Execution of novel explicit RKARMS(4,4) technique in determining initial configurations of extra-solar protoplanets formed by disk instability

Peer review under responsibility of National Research Institute of Astronomy and Geophysics.

Figures & data

Figure 1 Initial temperature–pressure profiles of some protoplanets.

Figure 2 Density distribution inside some initial protoplanets.

Figure 3 Error estimation of the proposed method through p, q and t with references to the protoplanets with masses $1M_J$ and $10M_J$; (a) for $1M_J$ and (b) for $10M_J$.

Table 1 Comparison of LTE, GTE, and error estimation for RKARMS(4,4) method.

RK-embedded algorithm	Local truncation error (LTE)	Global truncation error (GTE)	Error estimation (ERREST)
Explicit RK-embedded arithmetic root mean square method	${\mathit{LTE}}_{\mathit{AM}}-{\mathit{LTE}}_{\mathit{RMS}}⩽\frac{6457}{184320}{P}^4{\mathit{Qh}}^5=\left|{\mathit{LTE}}_{\mathit{AM}}-{\mathit{LTE}}_{\mathit{RMS}}\right|⩽\frac{6457}{184320}{P}^4{\mathit{Qh}}^5$	$\left|{\epsilon }_n\right|⩽\left(\frac{h^4}{164320\mathit{LD}}\right)\times M\left(e^{\mathit{DL}(x_n-x_0)}-1\right)$	ERREST = $\left|y_{\mathit{AM}}-y_{\mathit{RMS}}\right|\times \frac{6457}{184320}$

Table 2 Comparative distribution of thermodynamic variables inside a 1 Jupiter mass protoplanet.

$r/R$	Classical Runge–Kutta 4th order method			New explicit RKARMS(4,4) method
	P (dynes ${\text{cm}}^{-2}$)	T (K)	ρ (g cm^−3)	P (dynes ${\text{cm}}^{-2}$)	T (K)	ρ (g cm^−3)
0.99	$3.8954527\times {10}^{-07}$	$1.6102493\times {10}^{00}$	$6.4512556\times {10}^{-15}$	$3.8954527\times {10}^{-07}$	$1.6102493\times {10}^{00}$	$6.4512556\times {10}^{-15}$
0.90	$5.4656957\times {10}^{-03}$	$1.7901342\times {10}^{01}$	$8.1421533\times {10}^{-12}$	$5.3980124\times {10}^{-03}$	$1.7845786\times {10}^{01}$	$8.2236662\times {10}^{-12}$
0.80	$1.3290122\times {10}^{-01}$	$4.0709788\times {10}^{01}$	$8.7058176\times {10}^{-11}$	$1.3185159\times {10}^{-01}$	$4.0629732\times {10}^{01}$	$8.6930245\times {10}^{-11}$
0.70	$1.0706637\times {10}^{00}$	$7.0401377\times {10}^{01}$	$4.0555650\times {10}^{-10}$	$1.0644930\times {10}^{00}$	$7.0301521\times {10}^{01}$	$4.0473810\times {10}^{-10}$
0.60	$5.6569018\times {10}^{00}$	$1.0969770\times {10}^{02}$	$1.3751831\times {10}^{-09}$	$5.6320485\times {10}^{00}$	$1.0958258\times {10}^{02}$	$1.3770036\times {10}^{-09}$
0.50	$2.4262787\times {10}^{01}$	$1.6205682\times {10}^{02}$	$3.9925723\times {10}^{-09}$	$2.4183268\times {10}^{01}$	$1.6194018\times {10}^{02}$	$4.0165598\times {10}^{-09}$
0.40	$9.2395087\times {10}^{01}$	$2.3108465\times {10}^{02}$	$1.0662455\times {10}^{-08}$	$9.2199260\times {10}^{01}$	$2.3100654\times {10}^{02}$	$1.0713107\times {10}^{-08}$
0.30	$3.2328971\times {10}^{02}$	$3.1822282\times {10}^{02}$	$2.7091933\times {10}^{-08}$	$3.2309502\times {10}^{02}$	$3.1828741\times {10}^{02}$	$2.7168903\times {10}^{-08}$
0.20	$1.0297073\times {10}^{03}$	$4.1622901\times {10}^{02}$	$6.5972184\times {10}^{-08}$	$1.0316243\times {10}^{03}$	$4.1668254\times {10}^{02}$	$6.6508432\times {10}^{-08}$
0.10	$2.7969290\times {10}^{03}$	$4.9796902\times {10}^{02}$	$1.4978163\times {10}^{-07}$	$2.8170636\times {10}^{03}$	$4.9933180\times {10}^{02}$	$1.5070511\times {10}^{-07}$
0.01	$1.0954682\times {10}^{04}$	$6.1391756\times {10}^{02}$	$4.7584900\times {10}^{-07}$	$1.1489201\times {10}^{04}$	$6.2424341\times {10}^{02}$	$4.9146509\times {10}^{-07}$