<blockquote>
<p>Usually, how is phase=0 defined? It is relative to the line of apses perhaps with phase=0 at periastron?</p>
</blockquote>
<p>The true anomaly, <span class="math-container">$\theta$</span>, is the angle between the current location of the orbiting particle and its location in the orbit at which it is closest to the central body (called the <a href="https://en.wikipedia.org/wiki/Apsis" rel="nofollow noreferrer">periapsis</a>/periastron). The word &quot;phase&quot; is used variously in physics and astronomy, but based on your question, you may mean <span class="math-container">$\theta$</span> since it is <a href="https://en.wikipedia.org/wiki/True_anomaly#From_the_eccentric_anomaly" rel="nofollow noreferrer">defined</a> as being zero at <a href="https://en.wikipedia.org/wiki/Kepler_orbit#Keplerian_elements" rel="nofollow noreferrer">periastron.</a></p>
<blockquote>
<p>Are orbital phases of the black spot in the figure measured from some angle or time/period?</p>
</blockquote>
<p>It ultimately depends on how you want to compute the trajectory/orbit of the particle (i.e., the black &quot;spot&quot; in the OP image) that is moving about a primary body (i.e., the star in the OP image). Alternate values can be used instead of <span class="math-container">$\theta$</span>, for example <span class="math-container">$\displaystyle M$</span>, the mean anomaly.</p>

CC BY-SA 4.0
2021-05-08T08:44:58.967Z
The mean anomaly is proportional to time. The phase is also intended to be defined to be proportional to time and has nothing to do with eccentricity?
questionhang
CC BY-SA 4.0
2021-05-08T17:40:57.520Z
The true anomaly is not directly proportional to time. The true anomaly is related to the eccentric anomaly by $\cos\theta = (\cos E - e)/(1 - e\cos E)$.
Daddy Kropotkin
CC BY-SA 4.0
2021-05-07T17:44:32.807Z
2021-05-08T17:41:44.733Z
2021-05-08T17:41:44.733Z
Daddy Kropotkin
<p><a href="https://arxiv.org/abs/1102.4301" rel="nofollow noreferrer">Moreno et al.</a> in their paper <em>Eccentric binaries: Tidal flows and periastron events</em> [2011] define the orbital phase as from -0.5 to 0.5, where periastron is at phase 0, and apastron is at -0.5 and 0.5.  Orbital phase here is <span class="math-container">$\phi = \dfrac{t}{P}$</span>, where time <span class="math-container">$t=0$</span> at periastron and <span class="math-container">$P$</span> is the orbital period.  This definition allows them to plot the behavior of several binary star systems with different periods on the same x-axis. Here is a figure from their paper:</p>
<p><a href="https://i.sstatic.net/rjI07.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/rjI07.png" alt="enter image description here" /></a></p>
<p>We don't necessarily have to choose periastron as &quot;phase=0.&quot; For example, as we observe an eclipsing binary star system, the <a href="https://en.wikipedia.org/wiki/Light_curve" rel="nofollow noreferrer">light curve</a> will appear to pulse.  This is an effect due to the reduced overall system luminosity as one star passes in front of the other.</p>
<p>Here is an example from <a href="http://www.physics.sfasu.edu/astro/ebstar/ebstar.html" rel="nofollow noreferrer">Dan Bruton's website</a>:</p>
<p><a href="https://i.sstatic.net/uQpIK.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/uQpIK.png" alt="enter image description here" /></a></p>
<p>One can see that the length of 1 unit of phase is equal to the orbital period.  Here, the deep dips in magnitude correspond to when the lower luminosity star passes in front of the more luminous star, and the more shallow dips correspond to when the higher luminosity star passes in front of the less luminous star.  Since the light measurements are somewhat noisy, the technique of <a href="https://imagine.gsfc.nasa.gov/science/toolbox/timing2.html" rel="nofollow noreferrer">folding</a> is often used.  To fold a light curve, one segments the data set along the integer phase boundaries and then adds all the light curves together, to increase the signal to noise ratio.</p>
<p>Here is a corresponding <a href="https://en.wikipedia.org/wiki/Beta_Lyrae" rel="nofollow noreferrer">wikipedia</a> gif of the Beta Lyrae system using the CHARA array:</p>
<p><a href="https://i.sstatic.net/ze8R6.gif" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ze8R6.gif" alt="enter image description here" /></a></p>
<p>The origin (phase=0) in the above example is chosen to coincide with the occultation of one star by the other.  In general, the choice of origin will be made to highlight whatever feature the astronomer is interested in.  The eccentricity of the orbit won't drive the choice of origin for this technique. Instead, the choice of origin is driven by the light curve itself, which is a function of both the orbital system and the location of the observer.</p>

CC BY-SA 4.0
2021-05-07T18:35:09.350Z
This isn't (as I understand it) wholly what the question is asking about, the binary doesn't need to be eclipsing, and your answer doesn't address how phase is determined during the orbit of an eccentric binary. e.g. Where is phase 0.25?
ProfRob
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2021-05-07T19:42:51.423Z
@ProfRob Very insightful comment, as usual.  Edited.
Connor Garcia
CC BY-SA 4.0
2021-05-08T08:55:48.617Z
@ConnorGarcia   t/p, you mean the phase is proportional to time even if eccentricity!=0?
questionhang
CC BY-SA 4.0
2021-05-09T08:31:21.767Z
I reverse my previous comment. This is the only way that a phase can be defined observationally, where you may not know the eccentricity.
ProfRob
CC BY-SA 4.0
2021-05-07T18:23:19.143Z
2021-05-07T20:08:26.570Z
2021-05-07T20:08:26.570Z
Connor Garcia
<p>Are orbital phases of the black spot in the figure measured from some angle or time/period?</p>
<p>Usually, how is phase=0 defined? It is relative to the line of apses perhaps with phase=0 at periastron?</p>
<p><strong>Any of the anomalies below is defined as orbital phase? Please specify reference papers or books.</strong></p>
<p>E: eccentric anomaly</p>
<p>M: mean anomaly</p>
<p>u: true anomaly</p>
<p><a href="https://i.sstatic.net/P5bop.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/P5bop.png" alt="enter image description here" /></a></p>
<p><a href="https://zh.wikipedia.org/zh-cn/%E7%9C%9F%E8%BF%91%E9%BB%9E%E8%A7%92" rel="nofollow noreferrer">Source</a></p>

CC BY-SA 4.0
2021-05-06T21:52:16.610Z
I'm really not sure what you're asking. Have you read here https://en.wikipedia.org/wiki/Kepler_orbit#Development_of_the_laws ?
Daddy Kropotkin
CC BY-SA 4.0
2021-05-07T01:36:08.400Z
You mean `true anomaly` is what I want?
questionhang
CC BY-SA 4.0
2021-05-07T13:27:15.513Z
I don't know what you want, you've not made it clear at all.
Daddy Kropotkin
CC BY-SA 4.0
2021-05-07T14:38:42.927Z
I asked "How is phase defined and how is phase=0 defined".
questionhang
CC BY-SA 4.0
2021-05-07T15:05:37.283Z
What is "phase"? What is phase=0 mean? These are just words, you've provided no references and the image you added does not specify what you mean either. The poor quality of your question reflects how little effort you put into helping us understand it.
Daddy Kropotkin
CC BY-SA 4.0
2021-05-07T16:04:08.060Z
It is better now?
questionhang
CC BY-SA 4.0
2021-05-07T16:08:50.937Z
Yes, I think that's much better. Thank you. I gave +1
Daddy Kropotkin
CC BY-SA 4.0
2021-05-08T13:13:14.027Z
I still find it unclear whether you just want to know how phase zero is defined (body of the question) or how phase is calculated (title of the question).
ProfRob
CC BY-SA 4.0
2021-05-08T16:09:08.217Z
@uhoh The link was attached in the old post. Do you have it? I can not find it now.
questionhang
CC BY-SA 4.0
2021-05-08T16:09:44.033Z
@ProfRob phase=o is included by the definition of phase.
questionhang
CC BY-SA 4.0
2021-05-08T23:44:19.503Z
@questionhang Got it, yes! It's there in the [edit history](https://astronomy.stackexchange.com/revisions/43695/1)  (click here to get that https://i.sstatic.net/0KjqM.png) I've got it thanks! I guess I was looking at the blue image so much I missed the blue text right next to it. *Thanks!*
uhoh
<binary-star><terminology><stellar-dynamics><eclipsing-star-systems>
How is phase defined in a binary orbit with eccentricity?
questionhang
# How is phase defined in a binary orbit with eccentricity?

(Asked by: username_0 on 2021-05-06)

Are orbital phases of the black spot in the figure measured from some angle or time/period?

Usually, how is phase=0 defined? It is relative to the line of apses perhaps with phase=0 at periastron?

**Any of the anomalies below is defined as orbital phase? Please specify reference papers or books.**

E: eccentric anomaly

M: mean anomaly

u: true anomaly

[![enter image description here](https://i.sstatic.net/P5bop.png)](https://i.sstatic.net/P5bop.png)

[Source](https://zh.wikipedia.org/zh-cn/%E7%9C%9F%E8%BF%91%E9%BB%9E%E8%A7%92)

**Tags:** binary-star, terminology, stellar-dynamics, eclipsing-star-systems
> **Comments:**
> **username_1**: I'm really not sure what you're asking. Have you read here https://en.wikipedia.org/wiki/Kepler_orbit#Development_of_the_laws ?  
> **username_0**: You mean `true anomaly` is what I want?  
> **username_1**: I don't know what you want, you've not made it clear at all.  
> **username_0**: I asked "How is phase defined and how is phase=0 defined".  
> **username_1**: What is "phase"? What is phase=0 mean? These are just words, you've provided no references and the image you added does not specify what you mean either. The poor quality of your question reflects how little effort you put into helping us understand it.  
> **username_0**: It is better now?  
> **username_1**: Yes, I think that's much better. Thank you. I gave +1  
> **username_2**: I still find it unclear whether you just want to know how phase zero is defined (body of the question) or how phase is calculated (title of the question).  
> **username_0**: @uhoh The link was attached in the old post. Do you have it? I can not find it now.  
> **username_0**: @ProfRob phase=o is included by the definition of phase.  
> **username_3**: @questionhang Got it, yes! It's there in the [edit history](https://astronomy.stackexchange.com/revisions/43695/1)  (click here to get that https://i.sstatic.net/0KjqM.png) I've got it thanks! I guess I was looking at the blue image so much I missed the blue text right next to it. *Thanks!*  

---

## 2 answers

### Answer by username_1 on 2021-05-07. Score: 3

> Usually, how is phase=0 defined? It is relative to the line of apses perhaps with phase=0 at periastron?

The true anomaly, $\theta$, is the angle between the current location of the orbiting particle and its location in the orbit at which it is closest to the central body (called the [periapsis](https://en.wikipedia.org/wiki/Apsis)/periastron). The word "phase" is used variously in physics and astronomy, but based on your question, you may mean $\theta$ since it is [defined](https://en.wikipedia.org/wiki/True_anomaly#From_the_eccentric_anomaly) as being zero at [periastron.](https://en.wikipedia.org/wiki/Kepler_orbit#Keplerian_elements)

> Are orbital phases of the black spot in the figure measured from some angle or time/period?

It ultimately depends on how you want to compute the trajectory/orbit of the particle (i.e., the black "spot" in the OP image) that is moving about a primary body (i.e., the star in the OP image). Alternate values can be used instead of $\theta$, for example $\displaystyle M$, the mean anomaly.

> **Comments:**
> **username_0**: The mean anomaly is proportional to time. The phase is also intended to be defined to be proportional to time and has nothing to do with eccentricity?  
> **username_1**: The true anomaly is not directly proportional to time. The true anomaly is related to the eccentric anomaly by $\cos\theta = (\cos E - e)/(1 - e\cos E)$.  

---

### Answer by username_4 on 2021-05-07. Score: 2

[Moreno et al.](https://arxiv.org/abs/1102.4301) in their paper *Eccentric binaries: Tidal flows and periastron events* [2011] define the orbital phase as from -0.5 to 0.5, where periastron is at phase 0, and apastron is at -0.5 and 0.5. Orbital phase here is $\phi = \dfrac{t}{P}$, where time $t=0$ at periastron and $P$ is the orbital period. This definition allows them to plot the behavior of several binary star systems with different periods on the same x-axis. Here is a figure from their paper:

[![enter image description here](https://i.sstatic.net/rjI07.png)](https://i.sstatic.net/rjI07.png)

We don't necessarily have to choose periastron as "phase=0." For example, as we observe an eclipsing binary star system, the [light curve](https://en.wikipedia.org/wiki/Light_curve) will appear to pulse. This is an effect due to the reduced overall system luminosity as one star passes in front of the other.

Here is an example from [Dan Bruton's website](http://www.physics.sfasu.edu/astro/ebstar/ebstar.html):

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

One can see that the length of 1 unit of phase is equal to the orbital period. Here, the deep dips in magnitude correspond to when the lower luminosity star passes in front of the more luminous star, and the more shallow dips correspond to when the higher luminosity star passes in front of the less luminous star. Since the light measurements are somewhat noisy, the technique of [folding](https://imagine.gsfc.nasa.gov/science/toolbox/timing2.html) is often used. To fold a light curve, one segments the data set along the integer phase boundaries and then adds all the light curves together, to increase the signal to noise ratio.

Here is a corresponding [wikipedia](https://en.wikipedia.org/wiki/Beta_Lyrae) gif of the Beta Lyrae system using the CHARA array:

[![enter image description here](https://i.sstatic.net/ze8R6.gif)](https://i.sstatic.net/ze8R6.gif)

The origin (phase=0) in the above example is chosen to coincide with the occultation of one star by the other. In general, the choice of origin will be made to highlight whatever feature the astronomer is interested in. The eccentricity of the orbit won't drive the choice of origin for this technique. Instead, the choice of origin is driven by the light curve itself, which is a function of both the orbital system and the location of the observer.

> **Comments:**
> **username_2**: This isn't (as I understand it) wholly what the question is asking about, the binary doesn't need to be eclipsing, and your answer doesn't address how phase is determined during the orbit of an eccentric binary. e.g. Where is phase 0.25?  
> **username_4**: @ProfRob Very insightful comment, as usual.  Edited.  
> **username_0**: @ConnorGarcia   t/p, you mean the phase is proportional to time even if eccentricity!=0?  
> **username_2**: I reverse my previous comment. This is the only way that a phase can be defined observationally, where you may not know the eccentricity.  

---

