$m_1$</span> could be a M2 dwarf, and that should work out fine. Also remember, that in your chosen configuration, there's always one part of the planets orbit around <span class="math-container">$S_1$</span>, namely when it's exactly between <span class="math-container">$S_1$</span> and <span class="math-container">$S_2$</span>, where the whole planet experiences perpetual day, and depending on the choice of stellar type, might get quite cooked.</p>

<p>For more on three-body motion, I'd recommend diving into <strong>Murray &amp; Dermott: Solar system dynamics.</strong></p>

CC BY-SA 4.0
2019-09-27T11:11:22.260Z
2019-09-27T11:11:22.260Z
AtmosphericPrisonEscape
<p>The thing to calculate is the angular distance between the primary sun and the secondary. We can measure the day as an angle <span class="math-container">$\theta$</span> (say <span class="math-container">$\theta=0$</span> at dawn, <span class="math-container">$\theta=\pi/2$</span> for noon, <span class="math-container">$\theta=\pi$</span> for dusk). So we can set <span class="math-container">$\theta(t)=(2\pi/d) t$</span> where <span class="math-container">$d$</span> is the length of a day. Let us assume the secondary is an angle <span class="math-container">$\alpha$</span> ahead of the primary. </p>

<p>Then it will rise when the primary is at <span class="math-container">$\theta=-\alpha$</span> (dim day dawn), the bright day starts at <span class="math-container">$\theta=0$</span>, ends at <span class="math-container">$\theta=\pi-\alpha$</span> and becomes dim day, and then at <span class="math-container">$\theta=\pi$</span> it becomes night. If <span class="math-container">$\alpha$</span> is small there will be brief dim day periods and the rest is bright day or real night. If <span class="math-container">$\alpha$</span> is larger there will be more dim day and less bright day and night, until we reach <span class="math-container">$\alpha=\pi$</span> when there is constant dim day. Beyond this we return gradually to the original state. The image below shows the different states, with blue for night, green for dim day, and yellow for bright day. </p>

<p><a href="https://i.sstatic.net/SppmC.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/SppmC.png" alt="enter image description here"></a></p>

<p>How does <span class="math-container">$\alpha$</span> change? If the secondary orbits with the primary with a semimajor-axis <span class="math-container">$a$</span> and the reduced mass of the system is <span class="math-container">$\mu=M_1M_2/(M_1+M_2)$</span> the period is <span class="math-container">$P=2\pi \sqrt{a^3/G\mu}$</span>. I am now assuming it is a circular orbit and that <span class="math-container">$a$</span> is big compared to the planet's orbit around the primary. Then <span class="math-container">$\alpha(t)=2\pi/P= \sqrt{G\mu/a^3} t $</span>. This changes rather slowly, over a timespan of years. </p>

<p>So if we set <span class="math-container">$a=10$</span> AU and assume <span class="math-container">$\mu=1 M_\odot$</span> the period is 31.6228 years. So basically we will run through the above diagram from left to right every day, slowly rising over a period of 31 years. The amount of dim day rises linearly with time, until it is 100% halfway through, and then it declines linearly again. </p>

CC BY-SA 4.0
2019-09-27T11:51:38.233Z
2019-09-27T16:48:11.863Z
2019-09-27T16:48:11.863Z
Anders Sandberg
<p>I am a writer and space enthusiast. For a story I am currently writing I would like to have a planet which orbits a star which resembles the sun, which in turn orbits another star. In this situation there would be multiple types of "days" including "bright day," when both suns are visible, "dim day," when only the further sun is visible, and "day," where only the nearer sun is visible. Is there any way to determine how this cycle would work? Any formulas or papers done? If so, what information would I need to have to figure it out? Thank you.</p>

<orbit><binary-star><orbital-elements>
Night-Day Cycles in S-Type Planetary Systems
Taylor S
# Night-Day Cycles in S-Type Planetary Systems

(Asked by: username_0 on 2019-09-26)

I am a writer and space enthusiast. For a story I am currently writing I would like to have a planet which orbits a star which resembles the sun, which in turn orbits another star. In this situation there would be multiple types of "days" including "bright day," when both suns are visible, "dim day," when only the further sun is visible, and "day," where only the nearer sun is visible. Is there any way to determine how this cycle would work? Any formulas or papers done? If so, what information would I need to have to figure it out? Thank you.

**Tags:** orbit, binary-star, orbital-elements

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## 2 answers

### Answer by username_1 on 2019-09-27. Score: 3

Although this question would probably be more suited for the worldbuilding.se, you're asking for scientific references, so I guess this is ok to ask here.

What you describe is a hierarchical system, akin to Sun-Jupiter-Anymoon. I assume the planet orbits star $S\_1$ directly, and $S\_2$ is the larger star, and the planet will be abbreviated with $P$.

I don't know any references, but just from geometric considerations I'll state a formula here that should work in a quasi-2D setting (what I mean by this, later).
The surface illumination function $I(t)$ should be a simple sum of the two-star contributions $$I(t)=I\_1\,\sin(\omega\_0 t) + I\_2(t)\, \sin(\omega\_1\,t), $$ where $\omega\_0$ is the spin rotation frequency of P around itself. The amplitudes of the sinii are $I\_1$, the constant amplitude related to the constant distance of $S\_1$ to $P$, and $I\_2(t)$ being the irradiation amplitude coming $S\_2$. As example, $I\_1$ could be a number like the familiar $1360 \, kW/m^2$ for the sun.  
I wrote $I\_2(t)$ as a function of time, because distance from the planet to $S\_2$ varies over time, as it orbits $S\_1$. Therefore, as simplest approximation, I would assume that this is also a sinus, hence $$I\_2(t)=I\_{2,flct}\; \sin(\omega\_1\,t)+I\_{2,avg},$$ with $I\_{2,avg}$ being the average irradiation at $S\_1$'s orbit around $S\_2$, and $I\_{2,flct}$ is given by the minimum and maximum distance of the planet towards $S\_2$, and finally $\omega\_1$ being the orbital frequency of P around $S\_1$.

Once you have some numbers for the constants, you can use python, or any other routine, in order to investigate the behaviour of your day/night cycles. Here it is important that your routine can replace the negative contributions in $I(t)$ with zero values, otherwise the negative values of $I\_1$ could 'eat' the positive values of $I\_2(t)$ and instead of perpetual day you'd get a perpetual night from the formula, when P would be directly between $S\_1$ and $S\_2$. In python you would do this via some command like "I\_updated = [0 if i < 0 else i for i in I\_raw]", where "I\_raw" would be the values from the individual $S\_1$ or $S\_2$ contributions, before summing them up to $I(t)$.

If you want to know 'which sun is up' at any given moment, you can plot the sinii separately.

This quickly scribbled formula ignores possible 3D-effects on the illumination, such as [a possible Kozai resonance in your system](https://en.wikipedia.org/wiki/Kozai_mechanism), but also misrepresents 'shadowing' of $S\_2$ by $S\_1$. Therefore the formula represents a very slightly inclined orbital plane w.r.t $S\_1$.

Please observe that your system, as you want to build it, might not be stable for too long:  
For the hierarchical structure of your orbits to exist, the stars must have a mass ratio of at least $m\_2/m\_1\approx 23$. If $m\_1$ is a sun-like G0 star, then $m\_2$ will be an O9, which has a lifetime of only 4 million years on the main sequence.  
However, if you assign $m\_2$ to be the G0 star, then $m\_1$ could be a M2 dwarf, and that should work out fine. Also remember, that in your chosen configuration, there's always one part of the planets orbit around $S\_1$, namely when it's exactly between $S\_1$ and $S\_2$, where the whole planet experiences perpetual day, and depending on the choice of stellar type, might get quite cooked.

For more on three-body motion, I'd recommend diving into **Murray & Dermott: Solar system dynamics.**


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### Answer by username_2 on 2019-09-27. Score: 3

The thing to calculate is the angular distance between the primary sun and the secondary. We can measure the day as an angle $\theta$ (say $\theta=0$ at dawn, $\theta=\pi/2$ for noon, $\theta=\pi$ for dusk). So we can set $\theta(t)=(2\pi/d) t$ where $d$ is the length of a day. Let us assume the secondary is an angle $\alpha$ ahead of the primary.

Then it will rise when the primary is at $\theta=-\alpha$ (dim day dawn), the bright day starts at $\theta=0$, ends at $\theta=\pi-\alpha$ and becomes dim day, and then at $\theta=\pi$ it becomes night. If $\alpha$ is small there will be brief dim day periods and the rest is bright day or real night. If $\alpha$ is larger there will be more dim day and less bright day and night, until we reach $\alpha=\pi$ when there is constant dim day. Beyond this we return gradually to the original state. The image below shows the different states, with blue for night, green for dim day, and yellow for bright day.

[![enter image description here](https://i.sstatic.net/SppmC.png)](https://i.sstatic.net/SppmC.png)

How does $\alpha$ change? If the secondary orbits with the primary with a semimajor-axis $a$ and the reduced mass of the system is $\mu=M\_1M\_2/(M\_1+M\_2)$ the period is $P=2\pi \sqrt{a^3/G\mu}$. I am now assuming it is a circular orbit and that $a$ is big compared to the planet's orbit around the primary. Then $\alpha(t)=2\pi/P= \sqrt{G\mu/a^3} t $. This changes rather slowly, over a timespan of years.

So if we set $a=10$ AU and assume $\mu=1 M\_\odot$ the period is 31.6228 years. So basically we will run through the above diagram from left to right every day, slowly rising over a period of 31 years. The amount of dim day rises linearly with time, until it is 100% halfway through, and then it declines linearly again.


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