<p>This work deals about binary mergers, i.e. two stars with (different) gravitational masses <span class="math-container">$M_1$</span> and <span class="math-container">$M_2$</span> and accordingly two different compactness <span class="math-container">$C_1$</span> and <span class="math-container">$C_2$</span>, see below Eq.(4) for more details.</p>
<p>The meaning of the arrow in Eq.(6) is simply copy the complete previous (left) expression and replace any index 1 by 2 and vice versa. Thus, the second summand reads written out in detail
<span class="math-container">$$
\ldots + \left( \frac{a}{C_2} + b \frac{M^n_1}{M^n_2} +cC_2\right) M_2
$$</span></p>

CC BY-SA 4.0
2023-03-16T22:12:33.123Z
2023-03-16T22:12:33.123Z
Bernd
<p>I was reading <a href="https://arxiv.org/pdf/2002.07728.pdf" rel="noreferrer">this paper</a> on the ejecta mass of compact binary mergers and came across this notation in equation (6):</p>
<p><span class="math-container">$$\frac{M_\text{dyn}}{10^{-3}M_\odot}=\left(\frac{a}{C_1}+b\frac{M_2^n}{M_1^n}+cC_1\right)M_1+(1\leftrightarrow 2)$$</span></p>
<p>which is an equation of the dynamical ejecta mass of a binary neutron star merger in terms of the mass and compactness of the two neutron stars with coefficients <span class="math-container">$a,b,c,d$</span> fitted to simulations. My question is what does the notation <span class="math-container">$(1 \leftrightarrow 2)$</span> mean in this context?</p>

CC BY-SA 4.0
2023-02-28T17:34:38.200Z
Could be a sloppy way of writing "a number between 1 and 2". Do they reference other papers where this is coming from?
AtmosphericPrisonEscape
CC BY-SA 4.0
2023-02-28T19:05:59.837Z
@AtmosphericPrisonEscape [This](http://example.com) is the paper being referenced but I don't think it contains a clarification either.
hikari30
CC BY-SA 4.0
2023-03-01T03:47:07.583Z
@AtmosphericPrisonEscape Sorry, I just realized I did not include the actual link. Here's the link to the paper: https://arxiv.org/pdf/1612.03665.pdf
hikari30
CC BY-SA 4.0
2023-03-02T00:49:28.800Z
You cross-posted this on Physics SE, where you also got an answer.  Could you please close this question so it is not left as an unanswered question on the system.  In general cross-posting is discouraged.
StephenG - Help Ukraine
CC BY-SA 4.0
2023-03-02T05:10:29.117Z
@StephenG-HelpUkraine Will do, sorry about that. In case someone with the same confusion finds this question in the future, here's the link to the answer: https://physics.stackexchange.com/questions/741010/notation-in-zees-quantum-field-theory

In the case of this equation, you would add the term again but with the $1$-indices replaced with $2$'s.
hikari30
CC BY-SA 4.0
2023-03-02T05:17:56.953Z
I don't think I have the permission to close a question so I guess I will leave it here until a moderator closes it.
hikari30
CC BY-SA 4.0
2023-03-02T11:43:56.173Z
I'd suggest you copy the answer from Physics SE and put it here.  You can accept than answer as well and it will remove an unanswered question from the system.  In case you're wondering there is no problem with self-answering questions on SE.
StephenG - Help Ukraine
<binary-star><neutron-star>
Meaning of Bidirectional Arrow in a Fitting Formula
hikari30
# Meaning of Bidirectional Arrow in a Fitting Formula

(Asked by: username_0 on 2023-02-28)

I was reading [this paper](https://arxiv.org/pdf/2002.07728.pdf) on the ejecta mass of compact binary mergers and came across this notation in equation (6):

$$\frac{M\_\text{dyn}}{10^{-3}M\_\odot}=\left(\frac{a}{C\_1}+b\frac{M\_2^n}{M\_1^n}+cC\_1\right)M\_1+(1\leftrightarrow 2)$$

which is an equation of the dynamical ejecta mass of a binary neutron star merger in terms of the mass and compactness of the two neutron stars with coefficients $a,b,c,d$ fitted to simulations. My question is what does the notation $(1 \leftrightarrow 2)$ mean in this context?

**Tags:** binary-star, neutron-star
> **Comments:**
> **username_1**: Could be a sloppy way of writing "a number between 1 and 2". Do they reference other papers where this is coming from?  
> **username_0**: @AtmosphericPrisonEscape [This](http://example.com) is the paper being referenced but I don't think it contains a clarification either.  
> **username_0**: @AtmosphericPrisonEscape Sorry, I just realized I did not include the actual link. Here's the link to the paper: https://arxiv.org/pdf/1612.03665.pdf  
> **username_2**: You cross-posted this on Physics SE, where you also got an answer.  Could you please close this question so it is not left as an unanswered question on the system.  In general cross-posting is discouraged.  
> **username_0**: @StephenG-HelpUkraine Will do, sorry about that. In case someone with the same confusion finds this question in the future, here's the link to the answer: https://physics.stackexchange.com/questions/741010/notation-in-zees-quantum-field-theory  In the case of this equation, you would add the term again but with the $1$-indices replaced with $2$'s.  
> **username_0**: I don't think I have the permission to close a question so I guess I will leave it here until a moderator closes it.  
> **username_2**: I'd suggest you copy the answer from Physics SE and put it here.  You can accept than answer as well and it will remove an unanswered question from the system.  In case you're wondering there is no problem with self-answering questions on SE.  

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## 1 answers

### Answer by username_3 on 2023-03-16. Score: 2

This work deals about binary mergers, i.e. two stars with (different) gravitational masses $M\_1$ and $M\_2$ and accordingly two different compactness $C\_1$ and $C\_2$, see below Eq.(4) for more details.

The meaning of the arrow in Eq.(6) is simply copy the complete previous (left) expression and replace any index 1 by 2 and vice versa. Thus, the second summand reads written out in detail
$$
\ldots + \left( \frac{a}{C\_2} + b \frac{M^n\_1}{M^n\_2} +cC\_2\right) M\_2
$$


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