Revisiting the pulsational characteristics of the exoplanet host star $\beta$ Pictoris

TL;DR

This study revisits the pulsational properties of the exoplanet host star beta Pictoris using multi-passband photometry and spectropolarimetry, identifying pulsation modes and constraining stellar parameters with asteroseismology.

Contribution

It provides a detailed mode identification and stellar parameter estimation for beta Pictoris, incorporating multi-instrument data and 2D rotating models, and confirms the star's non-magnetic nature.

Findings

Identified 15 pulsation frequencies between 34 and 55 c/d.

Mode identification includes 3 l=1, 6 l=2, and 6 l=3 modes.

Determined stellar parameters: radius ~1.5 R_sun, mass ~1.8 M_sun, rotation rate 27% of break-up velocity.

Abstract

Exoplanet properties crucially depend on their host stars' parameters. In case the exoplanet host star shows pulsations, asteroseismology can be used for an improved description of the stellar parameters. We aim to revisit the pulsational properties of beta Pic and identify its pulsation modes from normalised amplitudes in five different passbands. We also investigate the potential presence of a magnetic field. We conduct a frequency analysis using three seasons of BRITE-Constellation observations in the BRITE blue and red filters, the ~620-day long bRing light curve and the nearly 8-year long SMEI photometric time series. We calculate normalised amplitudes using all passbands including previously published values obtained from ASTEP observations. We investigate the magnetic properties of beta Pic using spectropolarimetric observations conducted with the HARPSpol instrument. Using 2D rotating models, we fit the normalised amplitudes and frequencies through Monte Carlo Markov Chains. We identify 15 pulsation frequencies in the range from 34 to 55c/d, where two display clear amplitude variability. We use the normalised amplitudes in up to five passbands to identify the modes as three l = 1, six l = 2 and six l = 3 modes. beta Pic is shown to be non-magnetic with an upper limit of the possible undetected dipolar field of 300G. Multiple fits to the frequencies and normalised amplitudes are obtained including one with a near equator-on inclination for beta Pic, which corresponds to our expectations based on the orbital inclination of beta Pic b and the orientation of the circumstellar disk. This solution leads to a rotation rate of 27% of the Keplerian break-up velocity, a radius of 1.497+-0.025Rsun, and a mass of 1.797+-0.035Msun. The ~2% errors in radius and mass do not account for uncertainties in the models and a potentially erroneous mode-identification.

Figures

Tables

Table 1. Table 1: Properties of the multi-colour observations for β 𝛽 \beta Pictoris.

Obs ID	wavelength	obsstart	obsend	time base	1/T	N	res. noise	$f_{\mathrm{Nyquist}}$
	[nm]	[d]	[d]	[d]	[ d-1]	#	[ppm]	[ d-1]
BHr	550 – 700	16 March 2015	2 June 2015	78.323	0.013	44236	100	4181
BLb	390 – 460	15 Dec. 2016	21 June 2017	187.923	0.005	74306	170	2130
BTr	550 – 700	4 Nov. 2016	17 June 2017	224.573	0.004	53620	47	2089
BHr	550 – 700	7 Jan. 2017	30 Jan. 2017	23.722	0.042	13958	$\ast$	2130
BTr + BHr	550 – 700	4 Nov. 2016	17 June 2017	224.573	0.004	67578	40	2146
BHr	550 – 700	9 Nov. 2017	25 April 2018	167.335	0.006	53262	43	2127
SMEI	450 – 950	6 Feb. 2003	30 Dec. 2010	2884.609	0.0003	28623	92	7
bRing	463 – 639	2 Feb. 2017	16 Oct. 2018	620.249	0.002	68126	110	135

Table 2. Table 2: Pulsation frequencies, amplitudes, phases and signal-to-noise values derived from the BRITE-Constellation data sorted by increasing frequency and comparison to the literature.

No.	Frequency		$A_R$ 2015	$A_R$ 2016	$A_R$ 2017	${\varphi }_R$ 2015	${\varphi }_R$ 2016	${\varphi }_R$ 2017	S/NR 2015	S/NR 2016	S/NR 2017	cross ID
#	[ d-1]	[$\mu$Hz]	[mmag]	[mmag]	[mmag]
F1	34.4342(9)	398.543(11)	0.39(5)		0.27(3)	0.61(2)		0.37(2)	3.83		6.34	close to f10 in DM17
F2	38.1293(4)	441.311(4)		0.17(3)			0.38(2)			4.2		f11 in DM17
F3	39.0629(1)	452.117(1)		0.53(3)	0.49(3)		0.059(8)	0.926(9)		11.1	10.82	f5 in DM17
F4	43.5268(4)	503.783(4)		0.19(3)	0.18(3)		0.67(2)	0.83(2)		4.3	4.1	f9 in DM17
F5	45.2705(4)	523.964(4)		0.20(3)			0.48(2)			4.5		f12 in DM17
F6	46.5428(2)	538.690(2)		0.42(3)	0.48(3)		0.22(1)	0.515(9)		9.5	10.56	f6 in DM17
F7	47.2831(2)	547.258(2)		0.39(3)			0.73(1)			5.1		f7 in DM17, f3 in CK03b
F8	47.43924(5)	549.0653(6)	1.38(5)	1.45(3)	1.24(3)	0.786(6)	0.436(3)	0.816(4)	9.05	20.7	24.2	f1 in DM17, f1 in CK03b
F9	48.9185(3)	566.186(4)		0.22(3)			0.38(2)			5.3		f8 in DM17
F10	49.4161(4)	571.946(5)		0.17(3)			0.43(3)			4.3
F11	50.49210(8)	584.399(1)	0.55(5)	0.84(3)	1.06(3)	0.07(2)	0.022(5)	0.145(4)	5.33	19.9	23.9	f2 in DM17
F12	50.8312(4)	588.324(5)		0.15(3)			0.24(3)			3.9		f14 in DM17
F13	53.6917(2)	621.431(2)	1.12(5)	0.40(3)	1.27(3)	0.797(8)	0.75(1)	0.534(4)	10.43	9.5	16.38	f3 in DM17
F14	53.8915(6)	623.744(7)	0.64(5)			0.11(1)			6.55			f23 in DM17
F15	54.2372(2)	627.745(2)	0.40(5)	0.44(3)	0.51(3)	0.15(2)	0.63(1)	0.002(9)	4.55	10.3	10.08	f4 in DM17
No.	Frequency			$A_B$ 2016			${\varphi }_B$ 2016			S/NB 2016		cross ID
#	[ d-1]	[$\mu$Hz]		[mmag]
F3	39.0629(1)	452.117(1)		0.75(11)			0.47(2)			4.4		f5 in DM17
F4	43.5268(4)	503.783(4)		1.00(11)			0.67(2)			4.7		f9 in DM17
F8	47.43924(5)	549.0653(6)		2.30(11)			0.749(8)			13.4		f1 in DM17, f1 in CK03b
F11	50.49210(8)	584.399(1)		0.95(11)			0.69(2)			5.8		f2 in DM17

Table 3. Table 3: Pulsation frequencies, amplitudes, phases and signal-to-noise values derived from the SMEI and bRing data sorted by increasing frequency and comparison to the literature.

BRITE No.	Frequency	$A_{\mathrm{SMEI}}$	$A_{\mathrm{bRing}}$	${\varphi }_{\mathrm{SMEI}}$	${\varphi }_{\mathrm{bRing}}$	S/NSMEI	S/NbRing	cross ID
[ d-1]	[ d-1]		[mmag]
F1	34.39059(4)	0.36(8)	–	0.0(2)	–	4.1	–	close to f10 in DM17
F3	39.06307(3)	0.55(8)	0.8(2)	$-$0.3(1)	0.32(4)	5.5	5.1	f5 in DM17
Fc	44.68351(2)	0.67(8)	–	$-$0.1(1)	–	6.8	–	f13 in DM17
F6	46.5428(4)	–	0.5(2)	–	0.13(4)	–	5.8	f6 in DM17
F7	47.28348(4)	0.43(8)	–	$-$0.3(2)	–	4.5	–	f7 in DM17
F8	47.43920(2)	0.76(8)	1.1(2)	$-$0.4(1)	0.05(3)	7.8	6.9	f1 in DM17
F11	50.49182(4)	0.36(8)	0.8(2)	0.4(2)	0.23(3)	3.9	5.3	f2 in DM17
F13	53.67090(3)	0.45(8)	1.1(2)	0.1(1)	0.69(3)	4.0	6.5	f3 in DM17
F15	54.2372(3)	–	0.6(2)	–	0.94(4)	–	6.8	f4 in DM17

Table 4. Table 4: Correspondence between photometric bands used to observe β 𝛽 \beta Pic (“This study”) and those used to implement the limb-darkening law (“Claret (2000)”). The third column gives the degree of correlation between the two.

This study	Claret (2000)	Correlation
BRITE red	Johnson R	0.887
BRITE blue	Johnson B	0.857
ASTEP $i^{\prime }$	Johnson I	0.445
SMEI	Johnson R	0.729
bRing	Johnson V	0.609

Table 5. Table 5: List of classical constraints and least-squares fit.

Quantity	Value	Reference	Fit	MCMC runs
Mass ($M_{\odot }$)	${1.80}_{-0.04}^{+0.03}$	Wang et al. (2016)	$1.797\pm 0.035$	Truncated Gaussiana $1.797\pm 0.053$
Parallax, $\pi$ (mas)	$51.44\pm 0.12$	van Leeuwen (2007)	$51.45\pm 0.12$	–
Angular diameter, $\theta$ (mas)	$0.736\pm {0.019}^b$	Defrère et al. (2012)	$0.716\pm 0.012$	–
Radius ($R_{\odot }$)	$1.538\pm 0.040$	Deduced from $\pi$ and $\theta$	$1.497\pm 0.025$	Truncated Gaussiana $1.538\pm 0.040$
$\log g$ (dex)	$4.25\pm 0.10$	Lanz et al. (1995)	$4.343\pm 0.017$	Gaussian $4.25\pm 0.10$
Luminosity ($L_{\odot }$)	$8.47\pm {0.23}^c$	Crifo et al. (1997)	$8.62\pm 0.21$	–
$T_{\mathrm{eff}}$ (K)	$8143\pm 67$	Average from multiple papersd	$8090\pm 59$	–
$\upsilon \sin i$ ($\mathrm{km}{s}^{-1}$)	$124\pm 3$	Koen et al. (2003)	–	Gaussian $124\pm 3$

Table 6. Table 6: Best solutions obtained for different MCMC runs.

MCMC run	$\Omega /{\Omega }_K$	$i$ (in ∘)	$f$ (in d-1)	$R$ (in $R_{\odot }$)	$M$ (in $M_{\odot }$)	${\chi }_{\mathrm{ampl}.}^2$	${\chi }_{\mathrm{seismic}}^2$	${\chi }_{\mathrm{seismic}}^2\left(\mathrm{true}\right)$
True seismic	0.3903	45.3	0.96960	1.638	1.718	467.8	2615	$3.0\times {10}^6$
	$0.3903\pm 0.0002$	$45.3\pm 0.2$	$0.96960\pm 0.00005$	$1.638\pm 0.006$	$1.719\pm 0.020$
	$0.2732\pm 0.0001$	$86.1\pm 1.3$	$0.92894\pm 0.00006$	$1.652\pm 0.002$	$1.793\pm 0.007$
${\sigma }_F=0.01$	0.3977	42.9	1.0241	1.595	1.755	468.8	1017	$1.2\times {10}^7$
	$0.3977\pm 0.0002$	$42.9\pm 0.5$	$1.0241\pm 0.0001$	$1.595\pm 0.015$	$1.756\pm 0.049$
${\sigma }_F=0.05$	0.2625	80.4	1.0656	1.505	1.800	412.7	2257	$7.8\times {10}^7$
	$0.2624\pm 0.0011$	$82.3\pm 3.7$	$1.0656\pm 0.0004$	$1.501\pm 0.011$	$1.786\pm 0.041$
${\sigma }_F=0.05$ (2017/18)	0.2599	86.0	1.0630	1.505	1.794	508.8	2017	$2.8\times {10}^7$
	$0.2601\pm 0.0011$	$84.8\pm 3.2$	$1.0630\pm 0.0005$	$1.509\pm 0.009$	$1.809\pm 0.032$
${\sigma }_F=0.05$ (even)	0.2742	76.5	0.9978	1.570	1.776	449.1	4723	$1.3\times {10}^8$
	$0.2736\pm 0.0015$	$78.6\pm 3.6$	$0.9976\pm 0.0003$	$1.565\pm 0.015$	$1.758\pm 0.052$
${\sigma }_F=0.05$ (even, 2017/18)	0.2799	79.8	0.9298	1.636	1.737	528.9	7211	$9.0\times {10}^7$
	$0.2801\pm 0.0013$	$81.5\pm 3.6$	$0.9298\pm 0.0005$	$1.631\pm 0.012$	$1.720\pm 0.039$
${\sigma }_F=0.1$ (even)	0.2723	89.9	0.9258	1.658	1.802	412.3	6559	$6.5\times {10}^7$
	$0.2735\pm 0.0007$	$87.8\pm 1.6$	$0.9262\pm 0.0004$	$1.653\pm 0.004$	$1.785\pm 0.013$
${\sigma }_F=0.1$ (even, 2017/18)	0.2723	89.2	0.9258	1.658	1.802	515.9	8049	$6.2\times {10}^7$
	$0.2735\pm 0.0007$	$87.6\pm 1.6$	$0.9262\pm 0.0004$	$1.653\pm 0.004$	$1.785\pm 0.013$

Table 7. Table 7: Mode identifications in the form ( n , ℓ , m ) 𝑛 ℓ 𝑚 (n,\ell,m) for best and alternate configurations.

No.	True seismic		${\sigma }_F=0.01$	${\sigma }_F=0.05$	${\sigma }_F=0.05$	${\sigma }_F=0.05$	${\sigma }_F=0.05$	${\sigma }_F=0.1$	${\sigma }_F=0.1$
#	Best	Alternate			(2017/18)	(even)	(even, 2017/18)	(even)	(even, 2017/18)
F1	$(3,1,0)$	$(3,2,1)$	$(2,2,-2)$	$(1,3,-3)$	$(2,3,1)$	$(3,2,2)$	$(3,1,-1)$	$(2,3,-3)$	$(2,3,-3)$
F2	$(3,2,-1)$	$(4,2,2)$	$(4,3,3)$	$(3,3,3)$	$(3,3,3)$	$(3,2,0)$	$(4,2,2)$	$(4,2,2)$	$(4,2,2)$
F3	$(3,3,-1)$	$(4,3,3)$	$(4,1,1)$	$(3,1,-1)$	$(3,1,-1)$	$(3,3,1)$	$(3,3,-1)$	$(3,3,-1)$	$(3,3,-1)$
F4	$(5,1,1)$	$(4,2,-1)$	$(4,1,-1)$	$(3,3,0)$	$(3,3,0)$	$(4,1,-1)$	$(5,0,0)$	$(4,2,-2)$	$(4,2,-2)$
Fc	$(4,2,-2)$	$(5,2,2)$	$(5,3,3)$	$(3,3,-1)$	$(3,3,-1)$	$(4,2,0)$	$(5,2,2)$	$(5,2,2)$	$(5,2,2)$
F5	$(4,3,-1)$	$(5,3,3)$	$(5,2,2)$	$(4,3,3)$	$(4,3,3)$	$(4,3,1)$	$(4,3,-1)$	$(5,3,3)$	$(5,3,3)$
F6	$(5,3,2)$	$(5,1,0)$	$(4,3,0)$	$(4,1,-1)$	$(4,1,-1)$	$(5,0,0)$	$(5,1,-1)$	$(5,1,-1)$	$(5,1,-1)$
F7	$(5,2,1)$	$(5,2,1)$	$(4,2,-2)$	$(3,3,-3)$	$(4,3,2)$	$(4,2,-2)$	$(4,3,-3)$	$(4,3,-3)$	$(4,3,-3)$
F8	$(5,3,1)$	$(4,3,-3)$	$(4,3,-1)$	$(4,2,0)$	$(4,2,0)$	$(5,1,1)$	$(5,2,0)$	$(5,2,0)$	$(5,2,0)$
F9	$(6,3,3)$	$(5,3,1)$	$(5,3,2)$	$(4,3,1)$	$(4,3,1)$	$(4,3,-1)$	$(5,3,1)$	$(5,3,1)$	$(5,3,1)$
F11	$(6,1,1)$	$(5,2,-2)$	$(5,1,-1)$	$(4,2,-2)$	$(4,2,-2)$	$(5,1,-1)$	$(6,1,1)$	$(6,1,1)$	$(6,1,1)$
F12	$(5,3,0)$	$(5,3,0)$	$(5,1,0)$	$(5,1,1)$	$(4,3,0)$	$(4,3,-3)$	$(6,2,2)$	$(6,2,2)$	$(6,2,2)$
F13	$(6,3,1)$	$(6,2,1)$	$(5,3,0)$	$(5,1,0)$	$(4,3,-2)$	$(6,0,0)$	$(\mathrm{𝟓},\mathrm{𝟑},-\mathrm{𝟑})$	$(6,1,-1)$	$(\mathrm{𝟓},\mathrm{𝟑},-\mathrm{𝟑})$
F14	$(6,2,0)$	$(5,3,-3)$	$(6,0,0)$	$(4,3,-2)$	$(5,1,-1)$	$(5,2,-2)$	$(\mathrm{𝟔},\mathrm{𝟏},-\mathrm{𝟏})$	$(5,3,-3)$	$(\mathrm{𝟔},\mathrm{𝟏},-\mathrm{𝟏})$
F15	$(6,2,1)$	$(6,2,0)$	$(5,3,-1)$	$(5,2,1)$	$(5,2,1)$	$(6,1,1)$	$(6,2,0)$	$(6,2,0)$	$(6,2,0)$

Equations

$$f^2=\frac{M}{R^3}\frac{R_{\mathrm{mod}}^3}{M_{\mathrm{mod}}},{\Omega }_{\mathrm{K}}=f{\Omega }_{\mathrm{K},\mathrm{mod}},$$

$$R$$

$$J_1(R)={\left(\frac{R{\Omega }_{\mathrm{K}}-g}{{\sigma }_g}\right)}^2+{\left(\frac{R\Omega \sin i-v\sin i}{{\sigma }_v}\right)}^2$$

$$M=\frac{R^3{\Omega }_{\mathrm{K}}^2}{G}$$

$$J_2(\theta ,\pi ,M,T_{\mathrm{eff}})$$

$$L=4\pi \sigma R^2{T}_{\mathrm{eff}}^4,g=\frac{GM}{R^2},R=\frac{\theta d}{2\pi },$$

Full text

*Aims. *We aim to revisit the pulsational properties of $\beta$ Pic and identify its pulsation modes from normalised amplitudes in five different passbands. We also investigate the potential presence of a magnetic field.

*Methods. *We conduct a frequency analysis using three seasons of BRITE-Constellation observations in the BRITE filters, the $\sim$620-day long bRing light curve and the nearly 8-year long SMEI photometric time series. We calculate normalised amplitudes using all passbands including previously published values obtained from ASTEP observations. We investigate the magnetic properties of $\beta$ Pic using spectropolarimetric observations conducted with the HARPSpol instrument. Using 2D rotating models, we fit the normalised amplitudes and frequencies through Monte Carlo Markov Chains.

*Results. *We identify 15 pulsation frequencies in the range from 34 to 55  d-1, where two – F13 at 53.6917  d-1 and F11 at 50.4921  d-1 – display clear amplitude variability. We use the normalised amplitudes in up to five passbands to identify the modes as three $\ell=1$, six $\ell=2$ and six $\ell=3$ modes. $\beta$ Pic is shown to be non-magnetic with an upper limit of the possible undetected dipolar field of 300 Gauss.

*Conclusions. *Multiple fits to the frequencies and normalised amplitudes are obtained including one with a near equator-on inclination for $\beta$ Pic, which corresponds to our expectations based on the orbital inclination of $\beta$ Pic b and the orientation of the circumstellar disk. This solution leads to a rotation rate of $27$% of the Keplerian break-up velocity, a radius of 1.497$\pm$0.025 $R_{\odot}$, and a mass of 1.797$\pm$0.035 $M_{\odot}$. The $\sim$2% errors in radius and mass do not account for uncertainties in the models and a potentially erroneous mode-identification.

1 Introduction

The description of the formation, structure and evolution of young stars is one of the big challenges in stellar astrophysics. Early stellar evolution plays a crucial role in our understanding of the formation and evolution of exoplanets, whose properties depend on the accuracy of the inferred stellar parameters. In many cases, stellar activity (star spots, magnetic fields, pulsations, circumstellar material etc.) complicates the reliable determination of the physical properties of stars from the combination of spectroscopic observations and theoretical models and affects the investigation of the exoplanet properties. However, the presence of pulsations might also be beneficial because asteroseismic methods can be used to constrain the interior structure of exoplanet host stars. With it, more reliable results for the fundamental stellar parameters can be inferred which in turn affects the precision of the derived exoplanet properties. In this context, the $\beta$ Pictoris system is quite an interesting object.

$\beta$ Pictoris (HD 39060, spectral type A6 V) is a bright star (V = 3.86 mag) located relatively close to us at 19.76 pc distance (calculated using a parallax of 50.623 $\pm$ 0.334 mas as given in the Gaia DR2 catalog; Brown et al. 2018). It is a member of the $\beta$ Pic moving group (Mamajek & Bell 2014). The star $\beta$ Pic is surrounded by a gas and dust debris disk which is seen nearly edge-on; its outer extent varies from 1450 to 1835 Astronomical Units (AU) (Larwood & Kalas 2001). The observed warp of its inner disk suggested the presence of a planet in the system. Using observations with the VLT at Paranal and the NACO camera, Lagrange et al. (2009, 2010) directly imaged the giant gas planet, $\beta$ Pic b, for the first time. Although its orbital inclination is close to equator-on, i.e. 88.81$\pm 0.12^{\circ}$ as seen from Earth, it is sufficiently inclined that $\beta$ Pic b does not transit its host star (Wang et al. 2016). The reported four distinct belts around the star kinematically indicate the presence of other planets (Wahhaj et al. 2003). So far, no additional planet was detected (e.g., Lous et al. 2018).

The age of the $\beta$ Pic system was investigated by several authors using many different methods (for a review see Mamajek & Bell 2014). Currently, there seems to be agreement that the age of the $\beta$ Pic system is $\sim$23 Myr (Mamajek & Bell 2014). From both spectroscopic observations and the derived age, it is evident that the star is in its early main sequence stage of evolution (e.g., Zwintz et al. 2014).

$\delta$ Scuti pulsations in $\beta$ Pic were first discovered 2003 through ground-based photometric time series (Koen 2003) where three low-amplitude modes were identified. Subsequent spectroscopic time series observations obtained with the 1.9 m telescope at the South African Astronomical Observatory (SAAO) revealed 18 $\delta$ Scuti pulsation frequencies in the range from 24 to 71 d-1 from line profile variations (Koen et al. 2003). Recently, Mékarnia et al. (2017) reported 31 pulsation frequencies between 34.76 and 75.68 d-1derived from photometric time series obtained between March and September 2017 using the 40-cm ASTEP telescope at Concordia Station in Antarctica.

Asteroseismology has been successfully used to study different types of pulsating stars from the pre-main sequence (pre-MS) to the final stages of evolution (e.g., in white dwarfs) in a mass range from $\sim$0.5 to 40 $M_{\odot}$ with effective temperatures between $\sim$3 000 K and 100 000 K. $\delta$ Scuti stars are located in the lower part of the so-called classical instability strip where it intersects with the main sequence. They have spectral types from A2 to F2 (Rodríguez & Breger 2001) and masses between 1.5 and 4 $M_{\odot}$ (e.g., Aerts et al. 2010) and lie in the effective temperature range between 6300 and 8600 K (Uytterhoeven et al. 2011). $\delta$ Scuti pulsations can be found in the pre-main sequence, main sequence and post-main sequence evolutionary stages and are driven by the heat-engine ($\kappa$-) mechanism acting in the second helium ionization zone (e.g., Aerts et al. 2010). The pulsation modes are radial and non-radial pressure (p) modes with periods in the range from $\sim$18 minutes to 0.3 days.

Despite the fact that $\delta$ Scuti stars were one of the first types of stars discovered to pulsate, little progress has been made in interpreting their pulsation spectra due to their complexity. $\delta$ Scuti stars are mostly multiperiodic and can show very rich pulsation frequency spectra (e.g., Poretti et al. 2009). As they are moderate to fast rotators (Breger 2000), the influence of rotation on the pulsation frequencies cannot be neglected in theoretical models. One of the first effects of rotation is to split the frequencies of modes with the same $n$ and $\ell$ values but different $m$ values, thus removing their degeneracy. At slow rotation rates for uniform rotation profiles, these rotationally split modes should be observed as multiplets with nearly equidistant frequency spacings, but in fact there are only a few cases known where such clear rotational splittings were found (e.g., Kurtz et al. 2014). At faster rotation rates, the multiplets become non-equidistant as higher-order effects of rotation intervene (e.g., Saio 1981; Espinosa et al. 2004), and they start to overlap making the spectrum more complicated to interpret (Reese et al. 2006). Eventually, the frequency spectrum of acoustic modes takes on a new structure composed of overlapping independently-organized subspectra associated with different classes of modes (Lignières & Georgeot 2008, 2009).

Furthermore, the pulsation amplitudes of $\delta$ Scuti stars can be variable for different reasons. A good overview of this topic can be found in Bowman et al. (2016), where the authors explain the two intrinsic causes – (i) beating of a pair of close, unresolved pulsation frequencies and (ii) non-linearity and coupling of modes – and one extrinsic cause, i.e. binary or multiple systems, of amplitude modulations. In the same study, the authors reveal that 61.3% of their sample of 983 $\delta$ Scuti stars include at least one pulsation mode showing amplitude modulation which illustrates that this is a quite common effect in this group of stars.

Another complicating factor in the interpretation of $\delta$ Scuti type pulsations can be the presence of a magnetic field. $\delta$ Scuti and rapidly oscillating, chemically peculiar A (roAp) stars are located in the same region of the HR diagram. The roAp stars (Kurtz et al. 2006) show pulsation periods between 6 and 24 minutes (e.g., Smalley et al. 2015) and possess global magnetic fields, inhomogeneous surface distributions of some chemical elements and strong overabundances, including the rare-earth elements (Ryabchikova et al. 2004). While the presence of sometimes quite strong magnetic fields are typical for roAp stars, for only very few $\delta$ Scuti stars have magnetic fields been measured (e.g., Kurtz et al. 2008; Neiner & Lampens 2015) or were suggested due to the presence of the aforementioned rare-earth anomaly (Escorza et al. 2016).

In the present study we use photometric time series obtained by the BRITE-Constellation satellites in two filters in three observing seasons in combination with photometric time series observed with the SMEI satellite (Jackson et al. 2004; Howard et al. 2013), the bRing instrument (Stuik et al. 2017) and the previously published results by Mékarnia et al. (2017) based on data from the ASTEP telescope in Antarctica (Abe et al. 2013; Guillot et al. 2015; Mékarnia et al. 2016) to constrain the pulsational properties of $\beta$ Pic, identify its pulsation modes from the multiband photometry and investigate the presence of amplitude modulation. Additionally, we use HARPSpol (Piskunov et al. 2011) spectropolarimetric data to investigate the presence of a magnetic field.

2 Observations

2.1 BRITE-Constellation

BRITE-Constellation111http://www.brite-constellation.at consists of five 20-cm cube nanosatellites each carrying a 3-cm telescope and feeding an uncooled CCD (Weiss et al. 2014). Three BRITE satellites – i.e., BRITE-Toronto (BTr), Uni-BRITE (UBr) and BRITE-Heweliusz (BHr) – carry a custom-defined red filter (550 – 700 nm), and two satellites – i.e., BRITE-Austria (BAb) and BRITE-Lem (BLb) – a custom-defined blue filter (390 – 460 nm). More details on the detectors, pre-launch and in-orbit tests are described by Pablo et al. (2016). Popowicz et al. (2017) describe the pipeline that processes the observed images yielding the instrumental magnitudes which are delivered to the users.

BRITE-Constellation observes large fields with typically 15 to 20 stars brighter than $V=6$ mag including at least three targets brighter than $V=3$ mag. Each field is observed at least 15 minutes per each $\sim$100-minute orbit for up to half a year (Weiss et al. 2014).

BRITE-Constellation first obtained observations of $\beta$ Pic from 16 March - 2 June 2015 (BRITE Run ID: 08-VelPic-I-2015), yielding a total time base of 78.323 days using BHr in stare mode (Popowicz et al. 2017). Hence, the corresponding frequency resolution, 1/T, is 0.013  d-1. After the success of this first observing season, a longer observing run was conducted using BTr from 4 November 2016 to 17 June 2017 for a total of 224.573 days and BLb from 15 December 2016 to 21 June 2017 for 187.923 days (BRITE Run ID: 23-VelPic-II-2016). BHr was used from 7 January 2017 to 30 January 2017 for 24 days to cover a gap in the BTr observations. Recently, the BRITE-Constellation observations of the third season for $\beta$ Pic were completed. The red BHr satellite obtained time series of $\beta$ Pic between 9 November 2017 to 25 April 2018 for 167.335 days (BRITE Run ID: 33-VelPicIII-2017). The 2016/2017 and 2017/2018 observations were made using the chopping mode where the position of the target star within the CCD plane is constantly alternated between two positions about 20 pixels apart on the CCD (Popowicz et al. 2017). An overview of the BRITE-Constellation observations is given in Table 1. Publicly available BRITE-Constellation data can be retrieved from the BRITE Public Data Archive (https://brite.camk.edu.pl/pub/index.html).

2.2 bRing

bRing (which stands for ”the $\beta$ Pictoris b Ring project”) consists of two ground-based observatories which monitored $\beta$ Pic photometrically in particular during the expected transit of the Hill sphere of its giant exoplanet $\beta$ Pic b in 2017 – 2018 (Stuik et al. 2017). One bRing instrument is located in the Sutherland observing station of the South African Astronomical Observatory, the second bRing site is at the Siding Spring Observatory in Australia. Both telescopes take observations in the wavelength range from 463 to 639 nm. A detailed description of the design, operations and observing strategy of bRing is provided by Stuik et al. (2017). Both bRing instruments will continue the observations of $\beta$ Pic for as long as possible in the future.

The data set used here was taken between 2 February 2017 and 16 October 2018 with a cadence of $\sim$5 minutes using a combination of the data obtained by both bRing instruments.

2.3 Solar Mass Ejection Imager (SMEI)

The Solar Mass Ejection Imager (SMEI; Eyles et al. 2003; Jackson et al. 2004; Howard et al. 2013) was launched as a secondary payload onboard the Coriolis spacecraft in January 2003. Its main purpose was to monitor and predict space weather in the inner solar system. The orbital period of SMEI is $\sim$101.5 minutes (e.g., Eyles et al. 2003). The mission was terminated due to budgetary reasons in September 2011. However, the SMEI images have also been shown to yield high quality, long duration stellar photometry. SMEI obtained brightness measurements of nearly the full sky using three cameras with a field-of-view of 3 $\times$ 60 deg2 each. The photometric passband ranges from 450 to 950 nm. The data rate for the SMEI photometric time series for a single star is one measurement per each $\sim$101.5-minute orbit. Consequently, the Nyquist frequency of the SMEI data lies at 7.08  d-1.

Stellar time series obtained by SMEI can be extracted from the SMEI website333http://smei.ucsd.edu/new_smei/data&images/stars/timeseries.html. They have been used several times in the past for a common interpretation with BRITE-Constellation data (e.g., Baade et al. 2018a; Kallinger et al. 2017). In the case of $\beta$ Pic  only times and magnitudes were available from the SMEI website with no additional information about the instrumental settings.

The SMEI data for $\beta$ Pic comprise 28623 data points obtained between 6 February 2003 and 30 December 2010 for about eight years in total, which corresponds to a classical Rayleigh frequency resolution of 0.0003 d-1 (see Table 1).

2.4 Spectropolarimetry

$\beta$ Pic was observed in conjunction with the BRITE spectropolarimetric survey (Neiner & Lèbre 2014) with the HARPSpol spectropolarimeter (Piskunov et al. 2011) installed on the ESO 3.6-m telescope in La Silla (Chile). Observations were acquired on November 7, 2014, and are available through the ESO archive444archive.eso.org. A series of seven consecutive Stokes V sequences were obtained to increase the signal-to-noise ratio (SNR) of the co-added spectrum while avoiding saturation of the detector. Each sequence consisted of four sub-exposures of 246 s, with each sub-exposure in a different configuration of the polarimeter. This led to a total of almost 2 h of exposure for the seven sequences.

The usual bias, flat-field, and ThAr calibrations have been obtained the same night and applied to the data. The data were reduced using a modified version of the REDUCE software (Piskunov & Valenti 2002; Makaganiuk et al. 2011). This included automatic normalisation of the spectra to the intensity continuum level.

3 Photometric data reduction and frequency analysis

The frequency analysis of the BRITE, SMEI and bRing photometric time series was performed independently of each other using the software package Period04 (Lenz & Breger 2005) that combines Fourier and least-squares algorithms. Frequencies were then prewhitened and considered to be significant if their amplitudes exceeded 3.8 times the local noise level in the amplitude spectrum (Breger et al. 1993; Kuschnig et al. 1997). Frequency, amplitude and phase errors are calculated using the formulae given by Montgomery & Odonoghue (1999).

We verified the analysis using the iterative prewhitening method based on the Lomb-Scargle periodogram that is described by Van Reeth et al. (2015).

3.1 Analysis of the BRITE photometry

The raw BRITE photometry was corrected for instrumental effects. The corrections included outlier rejection, and both one- and two-dimensional decorrelations with all available parameters, in accordance with the procedure described by Pigulski (2018).

The data obtained in 2016/17 by BTr and BHr were combined to a single red filter data set. An overview of their properties is given in Table 1. Figure 1 shows the light curves obtained by BHr in 2015 (panel a), by BTr and BHr in 2016/2017 (panel b), by BLb in 2016/2017 (panel c) and by BHr in 2017/2018 (panel d) to the same Y axis scale. For $\beta$ Pic  with a $B$ magnitude of 4.03 and a $V$ magnitude of 3.86 significantly less flux is measured through the blue filter than through the red filter which is reflected by a more than a factor of four higher scatter, and hence, also a factor of four higher residual noise level in the frequency analysis (Table 1).

3.1.1 2015 data

The frequency analysis of the BHr 2015 data yielded eight intrinsic frequencies with a S/N ratio larger than 3.8. A comparison to the frequencies reported by Mékarnia et al. (2017) shows that there is agreement for six frequencies (F1, F8, F11, F13, F14 and F15; see Table 2 and grey triangles in Fig. 15). Additionally, we find two frequencies at 32.456  d-1 with an amplitude of 0.47$\pm$0.05 mmag and at 61.367  d-1 with an amplitude of 0.49$\pm$0.05 mmag to be statistically significant with a S/N ratio of 4.66 and 4.92. As these frequencies do not appear in any of our other data sets including those obtained by bRing, SMEI and ASTEP (Mékarnia et al. 2017) , their origin is presently unclear, and, hence, we treat them with caution and discard them from our further analysis.

The residual noise level after prewhitening of the eight frequencies is 100 ppm. The spectral window function and amplitude spectra are shown in Figure 15 in the Appendix A.

3.1.2 2016/2017 data

The analysis of the combined 2016/2017 BRITE red filter data set yielded 13 significant pulsation frequencies (Table 2). Only frequency F10 at 49.4161 d-1 was not reported by Mékarnia et al. (2017). Although frequency F4 at 43.5268 d-1 is close to three times the BRITE orbital frequency and we would normally have to omit it, we identify it as pulsational because it was already reported by Mékarnia et al. (2017).

Due to the significantly higher noise level of the blue filter data set, only four pulsation frequencies were identified from the BLb observations (Table 2). These four frequencies (F3, F4, F8, F11) are also found in the 2015, 2016/2017 and 2017/2018 red filter data sets and have also been reported by Mékarnia et al. (2017).

The residual noise level after prewhitening all frequencies is 40 ppm for the combined red filter and 170 ppm for the blue filter data. Figure 2 shows the amplitude spectra using the combined BTr and BHr data set (left) and the BLb data (right).

3.1.3 2017/2018 data

With the 2017/18 BHr data set we confirmed seven of the pulsation frequencies previously identified from the BRITE-Constellation 2016/2017 data and Mékarnia et al. (2017). Additionally, two frequencies at 34.085  d-1 and 52.960  d-1 with amplitudes of 0.27$\pm$0.03 mmag and 0.20$\pm$0.03 mmag that were not found in the 2015 and 2016/17 data sets before, are statistically significant with S/N values of 6.34 and 4.77, respectively. As these frequencies do not appear in any of the other observations, we discard them from any further investigation. The residual noise level after prewhitening all pulsation frequencies is 43 ppm.

The corresponding amplitude spectrum is shown in Figure 16 in the Appendix A.

3.1.4 Combined BRITE red filter data

Combining the three seasons of BRITE-Constellation red filter data yields a total time base of 1135 days corresponding to a Rayleigh frequency resolution, 1/T, of 0.0009  d-1. All frequencies listed in Table 2 can be found in the combined BRITE red filter data set; no additional peaks are statistically significant. The residual noise level calculated from 0 to 100  d-1 after prewhitening all significant frequencies lies at 36 ppm.

Despite the fact that BRITE-Constellation observations are sensitive in the low frequency domain as was illustrated, e.g., by Baade et al. (2018b) or Ramiaramanantsoa et al. (2018), there is no evidence for the presence of g-modes.

3.2 Analysis of the bRing photometry

The complete bRing light curve used for the present analysis has a total time base of more than 620 days (panel e in Fig. 1). The Nyquist frequency lies at 135.37  d-1. We could identify six of the previously reported pulsation frequencies (i.e., F2, F6, F8, F11, F13 and F15) from this data set (left side in Fig. 3). The residual noise level after subtracting all formally significant frequencies is at 110 ppm (Table 1 and panel c in Fig. 3). Panel a in Figure 3 shows the bRing spectral window.

3.3 Analysis of the SMEI photometry

The SMEI light curves are affected by strong instrumental effects, such as large yearly flux fluctuations. We corrected for this one-year periodicity of instrumental origin by phasing the raw data with a one-year period, calculating median values in 200 phase intervals, interpolating between these points and subtracting the interpolated light curve. In the next step we detrended the data repeatedly with simultaneous sigma clipping to remove outliers and suppress any instrumental signal at low frequencies. The detrending was done 30 times starting with a time interval to calculate the mean, T, of 100 days and a sigma of 5, and ending the procedure with T = 0.7 days and a sigma of 4. As a consequence of this method, frequencies lower than 0.5  d-1 are suppressed. The annual light curves of $\beta$ Pic  show very similar behaviour, hence the corresponding minor differences were subtracted by detrending. Panel f in Figure 1 shows the complete reduced SMEI light curve.

The sampling of the SMEI data is only about one data point per 1.7 hours (i.e., the orbital period of the satellite) which results in a Nyquist frequency of only 7.08  d-1. Hence, investigating SMEI data for the presence of $\delta$ Scuti -type pulsations in the range from about 30 to 70  d-1 as expected for $\beta$ Pic goes already beyond the Nyquist frequency, $f_{\rm Nyquist}$. As it was shown, e.g., for Kepler data by Murphy et al. (2013), it is similarly possible to do super-Nyquist asteroseismology using the SMEI data as real peaks remain as singlets even if they are above $f_{\rm Nyquist}$. Panel e in Figure 3 shows the complete amplitude spectrum of the SMEI data ranging up to 100  d-1. It can clearly be seen how the pulsation frequencies between 30 and 60  d-1remain single peaks that can be easily distinguished from aliases caused by $f_{\rm Nyquist}$. The dips in the noise around 42.5 and 52  d-1 are caused by the strong detrending.

The frequency analysis of the SMEI data yielded seven pulsation frequencies that are either present in the BRITE data or were reported by Mékarnia et al. (2017) or both (Table 3, panels d to f in Figure 3).

4 Spectropolarimetric analysis

4.1 Stokes profiles

We applied the Least Squares Deconvolution (LSD) method (Donati et al. 1997) to the HARPSpol data to produce seven mean LSD Stokes I, Stokes V, and N profiles. The seven sequences were then co-added to produce one final set of LSD profiles, shown in Fig. 4.

Performing LSD requires the use of a mask indicating the list of lines present in the spectrum and to be used in the averaging, their wavelengths, depths and Landé factors. To produce this mask, we started from a line list extracted from the VALD3 atomic database (Piskunov et al. 1995; Kupka et al. 1999) for a star with $T_{\rm eff}$ = 8200 K and $\log g$ = 4.0 (i.e., the stellar parameters of $\beta$ Pic taken from (Lanz et al. 1995). We retained only the lines with a predicted depth larger than 0.01. In addition, we rejected hydrogen lines, lines blended with H lines or interstellar lines, and regions affected by telluric absorption. Finally, we adapted the depth of the lines in the mask to the actual depth of the lines observed in the spectra with an automatic fitting routine. In total, we used 6391 spectral lines, with a mean wavelength of 503.8 nm and a mean Landé factor of 1.202.

The LSD N profile represents the Null polarisation and is used as a sanity check for the spectropolarimetric measurement. The fact that the N profile shows only noise (see Fig. 4, middle panel) indicates that the spectropolarimetric measurement has not been polluted by instrumental effects or stellar variability of non-magnetic origin.

The LSD Stokes V profile should indicate a Zeeman signature if a magnetic field was present in $\beta$ Pic. Since this profile also shows only noise (see Fig. 4, top panel), it indicates that $\beta$ Pic is not magnetic, at the level of precision of our measurement.

We can evaluate the detection of a magnetic field statistically with the False Alarm Probability (FAP). We check the presence of a signature in the LSD Stokes V profile inside the velocity range [$-$110:152] km s*-1*, compared to the mean noise level in the LSD Stokes V profile outside the line. We adopted the convention defined by Donati et al. (1997) that there is a definite or marginal magnetic detection if FAP $<$ 0.1%.

The FAP analysis of the LSD Stokes V profile leads to no magnetic detection (with a FAP = 99.99%). A similar analysis of the N profile also indicates no detection in N (as expected).

4.2 Longitudinal magnetic field measurement

A first quantitative estimate of the non-detection of a magnetic field in $\beta$ Pic can be obtained via the measurement of the (undetected) longitudinal magnetic field $B_{l}$.

To this aim, we used a center-of-gravity method (Rees & Semel 1979; Wade et al. 2000) and applied it to the Stokes V and N profiles. We integrated the profiles over the same velocity range as for the FAP analysis, i.e. [$-$110:152] km s*-1*. We obtained $B_{l}$ = $-$14 $\pm$ 20 G. We applied the same method to the N profile and obtained $N_{l}$ = 11 $\pm$ 20 G. Both values are compatible with 0, indicating that no field is detected in $\beta$ Pic. The error of 20 G shows that our measurement has a good precision.

4.3 Upper limit of the undetected magnetic field

Another quantitative estimate of the non-detection of a magnetic field in $\beta$ Pic can be obtained by determining the maximum strength of a magnetic field that could have remained hidden in the noise of our data.

Since $\beta$ Pic is an A6V star, its envelope is radiative and thus, if it hosts a magnetic field, it must be of fossil origin (Neiner et al. 2015b). Fossil fields are usually dipolar and tilted with respect to the rotation axis of the star (Grunhut & Neiner 2015). Therefore, to estimate the upper limit of a possibly undetected magnetic field, we assumed an oblique dipolar field.

We followed the method described in Neiner et al. (2015a): for various values of the polar magnetic field strength $B_{\rm pol}$, we calculated 1000 models of the LSD Stokes $V$ profile with random inclination angle $i$, obliquity angle $\beta$, and rotational phase, and a white Gaussian noise with a null average and a variance corresponding to the SNR of the observed profile. We first fitted the LSD $I$ profile, we then calculated local Stokes $V$ profiles assuming the weak-field case and we integrated over the visible hemisphere of the star. We used a projected rotational velocity of $v\sin i$ = 124 km s*-1* taken from (Lanz et al. 1995) and a limb-darkening coefficient of 0.6. In this way we obtained a synthetic Stokes $V$ profile for each model, which we normalised to the intensity continuum. We used the same Landé factors and wavelengths as in the LSD calculation.

We then computed the probability of detection of a field in these 1000 models by applying the Neyman-Pearson likelihood ratio test. We further calculated the rate of detections among the 1000 models depending on the field strength (see Fig. 5). We required a 90% detection rate to consider that the field should have statistically been detected. This translates into an upper limit for the possible undetected dipolar field strength for $\beta$ Pic of $B_{\rm pol}$ = 300 G. Using a 50% detection rate would bring the limit of the possible undetected dipolar field strength down to $B_{\rm pol}$ = 120 G.

5 Amplitude variability

$\delta$ Scuti stars can show variable pulsation amplitudes that are either of intrinsic (beating of unresolved frequencies, non-linearity or mode-coupling) or extrinsic (binarity and multiple systems) origin (for a detailed overview see Bowman et al. 2016). We examined the amplitude variability of $\beta$ Pic using the three seasons of BRITE-Constellation red filter data and the bRing observations.

5.1 Annual changes in the amplitudes

Using the three BRITE R filter data sets obtained in consecutive years and the bRing light curve, we can study the annual amplitude variability of $\beta$ Pic’s pulsation frequencies from 2015 to 2018. Only four of the 15 identified pulsation frequencies appear in the BRITE data sets of all three years: F8, F11, F13 and F15. Frequencies F1, F3, F4, and F6 are detectable in two of the three BRITE observing seasons and the other seven frequencies only appear in one year of BRITE data. The top panel in Figure 6 illustrates that the amplitudes of six of these frequencies remain rather stable or change only slightly from year to year (i.e., F1, F3, F4, F6, F8 and F15), while F11 shows a strong increase in amplitude and F13’s amplitude decreases significantly in the 2016/2017 observations and increases again in the 2017/2018 data set. A zoom into the BRITE R filter amplitude spectra around F11 and F13 illustrates this behaviour in the Fourier domain within the three seasons of BRITE-Constellation observations (Figure 7).

In a next step, we divided the 620-day long bRing light curve into two parts of equal length and studied the resulting behaviour of the amplitudes with respect to the center points in time, i.e., mid 2017 and mid 2018. Despite the higher noise level in the bRing data which translates into larger errors on the amplitudes, it is obvious that the amplitude for F13 increases during this period of time. As the errors of the other four pulsation amplitudes (i.e., F3, F6, F8, and F11) are quite large, no clear conclusion on variability or stability can be drawn from these data.

5.2 Amplitude variability within observing seasons

Mékarnia et al. (2017) investigated the presence of amplitude and phase changes for their 10 first frequencies during their 7-month long observations by dividing their data set into seven parts, each 30 days in length. They showed that only their frequency f3 (corresponding to our frequency F13) at 53.69138 d-1  changes its amplitude from 403 to 826$\pm$66 ppm (i.e., 0.403 to 0.826$\pm$0.066 mmag).

For the purposes of comparison, we conducted the same analysis as Mékarnia et al. (2017), i.e., calculating the amplitude behaviour using 30-day subsets. As the overall amplitude of our frequency F13 is significantly lower during the longest BRITE observing run in 2016/17 (see Figure 6) and, therefore, is buried in the noise (i.e., not significant) when calculating 30-day subsets, we chose the 2017/18 BHr data set for this comparative analysis. Using subsets of 30-days length with 20 days of overlap, we find the amplitude of F13 to increase from 965 ppm (i.e., 0.965 mmag) to a maximum of 1489 ppm (i.e., 1.489 mmag); see blue symbols and line in the top panel of Figure 8).

As the ASTEP observations were conducted in the time from March to September 2017 (Mékarnia et al. 2017), there is an overlap of about four months with the second BRITE data set from 2016/17 which was taken a bit earlier starting in November 2016 and running until June 2017. The overall amplitude of our F13 at 53.6917  d-1 in this data set is only 396 ppm (i.e., 0.396 mmag), being the lowest in our analysis. During the subsequent observations with ASTEP the amplitude seems to have already increased. When BRITE-Constellation picked up $\beta$ Pic again in November 2017 for the third season, the amplitude continued to increase. This effect is evident despite the fact that ASTEP and BHr are different instruments carrying different filters.

The other four highest-amplitude frequencies – F3, F8, F11 and F15 – vary to a much smaller extent or remain basically at a constant level during this season (top panel in Figure 8). These four frequencies have been also identified in Mékarnia et al. (2017), but not marked as showing variable amplitudes.

The pulsation phases for the five selected frequencies during the BRITE 2017/18 observations can clearly be regarded as stable (middle panel in Figure 8). As the changes in the amplitudes of $\beta$ Pic  are not correlated with changes in phases (bottom panel in Figure 8), we interpret the variability of the amplitudes as being intrinsic and not caused by beating of two or more unresolved modes.

In a final test we used the bRing data set for a comparable investigation of amplitude variability. Due to the higher noise in the data compared to the BRITE observations, we had to choose 100-day subsets with 50-day overlaps to detect frequencies F8, F11, and F13. Unfortunately, the uncertainties of the amplitudes derived in the 100-day subsets are too high for an analogous interpretation of amplitude variability which can be seen in Figure 18.

6 Asteroseismic interpretation

We used the pulsation frequencies and amplitudes derived from up to five passbands to identify the pulsation modes from comparing the observed and theoretical normalised amplitudes. The five filters are BRITE B, BRITE R, SMEI, and bRing (filter information is given in Table 1) together with the previously published 31 frequencies using the ASTEP instrument (Mékarnia et al. 2017) which uses a Sloan $i^{\prime}$ filter (passband from 695 to 844 nm). The transmission curves of the filters of all five instruments used in the present analysis are illustrated in Figure 9.

In order to carry out mode identification we used a sequence of seven $1.8\,M_{\odot}$ models from the Self-Consistent Field method (Jackson et al. 2005; MacGregor et al. 2007) with rotation rates ranging from [math] to $0.6\,\Omega_{\mathrm{K}}$ in increments of $0.1\,\Omega_{\mathrm{K}}$, where $\Omega_{\mathrm{K}}=\sqrt{GM/R^{3}}$ is the Keplerian break-up rotation rate. The initial mass for the models was chosen based on the value of $1.8\,M_{\odot}$ given by Wang et al. (2016). The models are Zero Age Main Sequence (ZAMS) models which is fairly realistic in the case of $\beta$ Pic which was identified as a 23 Myr-old ZAMS star (Mamajek & Bell 2014). We then calculated their low-degree acoustic mode pulsations for $n$ ranging from $1$ or $2$ to $10$, $\ell$ from [math] to $3$, and $m$ from $-3$ to $3$, using the adiabatic version of TOP (Two-dimension Oscillation Program, see Reese et al. 2006, 2009). Pseudo non-adiabatic mode visibilities (i.e., disk-integrated geometric factors) were derived using the approach described by Reese et al. (2013, 2017) where the non-adiabatic pulsation amplitudes and phases came from 1D pulsation calculations using the MAD code (Dupret 2001) in non-rotating stellar models which span the relevant range in effective temperature and gravity. These mode visibilities were calculated for all inclinations between $0^{\circ}$ and $90^{\circ}$ in increments of $1^{\circ}$. The intensities in different photometric bands at each point of the stellar surface were obtained by integrating a black body spectrum at the latitude-dependent effective temperature multiplied by the filter’s transmission curve, thus taking gravity darkening into account. Limb darkening was included by multiplying these intensities by a Claret law (Claret 2000) for the filter777The relevant transmission curves were downloaded from: http://www.aip.de/en/research/facilities/stella/instruments/data/johnson-ubvri-filter-curves which was the closest match to the photometric bands used here, as listed in Table 4. The correlations given in the third column of Table 4 are defined as $\int_{0}^{\infty}f_{1}(\lambda)f_{2}(\lambda)\mathrm{d}\lambda/\sqrt{\int_{0}^{\infty}f_{1}^{2}(\lambda)\mathrm{d}\lambda\int_{0}^{\infty}f_{2}^{2}(\lambda)\mathrm{d}\lambda}$, where $\lambda$ is the wavelength.

As a first step, we searched for the low-degree modes that individually provide the best match to the observed amplitudes. We did not attempt to match observed phase differences as the pseudo non-adiabatic calculations were not considered to be sufficiently reliable to provide accurate theoretical phase differences. We chose not to compare normalised amplitudes directly because this amounts to choosing one of the photometric bands as a reference band and normalising the amplitudes in the other bands with respect to the amplitude in this reference band. This can lead to difficulties if the amplitude in the chosen reference band is close to zero. Instead, we normalised the observed amplitudes so that the sum of their squares equals one. For the sake of consistency, the errors on the amplitudes are also normalised by the same factor. The theoretical amplitudes are then normalised so as to optimize the $\chi^{2}$ fit to the observations taking into account the errors. Also, given the relatively large increment on the rotation rate, the normalised amplitudes and frequencies were interpolated to intermediate rotation rates in increments of $0.01\,\Omega_{\mathrm{K}}$.

Figure 10 shows a comparison between the observed and best-fitting normalised amplitudes. A good match can be observed for most of the modes and a relatively low $\chi^{2}$ value is obtained. However, since the modes were fit individually, the inclinations and rotation rates obtained for the different modes do not match. Fig. 11 illustrates these results: dark blue symbols represent the best solutions (i.e., those illustrated in Fig. 10). The small light blue dots are other solutions that satisfy the criterion $\chi^{2}\leq(N_{bands}-1)$, where $\chi^{2}$ is the $\chi^{2}$-value on the amplitudes for that particular mode and $N_{bands}$ is the number of bands in which that mode is detected. On the right side of the latter relation, $(N_{bands}-1)$ was chosen and not $N_{bands}$ as the intrinsic amplitude of theoretical mode is a free parameter. If none of the solutions (including the best solution) satisfies the above criterion, then the best solution is plotted using a square rather than a star. Hence, the light blue solutions give an idea of the uncertainties on these solutions. Hence, in order to obtain a more coherent result it is necessary to fit the modes simultaneously using a fixed value of the inclination and rotation rate. The fit to the normalised amplitudes shown here are nonetheless useful as they represent the best fit one can hope to achieve using our set of theoretical mode visibilities.

We then fit the frequencies and normalised amplitudes simultaneously. In order to achieve this, we carried out Monte Carlo Markov Chains (MCMC) runs using the python emcee package (Foreman-Mackey et al. 2013) and using the rotation rate, $\Omega/\Omega_{\mathrm{K}}$, the inclination, $i$, and a dimensionless scale factor, $f$, on the frequencies as free parameters together with the free amplitudes of the 15 pulsation frequencies. These 18 free parameters are then used to fit 46 amplitudes in five passbands, 15 pulsation frequencies and four classic constraints (mass, radius, $\log g$, and ${\upsilon}\sin i$) corresponding to in total 65 observables. We assumed a uniform prior on $\Omega/\Omega_{\mathrm{K}}$ over the interval $[0.0,0.6]$ and on $f$ over the interval $[0.5,2]$. The prior on $i$ was proportional to $\sin i$ in accordance with what is expected for random orientations of the rotation axis, although as will be explained in what follows we sometimes restricted the range of $i$ values. The scale factor was introduced to compensate for the fact that the set of stellar models was limited to a single mass. Indeed, multiplying the frequencies by a dimensionless scale factor $f$ amounts to carrying out a homologous transformation in which the mean density of the model is multiplied by $f^{2}$ thus providing a poor substitute for modifying the mass. This leads to the following relations:

[TABLE]

where quantities with the subscript “$\mathrm{mod}$” refer to the model prior to scaling, and those without this subscript to the scaled model (which should hopefully be close to the actual properties of $\beta$ Pic). We note that since our models only depend on $\Omega/\Omega_{\mathrm{K}}$, this parameter also uniquely determines $\Omega_{\mathrm{K,\,mod}}$. Finally, when matching the theoretical modes to the observed ones, we were careful to avoid assigning the same identification to two different observed modes.

Once the values of $\Omega/\Omega_{\mathrm{K}}$, $f$, and $i$ are fixed, it is possible to deduce the radius of the scaled model using the observational constraints on $\log g$ and ${\upsilon}\sin i$ (recalled in Table 5 below) using two different methods. Indeed, one has the following relations:

[TABLE]

where we recall that $\Omega_{\mathrm{K}}=f\Omega_{\mathrm{K,\,mod}}$. We note that we have neglected possible differences between the polar and equatorial radius in the above expressions. Given that the values of $\Omega/\Omega_{\mathrm{K}}$, $f$, and $i$ are selected by the MCMC algorithm, there is no reason to assume that the two above expressions give a priori the same values of $R$. One could therefore eliminate one of the free parameters in the MCMC calculations. We, however, opted for a different approach which consists in keeping all of the free parameters, calculating $R$ using both of the above equations, and rejecting solutions for which the two values of $R$ differ by more than $20\,\%$. The final value of $R$ is then obtained by minimizing the following least-squares cost function:

[TABLE]

The mass is subsequently obtained via the Keplerian breakup rotation rate and the above radius:

[TABLE]

At this point, it is useful to discuss the classical constraints on $\beta$ Pic’s fundamental atmospheric parameters. In Table 5 we list various constraints found in the literature. We note that other values have been obtained for some of these parameters: $\log g=4.15$ dex according to Gray et al. (2006), the angular diameter $\theta=0.84\pm 0.12\pm 0.10$ mas including limb darkening according to Di Folco et al. (2004), $\theta=0.712\pm 0.010$ mas based on surface-brightness relations from Kervella et al. (2004) (see Defrère et al. 2012) and ${\upsilon}\sin i=130\pm 4$ $\mathrm{km\,s}^{-1}$ according to Royer et al. (2002). It is then interesting to investigate to what extent these values are self-consistent. For this, we minimized the following cost function in order to obtain a coherent set of parameters:

[TABLE]

where $d$ is one astronomical unit which is used to convert parallax into distance, quantities with the subscript “$\mathrm{obs}$” are the observed values from Table 5, $\sigma_{x}$, $x$ representing any one of these quantities and the associated uncertainties, and $\sigma_{M}$ the average of the two errors on mass from the table. As can be seen, the above cost function makes use of the relations:

[TABLE]

where $\pi$ represents the parallax in the third equation. The best fit is provided in the fourth column of Table 5. As can be seen, the observational constraints are mostly consistent. This would have been less true if some of the alternate values such as $\log g=4.15$ dex (Gray et al. 2006) or $\theta=0.84\pm 0.12\pm 0.10$ (Di Folco et al. 2004) had been used. The errors were estimated by carrying out 10000 Monte Carlo realizations of the observed errors before carrying out the above minimization and taking the standard deviation of the resultant parameters. In some cases, these are very similar to the observed errors (e.g. the mass), whereas in other cases, they are considerably smaller (e.g. $\log g$).

In the MCMC runs that follow, we used the observational constraints directly rather than using the results from the fit, but made sure the observational constraints encompassed the results from the fit. The error on the mass was increased by $50\,\%$ to account for possible uncertainties resulting from the limited time span over which $\beta$ Pic b has been observed compared to the estimated orbital period (thus affecting the stellar mass estimate). We excluded solutions with masses or radii more than $3\sigma$ away from the estimated values in order to avoid having the classic constraints drowned out by the seismic ones. Also, we preferred the ${\upsilon}\sin i$ value from Koen et al. (2003) over that of Royer et al. (2002) due to a slightly larger resolution and signal-to-noise ratio in the spectroscopic observations, although we do note that the two values are within $2\sigma$ of each other. We recall that the values and errors of ${\upsilon}\sin i$ and $\log g$ intervene when calculating the radius during the MCMC runs (see Eq. 4). These constraints are summarized in the last column of Table 5.

We first start with an MCMC run using the observational errors on the frequencies and on the amplitudes. Figure 12 shows the best solution999This is the solution with the lowest $\chi^{2}$ value. This does not necessarily imply that its mode identification is the most represented in the sample of solutions. Hence, it may not be the dominant colour in the triangle plots (Figs. 13 and 14). found from the run. The upper panel compares observed and theoretical amplitude in different photometric bands, whereas the lower panel compares the resultant spectra. A fairly good fit to the spectrum is achieved but at the expense of the normalised amplitudes. Indeed, the extremely small errors on the frequencies mean that the normalised amplitudes are hardly taken into account. The parameters of this fit are given in the first row of Table 6 and the corresponding mode identifications are provided both in Fig. 12 and the second column of Table 7.

In order to get a more complete picture of the posterior probability distribution resulting from the above constraints, we show a triangle plot of the distribution of solutions obtained via the MCMC run in Fig. 13. This figure contains scatter plots for pairs of variables and histograms for single variables. These are colour-coded according to the mode identification in the solution. The MCMC run produced several isolated groups of solutions in the parameter space, associated with different mode identifications. Accordingly, providing errors which cover all of these solutions is not meaningful – instead, in what follows, we provide errors along with the statistical averages for the group of solutions with the same identification as the best solution, i.e. the solution which minimizes our cost function. These are provided on the line below the best solution in Table 6 for each configuration. Nonetheless, some of the other groups of solutions can also be quite important. For instance, the turquoise group contains a number of solutions with inclinations ranging from $83$ to $90^{\circ}$, which seems more realistic given the orientation of the orbit of $\beta$ Pic b and the disk around the star. The statistical averages and standard deviations are provided on the third line of Table 7 (excluding the line with the headers). The corresponding mode identifications are provided in the third column of Table 7.

In order to increase the weight of the observed amplitudes in the fit, we carried out other MCMC runs using a uniform frequency tolerance, denoted $\sigma_{\mathrm{F}}$. This frequency tolerance acts as a trade-off parameter between fitting the spectrum and matching the amplitudes. If a small value is chosen for $\sigma_{\mathrm{F}}$, the MCMC algorithm will favor solutions where the frequencies are a good match to the observations, but the amplitudes will be a poor fit. Conversely, large values of $\sigma_{\mathrm{F}}$ lead to the opposite behavior. Figure 19 in Appendix C shows the solutions obtained for $\sigma_{\mathrm{F}}=0.01$ d-1 and $0.05$ d-1  respectively. The corresponding identifications are provided in these figures as well as in Table 7. The resultant parameters are given in lines 4 to 7 of Table 6 (excluding the header line). The different choices of $\sigma_{\mathrm{F}}$ have lead to rather different solutions but which both match the observations fairly well. Moreover, the solution for $\sigma_{\mathrm{F}}=0.05$ d-1  is fairly close to equator-on. This may be realistic for $\beta$ Pic due to the nearly edge-on configuration of its disk and the planet’s orbit, and the lack of plausible mechanisms able to misalign the system. Nonetheless, this solution includes a few odd modes, i.e. modes which are antisymmetric with respect to the equator, as can be seen in Table 7 (by calculating the parity of $\ell+m$). This seems somewhat unrealistic since such modes should cancel out for an equator-on configuration and thus not be visible. Accordingly, we carried out an MCMC run for $\sigma_{\mathrm{F}}=0.05$ d-1  including only even modes and restricting the inclination to values between $70^{\circ}$ and $90^{\circ}$ (rather than $0^{\circ}$ to $90^{\circ}$). The corresponding solution is also shown in Fig. 19, and the resultant parameters given in Table 6. Although this solution has fairly similar parameter values for $\Omega/\Omega_{\mathrm{K}}$ and $i$ as the previous solution including even and odd modes, this solution corresponds to a completely different set of mode identifications. Furthermore, the inclination $i=76.8^{\circ}$ is not entirely satisfactory as it is sufficiently far from equator-on to invalidate the exclusion of odd modes.

A further MCMC run is carried out using $\sigma_{\mathrm{F}}=0.1$ d-1 in search of solutions which are closer to equator-on. The best solution is shown in Fig. 12 and the corresponding parameters given in Table 6. This solution is much closer to equator on. In fact, the statistical average value for the inclination provided in Table 6 is biased by the fact that the inclination range is bounded by $90^{\circ}$ thus artificially leading to a lower average. This best solution contains several modes with a similar identification as was obtained for $\sigma_{\mathrm{F}}=0.05$ d-1 with even and odd modes, apart from an offset of $1$ on the radial order. Finally, Figure 14 provides a colour-coded triangle plot showing the distribution of solutions in parameter space. A diversity of identifications are obtained. Furthermore, the group associated with the best solution (light green-yellow) clearly peaks at $90^{\circ}$, thus favoring an equator-on configuration.

Another factor to be taken into account is the fact that a couple of the amplitudes change significantly in the BRITE R band between two observational runs. Indeed, the F11 mode at 50.49 d-1 has an amplitude of 0.84 mmag in 2016/17 and increases to 1.06 mmag in 2017/18. Likewise, the F13 mode at 53.69 d-1 has an amplitude of 0.40 mmag in 2016/17 in BRITE R and increases to 1.27 mmag in 2017/18. The fits carried out so far were primarily based on the 2016/17 data. We therefore carried out a few more MCMC runs using the 2017/18 data instead. The resultant best solutions as well as relevant statistical averages and standard deviations are listed in Table 6. The corresponding mode identifications are provided in Table 7. The choice of observing season does have some impact on the values of $\Omega/\Omega_{\mathrm{K}}$, $i$, and $f$, especially for $\sigma_{\mathrm{F}}=0.05$ d-1 using only even modes. The mode identification is completely modified for this particular case, but is hardly affected for $\sigma_{\mathrm{F}}=0.1$ d-1 with even modes.

The fact that modifying the amplitudes of two modes can affect the entire mode identification is not entirely surprising since the MCMC algorithm is optimizing the fit to all of the modes simultaneously. Also, in almost all cases, the seismic and amplitude-related values of $\chi^{2}$ are degraded. This shows that the models clearly provide a better match to the $2016/17$ run over the $2017/18$ run. One possible explanation for this worse fit is the fact that the observations in the various photometric bands are not simultaneous and are for the most part more representative of the $2016/17$ time period. Ideally, amplitude changes should proportionally be the same in the different bands over similar time periods, thus canceling out when calculating normalised amplitudes given that these only depend on mode geometry.

Overall, our favored solution is the one obtained for $\sigma_{\mathrm{F}}=0.1$ d-1, using only even modes. Indeed, this solution seems to be the most coherent in terms of stellar inclination, given the measured inclination of the planetary orbit and the circumstellar disk of 88.81$\pm 0.12^{\circ}$ (Wang et al. 2016). It also leads to the best fit with the normalised amplitudes. However, the price to pay is a relatively high $\chi^{2}$ value for the seismic component (although the theoretical frequencies come in the same order as the observed ones). Possible causes for this significant difference in the frequencies include shortcomings in the stellar models. In particular, the models are uniformly rotating. This does not seem very realistic because baroclinic effects are expected to lead to differential rotation as shown in more realistic models based on the ESTER code (Espinosa Lara & Rieutord 2013; Rieutord et al. 2016). This in turn will affect rotational splittings (Reese et al. 2009), thus modifying the frequencies of non-axisymmetric modes. Also, the differences between the fitted and observed mode amplitudes still remain relatively high even for the most favorable solution. Possible causes for this include the use of pseudo non-adiabatic mode visibilities rather than visibilities based on fully non-adiabatic calculations.

However, at this point it is not possible to obtain reliable fully non-adiabatic calculations. Indeed, one would need rapidly rotating models which solve the energy conservation equation in a consistent way – this is currently only achieved in the ESTER code. However, the ESTER code does not currently model sub-surface convective envelopes which is expected to be relevant in a star of this mass. Another limitation in the visibility calculations is the fact that they do not rely on realistic model atmospheres but rather on blackbody spectra as was pointed out earlier. Finally, another factor which needs to be considered is the set of modes used in the identification. Indeed, we restricted ourselves to modes with $\ell$ between [math] and $3$. However, visibility calculations at rapid rotation rates suggest that higher $\ell$ modes may become more visible (e.g. Lignières & Georgeot 2009). Carrying out fits with higher $\ell$ values does lead, as expected, to better fits with $\chi^{2}_{\mathrm{ampl}}$ below $400$ and/or $\chi^{2}_{\mathrm{seismic}}$ below $1000$ in some cases. Hence, an alternate approach may be to select modes based on their visibilities at the relevant rotation rate, rather than using a predefined set of modes as was done above. Nonetheless, non-linear mode coupling and saturation mechanisms may lead to intrinsic mode amplitudes that alter which modes are actually observed compared to what would be expected from geometric visibility factors.

7 Discussion and Conclusion

The exoplanet host star $\beta$ Pic  was already known to show p-mode pulsations of $\delta$ Scuti -type (Koen 2003; Koen et al. 2003; Mékarnia et al. 2017). As observations with the Kepler space telescope (Borucki et al. 2010) revealed that many $\delta$ Scuti  stars show both, p- and g-modes (Uytterhoeven et al. 2011), we also investigated the presence of g-modes in our $\beta$ Pic data sets which would be expected to lie in the frequency range between 0.3 and 3  d-1 (Aerts et al. 2010). BRITE-Constellation observations are known to be in particular sensitive to frequencies in this range (e.g., Baade et al. 2018b; Ramiaramanantsoa et al. 2018) due to the satellites’ observing strategy (Weiss et al. 2014).

Using both, our best data set alone (i.e., the BRITE R filter observations of 2016/17 which reach the lowest residual noise level) and a combination of three seasons of BRITE R filter observations, we do not find evidence for the presence of g-modes down to a residual noise level of 36 ppm. It is therefore evident that if g-modes exist in $\beta$ Pic, they must possess even lower amplitudes that remain undetected in the data sets analyzed here.

Our 15 identified pulsation frequencies correspond to the 14 highest amplitude frequencies in Mékarnia et al. (2017) which are numbered f1 to f14 and their frequency f23. As the residual noise level of 9.45 ppm is significantly lower in Mékarnia et al. (2017) compared to our best data set (i.e., the BRITE R filter observations obtained in 2016/17) with a residual noise level of 40 ppm, not all the pulsation frequencies reported earlier are identified to be significant in our analysis.

We used the amplitudes of 15 $\delta$ Scuti -type p-mode frequencies detected in up to five different passbands - BRITE B & R, SMEI, bRing and ASTEP $i^{\prime}$ - to calculate normalised amplitudes for an asteroseismic study of $\beta$ Pic and an identification of its pulsation modes. This analysis was complicated by the fact that two pulsation frequencies show a clear variability in our time series observations: The frequency F13 at 53.6917 d-1 first decreases between the 2015 and 2016/17 observations and then increases again between 2016/17 and 2018; the amplitude of frequency F11 at 50.4921 d-1 constantly increases in the observations obtained between 2015 and 2018. This behaviour could consistently be found in the BRITE-Constellation and the bRing data and confirms earlier reports by Mékarnia et al. (2017). All other pulsation modes have stable amplitudes within the observational errors.

In general, the variability of certain amplitudes should proportionally be the same in different bands over similar periods of time; thus amplitude variability should not impact the calculations of normalised amplitudes given that these only depend on mode geometry. In the present case, the observations obtained by BRITE-Constellation, bRing and ASTEP were not taken exactly simultaneously, but with overlapping periods of time, and the SMEI photometry was obtained years before. Hence, the variable amplitudes of the two modes affect the observed normalised amplitudes and thus the identification of the pulsation modes.

For our asteroseismic interpretation of the normalised amplitudes we used the most precise BRITE R filter data set (i.e., the $\sim$224-day long BRITE R filter observations obtained in 2016/17 with a residual noise level of 40 ppm), the simultaneous BRITE B filter observations of 2016/17, the overall amplitudes obtained from the bRing and SMEI data and the previously published ASTEP $i^{\prime}$ amplitudes. From this, our favored solution of the asteroseismic models was obtained for a relatively large value for the frequency tolerance $\sigma_{\mathrm{F}}=0.1$  d-1 on the frequencies, only including even modes. This leads to the best fit to the normalised amplitudes, while at the same time getting a near equator-on inclination of $i=89.1^{\circ}$, which is in agreement with our expectations based on the orbital inclination of $\beta$ Pic b as well as that of the circumstellar disk. Correspondingly, the pulsation modes were identified as three $\ell=1$, six $\ell=2$ and six $\ell=3$ modes. Our preferred model also yields a rotation rate of $\sim$27% of Keplerian breakup velocity, a radius of $1.497\pm 0.025$ $R_{\odot}$  and a mass of $1.797\pm 0.035$ $M_{\odot}$  corresponding to an error of $\sim$2% in stellar mass and less than 2% in stellar radius. These errors do not account for uncertainties in the models and for errors resulting from an erroneous mode identification. The fact that the difference between the observations and the theory remains high implies that the model errors could be quite significant. Hence, although the errors on mass and radius of $\beta$ Pic  are quite small, they only account for a small part of the true error. The choice of observing season only has a limited impact on the values of $\Omega/\Omega_{\mathrm{K}}$, $i$, and $f$ when assuming a relatively large observational frequency tolerance $\sigma_{\mathrm{F}}=0.1$  d-1 on the frequencies. Finally, the choice of what set of theoretical modes should be considered when fitting the observations remains an open question.

Our analysis yields an independent and more accurate determination of the stellar parameters based on the combination of classic constraints with the pulsational properties of $\beta$ Pictoris derived in multiple passbands. We illustrate that adding seismic constraints considerably reduces the set of acceptable theoretical models, hence, resulting in higher precision.

Mode identification in rapidly rotating $\delta$ Scuti stars is one of the outstanding problems in stellar physics (e.g., Goupil et al. 2005; Deupree 2011). Our work constitutes an important step in addressing this, hence illustrates the importance of good priors on the classic quantities. However, the seismic analysis still manages to further restrict the acceptable values for the mass and radius. Additionally, $\beta$ Pictoris is a $\delta$ Scuti type star where most pulsation amplitudes remain stable over many years, while a few change sometimes even significantly. Hence, it is another candidate for future studies of the physical reasons of amplitude variability versus stability in $\delta$ Scuti stars. Our search for g-modes in our data sets was motivated by the idea to use them to probe the near-core region of $\beta$ Pictoris; the detection of g-modes would allow us to investigate differential rotation in the stellar interiors using prograde and retrograde pulsation modes (e.g., Zwintz et al. 2017) and study the angular momentum distribution (e.g., Aerts et al. 2017).

The absence of a magnetic field (i.e., if there is a magnetic field, its strength has to be lower than 300 Gauss) might also be a crucial factor to be considered when studying the exoplanetary system and circumstellar disk around $\beta$ Pictoris.

Acknowledgements.

The authors thank Marc-Antoine Dupret for results from the MAD code which allowed us to calculate pseudo non-adiabatic mode visibilities. We thank P. Kervella and F. Royer for helpful discussions and clarifications concerning some of the constraints on $\beta$ Pictoris. We also express our gratitude to our referee, Dietrich Baade, for his very constructive comments. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. K.Z. acknowledges support by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung (FWF, project V431-NBL) and the Austrian Space Application Programme (ASAP) of the Austrian Research Promotion Agency (FFG). D.R.R. acknowledges the support of the French Agence Nationale de la Recherche (ANR) to the ESRR project under grant ANR-16-CE31-0007 as well as financial support from the Programme National de Physique Stellaire (PNPS) of the CNRS/INSU co-funded by the CEA and the CNES. A.Pi. acknowledges support from the NCN grant 2016/21/B/ST9/01126. APo was responsible for image processing and automation of photometric routines for the data registered by the BRITE nano-satellite constellation, and was supported by the statutory activities grant BK/200/RAU1/2018 t.3. GH thanks the Polish National Center for Science (NCN) for support through grant 2015/18/A/ST9/00578. The research of S.M.R. and A.F.J.M. has been supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. GAW acknowledges Discovery Grant support from the Natural Science and Engineering Research Council (NSERC) of Canada. MI was the recipient of an Australian Research Council Future Fellowship (FT130100235) funded by the Australian Government. SNM is a U.S. Department of Defense SMART scholar sponsored by the U.S. Navy through SSC-LANT. Part of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. E.E.M. and S.N.M. acknowledge support from the NASA NExSS program. The bRing observatory at Siding Springs, Australia was supported by a University of Rochester University Research Award.

beta Pictoris

Appendix A Supplementary figures for the frequency analysis of BRITE data

Figure 15 shows the amplitude spectrum of the original BHr data (grey) with the eight identified frequencies (red). The orbital frequency of BHr at 14.831 d-1 and its multiples are given as dashed dark-red lines. Note that some of the peaks with high amplitude are alias frequencies to one of the pulsation frequencies with the orbital frequency and disappear after prewhitening.

The amplitude spectrum using the BHr 2017/18 data is shown in Figure 16 where again the frequencies reported by Mékarnia et al. (2017) are marked as grey triangles, the orbital frequency of BHr and its multiples are identified as dark red dashed lines and the spectral window is given in the top panel.

The frequency analysis of the combined BRITE R filter data yielded the same pulsation frequencies as given in Table 2. The corresponding amplitude spectrum with the identified frequencies marked in red is shown in Figure 17.

Appendix B Supplementary material for amplitude variability using bRing data

Investigation of the amplitude variability using the bRing time series which consists of observations taken in South Africa and Australia. As the Australian bRing instrument had first light on December 1, 2017, corresponding to a JD of $\sim$2458091.0, we did not detect the three frequencies F8, F11, F13 in the subsets using the first about 200-days of the bRing light curve. Unfortunately the errors on the amplitudes calculated from the individual subsets are mostly too high for a reliable study of amplitude variability from this data set.

Appendix C Supplementary material for the asteroseismic interpretation

Figure 19 shows the best matches from the MCMC runs between observed and theoretical multi-colour amplitudes and spectra using $\sigma_{\mathrm{F}}=0.01$ and $\sigma_{\mathrm{F}}=0.05$  d-1 as tolerances on the frequencies and restricting the theoretical spectrum to even modes and the inclination to the $[70^{\circ},90^{\circ}]$ interval in the last case.

Table C.1. lists the complete theoretical model frequency spectra from our best fitting solutions. The frequencies in bold face are those which matched $\beta$ Pic’s pulsations. Those in italics were filtered out before the fitting process because they correspond to modes which are antisymmetric with respect to the equator, and only near equator-on solutions were being searched for in those particular MCMC runs. We note that these predicted frequencies are subject to large uncertainties in the models, and will change significantly if the input physics will be improved.

See pages - of longtable.pdf