On the scaling and spacing of extra-solar multi-planet systems

Abstract

We investigate whether certain extra-solar multi-planet systems simultaneously follow the scaling and spacing rules of the angular-momentum-deficit model. The masses and semi-major axes of exoplanets in ten multi-planet systems are considered. It is found that GJ 667C, HD 215152, HD 40307, and Kepler-79 systems are currently close to configurations of the angular-momentum-deficit model. In a gas-poor scenario, GJ 3293, HD 141399, and HD 34445 systems are those which had a configuration of the angular-momentum-deficit model in the past and get scattered away due to post gaseous effects. In addition, no matter in gas-free or gas-poor scenario, 55 Cnc, GJ 876, and WASP-47 systems do not follow the angular-momentum-deficit model. Therefore, our results reveal important formation histories of these multi-planet systems.

Appendices

Appendix A

The data of ten systems are listed here (Table 4), where $m$ is the planetary minimum mass with lower error bound $m_{-}$ and upper error bound $m_{+}$; $a$ is the planetary semi-major axis with lower error bound $a_{-}$ and upper error bound $a_{+}$.

Table 4 The planetary data

Appendix B

(1)

Let the parameter $p=X^{+x_{+}}_{-x_{-}}$, i.e. a value $X$ with upper error bound $x_{+}$ and lower error bound $x_{-}$; the parameter $q=Y^{+y_{+}}_{-y_{-}}$, i.e. a value $Y$ with upper error bound $y_{+}$ and lower error bound $y_{-}$. In order to determine the upper and lower error bounds of the parameter ratio $p/q$, we need the expression that when $|s| << 1$, ${1}/{(1+s)} \approx1-s$. From below

$$\frac{p}{q}=\frac{X^{+x_{+}}_{-x_{-}}}{Y^{+y_{+}}_{-y_{-}}}, $$

we know the maximum is

$$\begin{aligned} \frac{p}{q} =&\frac{X+x_{+}}{Y-y_{-}} =\frac{X}{Y}\frac{\left(1+\frac{x_{+}}{X}\right)}{\left(1-\frac{y_{-}}{Y}\right)} \approx\frac{X}{Y}\left(1+\frac{x_{+}}{X} \right) \left(1+\frac{y_{-}}{Y}\right) \\ \approx&\frac{X}{Y}\left(1+\frac{x_{+}}{X}+\frac{y_{-}}{Y}\right), \end{aligned}$$

(8)

and the upper error bound is

$$ \frac{X}{Y}\left(\frac{x_{+}}{X}+\frac{y_{-}}{Y}\right). $$

(9)

Similarly, the minimum is

$$\begin{aligned} \frac{p}{q} =&\frac{X-x_{-}}{Y+y_{+}} =\frac{X}{Y}\frac{\left(1-\frac{x_{-}}{X}\right)}{\left(1+\frac{y_{+}}{Y}\right)} \approx\frac{X}{Y}\left(1-\frac{x_{-}}{X} \right) \left(1-\frac{y_{+}}{Y}\right) \\ \approx&\frac{X}{Y}\left(1-\frac{x_{-}}{X}-\frac{y_{+}}{Y}\right), \end{aligned}$$

(10)

and the lower error bound is

$$ \frac{X}{Y}\left(\frac{x_{-}}{X}+\frac{y_{+}}{Y}\right). $$

(11)

(2)

Given that the value of $x$ has an upper error bound $x_{+}$ and a lower error bound $x_{-}$, the upper and lower error bounds of a general function $f(x)$ can be determined through Taylor series. When $x_{+}$ and $x_{-}$ are small, using $\Delta x$ to denote $x_{+}$ or $-x_{-}$, we have

$$ f(x+\Delta x)\approx f(x)+f'(x)\Delta x +{\rm O}((\Delta x)^{2}). $$

(12)

The term $f'(x)\Delta x$ is thus used as the error estimation. It equals to $f'(x)x_{+}$ or $-f'(x)x_{-}$. When $f'(x)$ is positive, the upper error bound of $f(x)$ is $f'(x)x_{+}$, and the lower error bound of $f(x)$ is $f'(x)x_{-}$. When $f'(x)$ is negative, the upper error bound of $f(x)$ is $-f'(x)x_{-}$, and the lower error bound of $f(x)$ is $-f'(x)x_{+}$. For example, when the function $f(x)=\ln(x)$, $f'(x)>0$, the upper and lower error bounds are ${x_{+}}/{x}$ and ${x_{-}}/{x}$.

(3)

Given that the value of $x$ has an upper error bound $x_{+}$ and a lower error bound $x_{-}$, the value of $y$ has an upper error bound $y_{+}$ and a lower error bound $y_{-}$, the upper and lower error bounds of a general function $f(x,y)$, where $x$ and $y$ are two independent variables, can be determined through Taylor series. When $x_{+}$, $x_{-}$, $y_{+}$, and $y_{-}$ are small, using $\Delta x$ to denote $x_{+}$ or $-x_{-}$ and $\Delta y$ to denote $y_{+}$ or $-y_{-}$, we have

$$\begin{aligned} & f(x+\Delta x, y+\Delta y) \\ & \quad \approx f(x,y) +\frac{\partial f(x,y)}{\partial x}\Delta x \\ &\qquad {} +\frac{\partial f(x,y)}{\partial y}\Delta y +\mathrm{O}( (\Delta x)^{2}, (\Delta y)^{2}, (\Delta x)(\Delta y) ). \end{aligned}$$

(13)

The summation of below two terms, i.e.

$$ \frac{\partial f(x,y)}{\partial x}\Delta x +\frac{\partial f(x,y)}{\partial y}\Delta y, $$

(14)

is thus used as the error estimation. Its four possible values can be determined after $\Delta x$ is substituted by $x_{+}$ or $-x_{-}$, and $\Delta y$ is substituted by $y_{+}$ or $-y_{-}$. The maximum value would be positive, and the minimum value would be negative. The upper error bound of $f(x,y)$ is then equal to this maximum, and the lower error bound of $f(x,y)$ is the absolute value of the minimum.