The JADE code: Coupling secular exoplanetary dynamics and photo-evaporation

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Volume 647 (March 2021)

A&A, 647 (2021) A40

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Table A.1

Coefficients involved in the expression of the octupolar and hexadecapolar contributions of the perturber’s force.

Coefficient	Expression
α_e	$10 (1 - e^{2}) [{\hat{Q}}_{e} (Ê_{h} {\hat{Q}}_{q} + Ê_{q} {\hat{Q}}_{h}) + Ê_{e} (3 Ê_{q} Ê_{h} + {\hat{Q}}_{q} {\hat{Q}}_{h})]$ $10\left(1-e^2\right)\left[\hat{Q}_{\textrm{e}}\left((((hatE)))_{\textrm{h}}\hat{Q}_{\textrm{q}}+(((hatE)))_{\textrm{q}}\hat{Q}_{\textrm{h}}\right)+(((hatE)))_{\textrm{e}}\left(3(((hatE)))_{\textrm{q}}(((hatE)))_{\textrm{h}}+\hat{Q}_{\textrm{q}}\hat{Q}_{\textrm{h}}\right)\right]$
α_q	$- [Ê_{h} (- 1 + 8 e^{2} + (45 + 60 e^{2}) {\hat{H}}_{h}^{2} - 5 (2 + 5 e^{2}) {\hat{Q}}_{q}^{2}) + 5 (2 + 5 e^{2}) Ê_{q} (3 {\hat{H}}_{q} {\hat{H}}_{h} + {\hat{Q}}_{q} {\hat{Q}}_{h})]$ $-\left[(((hatE)))_{\textrm{h}}\left(-1+8e^2+\left(45+60e^2\right)\hat{H}_{\textrm{h}}^2-5\left(2+5e^2\right)\hat{Q}_{\textrm{q}}^2\right)+5\left(2+5e^2\right)(((hatE)))_{\textrm{q}}\left(3\hat{H}_{\textrm{q}}\hat{H}_{\textrm{h}}+\hat{Q}_{\textrm{q}}\hat{Q}_{\textrm{h}}\right)\right]$
α_h	$[- 10 (1 + 6 e^{2}) Ê_{h} {\hat{Q}}_{q} {\hat{Q}}_{h} + Ê_{q} (- 11 + 15 {\hat{H}}_{h}^{2} + 10 {\hat{Q}}_{h}^{2} + 3 e^{2} (- 29 + 35 {\hat{H}}_{q}^{2} + 30 {\hat{H}}_{h}^{2} + 20 {\hat{Q}}_{h}^{2}))]$ $\left[-10\left(1+6e^2\right)(((hatE)))_{\textrm{h}}\hat{Q}_{\textrm{q}}\hat{Q}_{\textrm{h}}+(((hatE)))_{\textrm{q}}\left(-11+15\hat{H}_{\textrm{h}}^2+10\hat{Q}_{\textrm{h}}^2+3e^2\left(-29+35\hat{H}_{\textrm{q}}^2+30\hat{H}_{\textrm{h}}^2+20\hat{Q}_{\textrm{h}}^2\right)\right)\right]$

β_e	$[10 (1 + 6 e^{2}) Ê_{h} {\hat{Q}}_{q} {\hat{Q}}_{h} - Ê_{q} (- 11 + 15 {\hat{H}}_{h}^{2} + 10 {\hat{Q}}_{h}^{2} + 3 e^{2} (- 29 + 35 {\hat{H}}_{q}^{2} + 30 {\hat{H}}_{h}^{2} + 20 {\hat{Q}}_{h}^{2}))]$ $\left[10\left(1+6e^2\right)(((hatE)))_{\textrm{h}}\hat{Q}_{\textrm{q}}\hat{Q}_{\textrm{h}}-(((hatE)))_{\textrm{q}}\left(-11+15\hat{H}_{\textrm{h}}^2+10\hat{Q}_{\textrm{h}}^2+3e^2\left(-29+35\hat{H}_{\textrm{q}}^2+30\hat{H}_{\textrm{h}}^2+20\hat{Q}_{\textrm{h}}^2\right)\right)\right]$
β_q	$[(- 10 + 30 e^{2}) Ê_{h} {\hat{Q}}_{e} {\hat{Q}}_{h} + Ê_{e} (- 11 + 15 {\hat{H}}_{h}^{2} + 10 {\hat{Q}}_{h}^{2} + 3 e^{2} (- 17 + 35 {\hat{H}}_{q}^{2} + 20 {\hat{H}}_{h}^{2} - 10 {\hat{Q}}_{h}^{2}))]$ $\left[\left(-10+30e^2\right)(((hatE)))_{\textrm{h}}\hat{Q}_{\textrm{e}}\hat{Q}_{\textrm{h}}+(((hatE)))_{\textrm{e}}\left(-11+15\hat{H}_{\textrm{h}}^2+10\hat{Q}_{\textrm{h}}^2+3e^2\left(-17+35\hat{H}_{\textrm{q}}^2+20\hat{H}_{\textrm{h}}^2-10\hat{Q}_{\textrm{h}}^2\right)\right)\right]$
β_h	$10 e^{2} [{\hat{Q}}_{e} (Ê_{h} {\hat{Q}}_{q} + Ê_{q} {\hat{Q}}_{h}) + Ê_{e} (3 Ê_{q} Ê_{h} + {\hat{Q}}_{q} {\hat{Q}}_{h})]$ $10e^2\left[\hat{Q}_{\textrm{e}}\left((((hatE)))_{\textrm{h}}\hat{Q}_{\textrm{q}}+(((hatE)))_{\textrm{q}}\hat{Q}_{\textrm{h}}\right)+(((hatE)))_{\textrm{e}}\left(3(((hatE)))_{\textrm{q}}(((hatE)))_{\textrm{h}}+\hat{Q}_{\textrm{q}}\hat{Q}_{\textrm{h}}\right)\right]$

γ_e	$(1 - e^{2}) [98 e^{2} e_{pert}^{2} {\hat{H}}_{q}^{2} {\hat{Q}}_{q} {\hat{Q}}_{h} + {\hat{H}}_{h} {\hat{H}}_{q} (e^{2} (e_{pert}^{2} (210 {\hat{H}}_{h}^{2} + 98 {\hat{Q}}_{q}^{2} + 84 {\hat{Q}}_{h}^{2} - 221) + 84 {\hat{H}}_{h}^{2} - 78)$ $\Big(1-e^2\Big)\Big[98e^2e_{\textrm{pert}}^2\hat{H}_{\textrm{q}}^2\hat{Q}_{\textrm{q}}\hat{Q}_{\textrm{h}}+\hat{H}_{\textrm{h}}\hat{H}_{\textrm{q}}\Big(e^2\Big(e_{\textrm{pert}}^2\Big(210\hat{H}_{\textrm{h}}^2+98\hat{Q}_{\textrm{q}}^2+84\hat{Q}_{\textrm{h}}^2-221\Big)+84\hat{H}_{\textrm{h}}^2-78\Big)$
	$+ e_{pert}^{2} (35 {\hat{H}}_{h}^{2} + 14 {\hat{Q}}_{h}^{2} - 17) + 14 {\hat{H}}_{h}^{2} - 6) + 2 e_{pert}^{2} {\hat{Q}}_{q} {\hat{Q}}_{h} ((42 e^{2} + 7) {\hat{H}}_{h}^{2} - 13 e^{2} - 1) + 49 e^{2} (5 e_{pert}^{2} + 2) {\hat{H}}_{h} {\hat{H}}_{q}^{3}]$ $+e_{\textrm{pert}}^2\Big(35\hat{H}_{\textrm{h}}^2+14\hat{Q}_{\textrm{h}}^2-17\Big)+14\hat{H}_{\textrm{h}}^2-6\Big)+2e_{\textrm{pert}}^2\hat{Q}_{\textrm{q}}\hat{Q}_{\textrm{h}}\Big(\Big(42e^2+7\Big)\hat{H}_{\textrm{h}}^2-13e^2-1\Big)+49e^2\Big(5e_{\textrm{pert}}^2+2\Big)\hat{H}_{\textrm{h}}\hat{H}_{\textrm{q}}^3\Big]$
γ_q	$- [{\hat{H}}_{e} (98 e^{2} (2 e^{2} + 1) e_{pert}^{2} {\hat{H}}_{h} {\hat{Q}}_{q}^{2} + 98 e^{2} (2 e^{2} + 1) e_{pert}^{2} {\hat{H}}_{q} {\hat{Q}}_{q} {\hat{Q}}_{h} + 49 e^{2} (2 e^{2} + 1) (5 e_{pert}^{2} + 2) {\hat{H}}_{q}^{2} {\hat{H}}_{h}$ $-\Big[\hat{H}_{\textrm{e}}\Big(98e^2\Big(2e^2+1\Big)e_{\textrm{pert}}^2\hat{H}_{\textrm{h}}\hat{Q}_{\textrm{q}}^2+98e^2\Big(2e^2+1\Big)e_{\textrm{pert}}^2\hat{H}_{\textrm{q}}\hat{Q}_{\textrm{q}}\hat{Q}_{\textrm{h}}+49e^2\Big(2e^2+1\Big)\Big(5e_{\textrm{pert}}^2+2\Big)\hat{H}_{\textrm{q}}^2\hat{H}_{\textrm{h}}$
	$+ 14 (22 e^{4} + 19 e^{2} + 1) e_{pert}^{2} {\hat{H}}_{h} {\hat{Q}}_{h}^{2} + 7 (8 e^{4} + 12 e^{2} + 1) (5 e_{pert}^{2} + 2) {\hat{H}}_{h}^{3} - {\hat{H}}_{h} (76 e^{4} + 86 e^{2}$ $+14\Big(22e^4+19e^2+1\Big)e_{\textrm{pert}}^2\hat{H}_{\textrm{h}}\hat{Q}_{\textrm{h}}^2+7\Big(8e^4+12e^2+1\Big)\Big(5e_{\textrm{pert}}^2+2\Big)\hat{H}_{\textrm{h}}^3-\hat{H}_{\textrm{h}}\Big(76e^4+86e^2$
	$+ (346 e^{4} + 309 e^{2} + 17) e_{pert}^{2} + 6)) - 2 e_{pert}^{2} {\hat{Q}}_{e} {\hat{Q}}_{h} (- 20 e^{4} - 2 e^{2} + 7 (6 e^{4} - 5 e^{2} - 1) {\hat{H}}_{h}^{2} + 1)]$ $+\Big(346e^4+309e^2+17\Big)e_{\textrm{pert}}^2+6\Big)\Big)-2e_{\textrm{pert}}^2\hat{Q}_{\textrm{e}}\hat{Q}_{\textrm{h}}\Big(-20e^4-2e^2+7\Big(6e^4-5e^2-1\Big)\hat{H}_{\textrm{h}}^2+1\Big)\Big]$
γ_h	$- 7 e^{2} [2 e_{pert}^{2} {\hat{Q}}_{e} ((7 - 7 e^{2}) {\hat{H}}_{q} {\hat{H}}_{h} {\hat{Q}}_{h} + (e^{2} + 2) {\hat{Q}}_{q}) + {\hat{H}}_{e} ({\hat{H}}_{q} (e^{2} (e_{pert}^{2} (70 {\hat{H}}_{h}^{2} + 84 {\hat{Q}}_{q}^{2} + 42 {\hat{Q}}_{h}^{2} - 95) + 28 {\hat{H}}_{h}^{2} - 22)$ $-7e^2\Big[2e_{\textrm{pert}}^2\hat{Q}_{\textrm{e}}\Big(\Big(7-7e^2\Big)\hat{H}_{\textrm{q}}\hat{H}_{\textrm{h}}\hat{Q}_{\textrm{h}}+\Big(e^2+2\Big)\hat{Q}_{\textrm{q}}\Big)+\hat{H}_{\textrm{e}}\Big(\hat{H}_{\textrm{q}}\Big(e^2\Big(e_{\textrm{pert}}^2\Big(70\hat{H}_{\textrm{h}}^2+84\hat{Q}_{\textrm{q}}^2+42\hat{Q}_{\textrm{h}}^2-95\Big)+28\hat{H}_{\textrm{h}}^2-22\Big)$
	$+ 7 (5 e_{pert}^{2} + 2) {\hat{H}}_{h}^{2} - e_{pert}^{2} - 2) + 14 (2 e^{2} + 1) e_{pert}^{2} {\hat{H}}_{h} {\hat{Q}}_{q} {\hat{Q}}_{h} + 21 e^{2} (5 e_{pert}^{2} + 2) {\hat{H}}_{q}^{3})]$ $+7\Big(5e_{\textrm{pert}}^2+2\Big)\hat{H}_{\textrm{h}}^2-e_{\textrm{pert}}^2-2\Big)+14\Big(2e^2+1\Big)e_{\textrm{pert}}^2\hat{H}_{\textrm{h}}\hat{Q}_{\textrm{q}}\hat{Q}_{\textrm{h}}+21e^2\Big(5e_{\textrm{pert}}^2+2\Big)\hat{H}_{\textrm{q}}^3\Big)\Big]$

δ_e	$7 [2 e_{pert}^{2} {\hat{Q}}_{e} ((7 - 7 e^{2}) {\hat{H}}_{q} {\hat{H}}_{h} {\hat{Q}}_{h} + (e^{2} + 2) {\hat{Q}}_{q}) + {\hat{H}}_{e} ({\hat{H}}_{q} (e^{2} (e_{pert}^{2} (70 {\hat{H}}_{h}^{2} + 84 {\hat{Q}}_{q}^{2} + 42 {\hat{Q}}_{h}^{2} - 95) + 28 {\hat{H}}_{h}^{2} - 22)$ $7\Big[2e_{\textrm{pert}}^2\hat{Q}_{\textrm{e}}\Big(\Big(7-7e^2\Big)\hat{H}_{\textrm{q}}\hat{H}_{\textrm{h}}\hat{Q}_{\textrm{h}}+\Big(e^2+2\Big)\hat{Q}_{\textrm{q}}\Big)+\hat{H}_{\textrm{e}}\Big(\hat{H}_{\textrm{q}}\Big(e^2\Big(e_{\textrm{pert}}^2\Big(70\hat{H}_{\textrm{h}}^2+84\hat{Q}_{\textrm{q}}^2+42\hat{Q}_{\textrm{h}}^2-95\Big)+28\hat{H}_{\textrm{h}}^2-22\Big)$
	$+ 7 (5 e_{pert}^{2} + 2) {\hat{H}}_{h}^{2} - e_{pert}^{2} - 2) + 14 (2 e^{2} + 1) e_{pert}^{2} {\hat{H}}_{h} {\hat{Q}}_{q} {\hat{Q}}_{h} + 21 e^{2} (5 e_{pert}^{2} + 2) {\hat{H}}_{q}^{3})]$ $+7\Big(5e_{\textrm{pert}}^2+2\Big)\hat{H}_{\textrm{h}}^2-e_{\textrm{pert}}^2-2\Big)+14\Big(2e^2+1\Big)e_{\textrm{pert}}^2\hat{H}_{\textrm{h}}\hat{Q}_{\textrm{q}}\hat{Q}_{\textrm{h}}+21e^2\Big(5e_{\textrm{pert}}^2+2\Big)\hat{H}_{\textrm{q}}^3\Big)\Big]$
δ_q	$- [588 e^{2} e_{pert}^{2} {\hat{H}}_{q}^{2} {\hat{Q}}_{q}^{2} + 196 (4 e^{2} + 1) e_{pert}^{2} {\hat{H}}_{h} {\hat{H}}_{q} {\hat{Q}}_{q} {\hat{Q}}_{h} + 56 (4 e^{2} + 3) e_{pert}^{2} {\hat{H}}_{h}^{2} {\hat{Q}}_{h}^{2} + 147 e^{2} (5 e_{pert}^{2} + 2) {\hat{H}}_{q}^{4}$ $-\Big[588e^2e_{\textrm{pert}}^2\hat{H}_{\textrm{q}}^2\hat{Q}_{\textrm{q}}^2+196\Big(4e^2+1\Big)e_{\textrm{pert}}^2\hat{H}_{\textrm{h}}\hat{H}_{\textrm{q}}\hat{Q}_{\textrm{q}}\hat{Q}_{\textrm{h}}+56\Big(4e^2+3\Big)e_{\textrm{pert}}^2\hat{H}_{\textrm{h}}^2\hat{Q}_{\textrm{h}}^2+147e^2\Big(5e_{\textrm{pert}}^2+2\Big)\hat{H}_{\textrm{q}}^4$
	$+ 49 (4 e^{2} + 1) (5 e_{pert}^{2} + 2) {\hat{H}}_{h}^{2} {\hat{H}}_{q}^{2} - 7 {\hat{H}}_{q}^{2} (2 e^{2} (53 e_{pert}^{2} + 22) + e_{pert}^{2} + 2) + 14 (4 e^{2} + 3) (5 e_{pert}^{2} + 2) {\hat{H}}_{h}^{4}$ $+49\Big(4e^2+1\Big)\Big(5e_{\textrm{pert}}^2+2\Big)\hat{H}_{\textrm{h}}^2\hat{H}_{\textrm{q}}^2-7\hat{H}_{\textrm{q}}^2\Big(2e^2\Big(53e_{\textrm{pert}}^2+22\Big)+e_{\textrm{pert}}^2+2\Big)+14\Big(4e^2+3\Big)\Big(5e_{\textrm{pert}}^2+2\Big)\hat{H}_{\textrm{h}}^4$
	$- {\hat{H}}_{h}^{2} (4 e^{2} (75 e_{pert}^{2} + 38) + 211 e_{pert}^{2} + 86) + 28 (e^{2} + 1) e_{pert}^{2} {\hat{Q}}_{q}^{2} + 4 (20 e^{2} + 1) e_{pert}^{2} {\hat{Q}}_{h}^{2} + 15 e^{2} e_{pert}^{2} + 46 e^{2} - e_{pert}^{2} + 10]$ $-\hat{H}_{\textrm{h}}^2\Big(4e^2\Big(75e_{\textrm{pert}}^2+38\Big)+211e_{\textrm{pert}}^2+86\Big)+28\Big(e^2+1\Big)e_{\textrm{pert}}^2\hat{Q}_{\textrm{q}}^2+4\Big(20e^2+1\Big)e_{\textrm{pert}}^2\hat{Q}_{\textrm{h}}^2+15e^2e_{\textrm{pert}}^2+46e^2-e_{\textrm{pert}}^2+10\Big]$
δ_h	$- [98 e^{2} e_{pert}^{2} {\hat{H}}_{q}^{2} {\hat{Q}}_{q} {\hat{Q}}_{h} + {\hat{H}}_{h} {\hat{H}}_{q} (e^{2} (e_{pert}^{2} (210 {\hat{H}}_{h}^{2} + 98 {\hat{Q}}_{q}^{2} + 84 {\hat{Q}}_{h}^{2} - 221) + 84 {\hat{H}}_{h}^{2} - 78) + e_{pert}^{2} (35 {\hat{H}}_{h}^{2} + 14 {\hat{Q}}_{h}^{2} - 17)$ $-\Big[98e^2e_{\textrm{pert}}^2\hat{H}_{\textrm{q}}^2\hat{Q}_{\textrm{q}}\hat{Q}_{\textrm{h}}+\hat{H}_{\textrm{h}}\hat{H}_{\textrm{q}}\Big(e^2\Big(e_{\textrm{pert}}^2\Big(210\hat{H}_{\textrm{h}}^2+98\hat{Q}_{\textrm{q}}^2+84\hat{Q}_{\textrm{h}}^2-221\Big)+84\hat{H}_{\textrm{h}}^2-78\Big)+e_{\textrm{pert}}^2\Big(35\hat{H}_{\textrm{h}}^2+14\hat{Q}_{\textrm{h}}^2-17\Big)$
	$+ 14 {\hat{H}}_{h}^{2} - 6) + 2 e_{pert}^{2} {\hat{Q}}_{q} {\hat{Q}}_{h} ((42 e^{2} + 7) {\hat{H}}_{h}^{2} - 13 e^{2} - 1) + 49 e^{2} (5 e_{pert}^{2} + 2) {\hat{H}}_{h} {\hat{H}}_{q}^{3}]$ $+14\hat{H}_{\textrm{h}}^2-6\Big)+2e_{\textrm{pert}}^2\hat{Q}_{\textrm{q}}\hat{Q}_{\textrm{h}}\Big(\Big(42e^2+7\Big)\hat{H}_{\textrm{h}}^2-13e^2-1\Big)+49e^2\Big(5e_{\textrm{pert}}^2+2\Big)\hat{H}_{\textrm{h}}\hat{H}_{\textrm{q}}^3\Big]$

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