Rotational Splitting and Asteroseismic Modeling of the δ Scuti Star EE Camelopardalis

The effective temperature T_eff of EE Cam is ${T}_{\mathrm{eff}}={6469}_{-73}^{+65}$ K and the [Fe/H] abundance is $[\mathrm{Fe}/{\rm{H}}]={0.24}_{-0.13}^{+0.12}$ dex according to the catalog by Ammons et al. (2006). Based on the Hipparcos parallax 4.34 ± 0.63 mas of van Leeuwen (2007), Breger et al. (2015) estimated the luminosity of EE Cam to be $\mathrm{log}L/{L}_{\odot }={1.53}_{-0.12}^{+0.14}$. The catalog of Nordström et al. (2004) shows values of T_eff and [Fe/H] being 6530 K and 0.06 respectively. In our work, we adopt a higher uncertainty of 200 K (e.g., Breger et al. 2015) for the effective temperature T_eff of Ammons et al. (2006). Meanwhile, we use a large range for the value of [Fe/H], which varys from 0 to 0.36 dex, to cover the results of Nordström et al. (2004) and Ammons et al. (2006).

We compute our theoretical models with the Modules for Experiments in Stellar Astrophysics (MESA; Paxton et al. 2011, 2013). The submodule "pulse" of version 6596 is used to compute stellar evolutionary models and to compute their corresponding oscillation frequencies (Christensen-Dalsgaard 2008; Paxton et al. 2011, 2013). Our theoretical models are constructed on basis of the OPAL opacity table GS98 (Grevesse & Sauval 1998) series. The Eddington gray atmosphere T–τ relation in the atmosphere integration is used. The mixing-length theory of Böhm-Vitense (1958) is chosen to treat convection. Effects of element diffusion, convective overshooting, and rotation are not included in our calculations.

In our calculations, we fix the mixing-length parameter α to the solar value of 1.80 and set the initial helium fraction Y = 0.245 + 1.54Z (e.g., Dotter et al. 2008; Thompson et al. 2014), as a function of the metallicity Z. The value of Z varies from 0.015 to 0.035 with a step of 0.001. The stellar mass M varies from 1.5 M_⊙ to 2.5 M_⊙ with a step of 0.01 M_⊙.

Figure 1 illustrates the evolutionary tracks of the theoretical models on the Hertzsprung–Russell Diagram. In this figure, the rectangle marks the 1σ error box of the effective temperature T_eff and the luminosity log $\mathrm{log}L/{L}_{\odot }$ (i.e., 6269 K < T_eff < 6669 K and 1.41 < $\mathrm{log}L/{L}_{\odot }$ < 1.67). We calculate frequencies of oscillation modes with ℓ = 0, 1, and 2 for every stellar model falling inside the error box and fit them to the observed frequencies according to

$$\begin{eqnarray}&&{\chi }^{2}=\displaystyle \frac{1}{k}\sum \,(| {\nu }_{i}^{\mathrm{theo}}-{\nu }_{i}^{\mathrm{obs}}{| }^{2}).\end{eqnarray} \tag{ 3 }$$

In Equation (3), ${\nu }_{i}^{\mathrm{theo}}$ denotes the theoretical frequency, ${\nu }_{i}^{\mathrm{obs}}$ denotes the observed frequency, and k denotes the number of the observed frequencies.

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In Section 2, we identify the observed frequencies based on the regularities of rotational splitting. In particular for the oscillation frequencies (f₂₅, f₁₁, f₂₈) in Multiplet 1 and (f₁₀, f₃₂, f₃₇) in Multiplet 6, mode identifications are unique and their m = 0 components are detected. When we do model fittings, we only use the two central components, f₁₁, and f₃₂, as well as the the fundamental radial mode f₁. Breger et al. (2015) identified f₁ as a radial mode based on the analyses of theoretical phase differences and amplitude ratios, and then suggested f₁ as the fundamental radial mode based on a detailed analysis of pulsation instability. We use the identification of f₁ as the fundamental radial mode in our calculations.

In Figure 2, we illustrate the changes of $1/{\chi }^{2}$ as a function of the effective temperature T_eff for grid models. Each curve in Figure 2 corresponds to one evolutionary track in Figure 1. It can be clearly seen that $1/{\chi }^{2}$ of the theoretical model with Z = 0.028 and M = 2.04 M_⊙ is much larger than those of other theoretical models. We therefore choose the model with the minimum value of ${\chi }^{2}=0.00035$ as the best-fitting model, and mark it with a filled circle in Figure 2. The fundamental parameters of the best-fitting model are M = 2.04 M_⊙, Z = 0.028, T_eff = 6433 K, $\mathrm{log}L/{L}_{\odot }=1.416$, R = 4.120 R_⊙, and log g = 3.518.

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Theoretical oscillation frequencies deduced from the best-fitting model are listed in Table 2, in which n_p is the number of radial nodes in propagation cavity of p modes, and n_g is the number of radial nodes in propagation cavity of g modes. The parameter β_ℓ,n measures the size of rotational splitting. Its general expression for a uniformly rotating star is described by Christensen-Dalsgaard (2003) as

$$\begin{eqnarray}&&{\beta }_{{\ell },n}=\displaystyle \frac{{\int }_{0}^{R}({\xi }_{r}^{2}+{L}^{2}{\xi }_{h}^{2}-2{\xi }_{r}{\xi }_{h}-{\xi }_{h}^{2}){r}^{2}\rho {dr}}{{\int }_{0}^{R}({\xi }_{r}^{2}+{L}^{2}{\xi }_{h}^{2}){r}^{2}\rho {dr}},\end{eqnarray} \tag{ 4 }$$

where ${L}^{2}={\ell }({\ell }+1)$, ρ is the local density, and where ξ_r and ξ_h are the radial displacement and the horizontal displacement, respectively. For high-order g modes, ${\beta }_{{\ell },n}$ can be simplified into $1-\tfrac{1}{{\ell }({\ell }+1)}$, which is in accordance with the term in Equation (1). In Figure 3, we show the theoretical values of ${\beta }_{{\ell },n}$ for theoretical oscillation frequencies. As shown in the figure, most of the ${\beta }_{{\ell },n}$ are in accordance with the asymptotic value 0.5 for ℓ = 1 modes and 0.833 for ℓ = 2 modes. They show more pronounced g-mode characters in the star. Meanwhile, it can be found that ${\beta }_{{\ell },n}$ of several oscillation modes clearly deviate from the asymptotic value. They show more pronounced p-mode characters in the star.

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Table 2.  Theoretical Frequencies Derived from the Best-fitting Model

${\nu }^{\mathrm{theo}}({\ell },{n}_{p},{n}_{g})$	${\beta }_{{\ell },n}$	${\nu }^{\mathrm{theo}}({\ell },{n}_{p},{n}_{g})$	${\beta }_{{\ell },n}$	${\nu }^{\mathrm{theo}}({\ell },{n}_{p},{n}_{g})$	${\beta }_{{\ell },n}$	${\nu }^{\mathrm{theo}}({\ell },{n}_{p},{n}_{g})$	${\beta }_{{\ell },n}$
(μHz)		(μHz)		(μHz)		(μHz)
57.123(0, 0, 0)	⋯	62.767(1, 1, −47)	0.504	35.313(2, 0, −148)	0.834	58.223(2, 0, −88)	0.850
74.986(0, 1, 0)	⋯	64.110(1, 1, −46)	0.502	35.553(2, 0, −147)	0.834	58.806(2, 0, −87)	0.846
95.185(0, 2, 0)	⋯	65.526(1, 1, −45)	0.502	35.765(2, 0, −146)	0.835	59.437(2, 0, −86)	0.842
116.820(0, 3, 0)	⋯	67.006(1, 1, −44)	0.502	35.952(2, 0, −145)	0.835	60.104(2, 0, −85)	0.839
138.568(0, 4, 0)	⋯	68.537(1, 1, −43)	0.503	36.169(2, 0, −144)	0.834	60.795(2, 0, −84)	0.838
		70.094(1, 1, −42)	0.507	36.415(2, 0, −143)	0.834	61.505(2, 1, −84)	0.838
30.091(1, 0, −100)	0.500	71.639(1, 1, −41)	0.516	36.674(2, 0, −142)	0.834	62.228(2, 1, −83)	0.837
30.403(1, 0, −99)	0.500	73.156(1, 1, −40)	0.533	36.943(2, 0, −141)	0.833	62.957(2, 1, −82)	0.838
30.720(1, 0, −98)	0.500	74.708(1, 1, −39)	0.552	37.219(2, 0, −140)	0.833	63.682(2, 1, −81)	0.838
31.041(1, 0, −97)	0.500	76.354(1, 1, −38)	0.579	37.499(2, 0, −139)	0.833	64.394(2, 1, −80)	0.837
31.368(1, 0, −96)	0.500	77.946(1, 1, −37)	0.641	37.783(2, 0, −138)	0.833	65.109(2, 1, −79)	0.836
31.700(1, 0, −95)	0.500	79.322(1, 2, −37)	0.612	38.069(2, 0, −137)	0.833	65.857(2, 1,−78)	0.834
32.034(1, 0, −94)	0.500	81.043(1, 2, −36)	0.528	38.358(2, 0, −136)	0.833	66.655(2, 1, −77)	0.834
32.363(1, 0, −93)	0.500	83.174(1, 2, −35)	0.508	38.646(2, 0, −135)	0.834	67.500(2, 1, −76)	0.833
32.685(1, 0, −92)	0.500	85.536(1, 2, −34)	0.504	38.928(2, 0, −134)	0.834	68.378(2, 1, −75)	0.833
33.014(1, 0, −91)	0.500	88.075(1, 2, −33)	0.503	39.190(2, 0, −133)	0.834	69.274(2, 1, −74)	0.833
33.362(1, 0, −90)	0.500	90.773(1, 2, −32)	0.506	39.436(2, 0, −132)	0.834	70.165(2, 1, −73)	0.834
33.727(1, 0, −89)	0.500	93.591(1, 2, −31)	0.516	39.699(2, 0, −131)	0.834	71.015(2, 1, −72)	0.836
34.107(1, 0, −88)	0.500	96.305(1, 2, −30)	0.601	39.989(2, 0, −130)	0.833	71.835(2, 1, −71)	0.839
34.499(1, 0, −87)	0.500	98.085(1, 2, −29)	0.750	40.299(2, 0, −129)	0.833	72.713(2, 1, −70)	0.839
34.902(1, 0, −86)	0.500	100.413(1, 3, −29)	0.569	40.621(2, 0, −128)	0.833	73.682(2, 1, −69)	0.839
35.315(1, 0, −85)	0.500	103.547(1, 3, −28)	0.527	40.953(2, 0, −127)	0.833	74.712(2, 1, −68)	0.839
35.738(1, 0, −84)	0.500	106.822(1, 3, −27)	0.520	41.293(2, 0, −126)	0.833	75.780(2, 1, −67)	0.839
36.171(1, 0, −83)	0.500	110.184(1, 3, −26)	0.519	41.639(2, 0, −125)	0.833	76.874(2, 1, −66)	0.841
36.611(1, 0, −82)	0.500	113.868(1, 3, −25)	0.523	41.989(2, 0, −124)	0.833	77.982(2, 1, −65)	0.843
37.058(1, 0, −81)	0.499	117.968(1, 3, −24)	0.553	42.338(2, 0, −123)	0.833	79.089(2, 1, −64)	0.847
37.507(1, 0, −80)	0.499	121.376(1, 3, −23)	0.804	42.684(2, 0, −122)	0.833	80.185(2, 1, −63)	0.849
37.958(1, 0, −79)	0.499	123.691(1, 4, −23)	0.596	43.021(2, 0, −121)	0.833	81.290(2, 1, −62)	0.848
38.409(1, 0, −78)	0.499	128.581(1, 4, −22)	0.511	43.343(2, 0, −120)	0.833	82.454(2, 2, −62)	0.844
38.870(1, 0, −77)	0.499	134.355(1, 4, −21)	0.508	43.649(2, 0, −119)	0.833	83.701(2, 2, −61)	0.840
39.353(1, 0, −76)	0.499	140.605(1, 4, −20)	0.554	43.974(2, 0, −118)	0.833	85.021(2, 2, −60)	0.838
39.864(1, 0, −75)	0.499	143.267(1, 4, −19)	0.921	44.337(2, 0, −117)	0.833	86.392(2, 2, −59)	0.837
40.400(1, 0, −74)	0.499	148.171(1, 5, −19)	0.515	44.726(2, 0, −116)	0.833	87.775(2, 2, −58)	0.837
40.958(1, 0, −73)	0.499	⋯	⋯	45.128(2, 0, −115)	0.833	89.082(2, 2, −57)	0.839
41.534(1, 0, −72)	0.499	30.084(2, 0, −174)	0.833	45.537(2, 0, −114)	0.833	90.280(2, 2, −56)	0.841
42.125(1, 0, −71)	0.499	30.264(2, 0, −173)	0.834	45.951(2, 0, −113)	0.833	91.607(2, 2, −55)	0.839
42.729(1, 0, −70)	0.499	30.438(2, 0, −172)	0.834	46.370(2, 0, −112)	0.833	93.140(2, 2, −54)	0.839
43.342(1, 0, −69)	0.499	30.570(2, 0,−171)	0.837	46.795(2, 0, −111)	0.833	94.796(2, 2, −53)	0.839
43.964(1, 0, −68)	0.499	30.693(2, 0, −170)	0.835	47.223(2, 0, −110)	0.833	96.521(2, 2, −52)	0.841
44.593(1, 0, −67)	0.499	30.870(2, 0, −169)	0.834	47.647(2, 0, −109)	0.832	98.262(2, 2, −51)	0.846
45.228(1, 0, −66)	0.499	31.062(2, 0, −168)	0.834	48.051(2, 0, −108)	0.831	99.951(2, 2, −50)	0.854
45.873(1, 0, −65)	0.499	31.257(2, 0, −167)	0.833	48.434(2, 0, −107)	0.830	101.599(2, 2, −49)	0.856
46.539(1, 0, −64)	0.499	31.452(2, 0, −166)	0.833	48.831(2, 0, −106)	0.831	103.365(2, 2, −48)	0.850
47.236(1, 0, −63)	0.499	31.651(2, 0, −165)	0.833	49.270(2, 0, −105)	0.832	105.315(2, 3, −48)	0.844
47.968(1, 0, −62)	0.499	31.855(2, 0, −164)	0.833	49.741(2, 0, −104)	0.832	107.396(2, 3, −47)	0.841
48.737(1, 0, −61)	0.500	32.065(2, 0, −163)	0.833	50.232(2, 0, −103)	0.833	109.533(2, 3, −46)	0.841
49.541(1, 0, −60)	0.500	32.278(2, 0, −162)	0.833	50.737(2, 0, −102)	0.833	111.514(2, 3, −45)	0.849
50.380(1, 0, −59)	0.500	32.492(2, 0, −161)	0.834	51.249(2, 0, −101)	0.833	113.148(2, 3, −44)	0.852
51.250(1, 0, −58)	0.501	32.702(2, 0, −160)	0.834	51.766(2, 0, −100)	0.833	115.191(2, 3, −43)	0.844
52.147(1, 0, −57)	0.501	32.895(2, 0, −159)	0.835	52.281(2, 0, −99)	0.832	117.683(2, 3, −42)	0.841
53.065(1, 0, −56)	0.503	33.041(2, 0, −158)	0.836	52.782(2, 0, −98)	0.832	120.369(2, 3, −41)	0.842
53.993(1, 0, −55)	0.506	33.203(2, 0, −157)	0.834	53.263(2, 0, −97)	0.832	123.145(2, 3, −40)	0.846
54.909(1, 0, −54)	0.516	33.412(2, 0, −156)	0.834	53.737(2, 0, −96)	0.834	125.894(2, 3, −39)	0.855
55.798(1, 0, −53)	0.532	33.635(2, 0, −155)	0.833	54.234(2, 0, −95)	0.835	128.436(2, 4, −39)	0.865
56.699(1, 0, −52)	0.550	33.865(2, 0, −154)	0.833	54.769(2, 0, −94)	0.836	130.908(2, 4, −38)	0.857
57.649(1, 0, −51)	0.577	34.099(2, 0, −153)	0.833	55.338(2, 0, −93)	0.836	133.694(2, 4, −37)	0.848
58.557(1, 0, −50)	0.631	34.335(2, 0, −152)	0.833	55.929(2, 0, −92)	0.837	136.539(2, 4, −36)	0.857
59.358(1, 1, −50)	0.601	34.575(2, 0, −151)	0.833	56.528(2, 0, −91)	0.839	139.175(2, 4, −35)	0.865
60.334(1, 1, −49)	0.529	34.819(2, 0, −150)	0.833	57.115(2, 0, −90)	0.844	142.286(2, 4, −34)	0.853
61.499(1, 1, −48)	0.509	35.066(2, 0, −149)	0.833	57.674(2, 0, −89)	0.849	146.036(2, 4, −33)	0.846

Note. ${\nu }_{\mathrm{theo}}$ is the calculated frequency in units of μHz. n_p denotes the number of radial nodes in propagation cavity of p-mode. n_g denotes the number of radial nodes in propagation cavity of g-mode. ${\beta }_{{\ell },n}$ is one parameter  that measures the size of rotational splitting.

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In Table 3, we show comparisons of the theoretical frequencies and the observed frequencies. The theoretical frequencies with $m\ne 0$ are deduced from m = 0 modes in Table 2 based on the parameter ${\beta }_{{\ell },n}$. It can be seen in Table 3 that m = 0 components in Multiplets 1, 2, 3, 4, 6, and 10 are detected. However, m = 0 components in Multiplets 5, 7, 8, 9, 11, 12, 13, 14, and 15 have not been detected. The filled circles in Figure 3 mark m = 0 components in these multiplets of Table 3. As shown in Figure 3, the values of ${\beta }_{{\ell },n}$ for oscillation modes with m = 0 in Multiplets 1, 2, 3, 4, 6, and 10 are in good agreement with the asymptotic value of g modes. For Multiplets 7, 8, 9, 11, 12, 13, 14, and 15, the values of ${\beta }_{{\ell },n}$ of their corresponding m = 0 modes also agree well with the asymptotic value. The above analyses also show that performing mode identifications on basis of the regularities of g modes is self-consistent.

Table 3.  Comparsions of the Theoretcial Frequencies and the Observed Multiplets of Table 1

Multiplet	ID	${\nu }^{\mathrm{obs}}$	${\nu }^{\mathrm{theo}}({\ell },m)$	${\rm{\Delta }}\nu$	Multiplet	ID	${\nu }^{\mathrm{obs}}$	${\nu }^{\mathrm{theo}}({\ell },m)$	${\rm{\Delta }}\nu$
		(μHz)	(μHz)	(μHz)			(μHz)	(μHz)	(μHz)
	f25	110.510	110.571(1, −1)	0.061		f36	127.547	128.271(2, −2)	0.724
					8
1	f11	113.886	113.868(1, 0)	0.018		f38	132.804	133.723(2, −1)	0.919
	f28	117.063	117.164(1, +1)	0.101
						f30	119.420	120.104(2, −2)	0.684
					9
	f6	57.147	56.999(1, −1)	0.148		f33	124.662	125.506(2, −1)	0.844
2
	f2	60.345	60.334(1, 0)	0.011
						f27	115.266	115.191(2, 0)	0.075
					10
	f20	85.341	85.536(1, 0)	0.195		f35	125.866	125.830(2, +2)	0.036
3
	f21	88.719	88.712(1, +1)	0.007
						f18	79.056	79.678(2, −2)	0.622
					11
	f22	93.218	93.591(1, 0)	0.373		f8	95.664	95.581(2, +1)	0.083
4
	f3	96.445	96.843(1, +1)	0.398
						f19	81.195	81.117(2, −1)	0.078
					12
	f5	55.144	54.580(1, −1)	0.564		f9	97.869	96.943(2, +2)	0.926
5
	f15	61.582	62.534(1, +1)	0.952
						f16	69.725	69.482(2, −2)	0.243
					13
	f10	109.677	109.755(2, −2)	0.078		f7	91.500	90.887(2, +2)	0.613
6	f32	120.354	120.369(2, 0)	0.015
						f24	98.519	98.931(2, −2)	0.412
	f37	131.031	130.983(2, +2)	0.048	14
						f31	119.707	120.134(2, +2)	0.427
	f4	50.001	50.231(2, −2)	0.230
						f34	125.531	125.736(2, −2)	0.205
7	f14	55.310	55.513(2, −1)	0.203	15
						f39	147.122	147.342(2, +2)	0.220
	f17	71.803	71.359(2, +2)	0.444

Note. ${\nu }^{\mathrm{obs}}$ is the observed frequencies in units of μHz, ${\nu }^{\mathrm{theo}}$ is the theoretical frequencies in units of μHz. ${\rm{\Delta }}\nu =| {\nu }^{\mathrm{obs}}-{\nu }^{\mathrm{theo}}|$.

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Furthermore, we give possible mode identifications for f₁₂, f₁₃, and f₂₃, and list them in Table 4. For f₁₂, we find that the frequency (1, 4, −23, +1) 127.447 μHz may be its possible model counterpart. Moreover, we notice in Table 4 that both of (2, 2, −61, +2) 94.290 μHz and (2, 2, −57, +1) 94.370 μHz are possible model counterparts for f₂₃. As discussed in Section 2, the spherical harmonic degree of f₁₅ allows for two possibilities: ℓ = 1 or ℓ = 2. For the former case, f₅ and f₁₅ constitute one incomplete triplet (Multiplet 5 in Table 1). It can be seen in Table 3 that (1, 0, −50, −1) 54.580 μHz and (1, 0, − 50, +1) 62.534 μHz are perhaps their model counterparts. For this case, we notice in Figure 3 that ${\beta }_{{\ell },n}$ of the $m\ne 0$ component (1, 0, −50, 0) 58.557 μHz clearly deviated from the asymptotic value 0.5. For the latter case, f₁₃ and f₁₅ constitute one incomplete quintuplet, and (2, 0, −102, −2) 40.236 μHz and (2, 0, −102, +2) 61.238 μHz may be their mode counterparts. For this case, ${\beta }_{{\ell },n}$ of the central component (2, 0, −102, 0) 50.737 μHz is 0.833, which agrees well with the asymptotic value.

Table 4.  Possible Mode Identifications for the Three Isolated Pulsation Frequencies Bases on the Best-fitting Model

ID	${\nu }^{\mathrm{obs}}$	${\nu }^{\mathrm{theo}}({\ell },{n}_{{\rm{p}}},{n}_{{\rm{g}}},m)$	${\rm{\Delta }}\nu$
	(μHz)	(μHz)	(μHz)
f1	57.101	57.123(0, 0, 0)	0.022
f12	127.707	127.447(1, 4, −23, +1)	0.260
f13	39.893	39.864(1, 0, −75, 0)	0.029
		39.825(2, 0, −151, +1)	0.068
		39.989(2, 0, −130, 0)	0.096
		39.878(2, 0, −115, −1)	0.015
f23	94.319	94.290(2, 2, −61, +2)	0.029
		94.370(2, 2, −57, +1)	0.051

Note. ${\rm{\Delta }}\nu =| {\nu }^{\mathrm{obs}}-{\nu }^{\mathrm{theo}}|$.

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When we do model fittings, we only use the fundamental radial mode f₁ and two nonraidal oscillation modes, f₁₁ and f₃₂, to fit with theoretical calculated frequencies. Figure 4 illustrates the profiles of Brunt−Väisälä frequency N and Lamb frequency L_ℓ (ℓ = 1, 2) for the best-fitting model. Figure 5 illustrates the scaled radial displacement eigenfunctions for the fundamental radial mode f₁ and two nonradial oscillation modes f₁₁ and f₃₂. We adopt the default boundary of the helium core of MESA and mark the position of the hydrogen fraction X_cb = 0.01 in Figures 4 and 5 with the vertical lines. The outer zone is the envelope of the star, and the inner zone is the helium core. As shown Figure 5, the fundamental radial mode f₁ mainly propagates in the stellar envelope, and then characterizes the features of the stellar envelope. However, for the two nonradial oscillation modes f₁₁ and f₃₂, they have pronounced features of mixed modes. Namely, distinct g-mode features appear in the helium core and p-mode features appear in the stellar envelope. Thus, the two nonradial oscillation modes can characterize the features of the helium core.

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To investigate more detailed information on the structure of EE Cam, we introduce two asteroseismic quantities: the acoustic radius τ₀ and the period separation Π₀. Both τ₀ and Π₀ are independent of ℓ. The acoustic radius τ₀ is the sound travel time for a sound wave from the core of the star to the surface. Aerts et al. (2010) define the acoustic radius τ₀ as

$$\begin{eqnarray}&&{\tau }_{0}={\int }_{0}^{R}\displaystyle \frac{{dr}}{{c}_{s}},\end{eqnarray} \tag{ 5 }$$

where c_s denotes the adiabatic sound speed. The value of c_s inside the helium core is much larger than that inside the stellar envelope, thus the features of the envelope of the star can be characterized by the acoustic radius τ₀. The expression of Π₀ is described as

$$\begin{eqnarray}&&{{\rm{\Pi }}}_{0}=2{\pi }^{2}{\left({\int }_{0}^{R}\displaystyle \frac{N}{r}{dr}\right)}^{-1}\end{eqnarray} \tag{ 6 }$$

(Unno et al. 1979; Tassoul 1980; Aerts et al. 2010). In Equation (6), N denotes the Brunt−Väisälä frequency. The period separation Π₀ is dominated by the behavior of the Brunt−Väisälä frequency inside the helium core of the star. Then the features of the helium core of the star can be characterized by Π₀.

According to the analyses of Chen et al. (2016), both the stellar envelope and the helium core need to match the actual structure of EE Cam in order to fit the three pulsation modes (f₁, f₁₁, and f₃₂). In Figure 2, it is very evident that $1/{\chi }^{2}$ of theoretical model with Z = 0.028 and M = 2.04 M_⊙ is much higher than those of other theoretical models. Hence, τ₀ and Π₀ are determined to be 5.80 hr and 463.7 s, respectively. The size of the helium core of EE Cam is estimated to be about M_He = 0.181 M_⊙ and R_He = 0.0796 R_⊙.

In addition, the rotational period P_rot of EE Cam is determined to be ${1.84}_{-0.05}^{+0.07}$ days. The theoretical radius R of the best-fitting model is 4.12 R_⊙. According to the formula ${\upsilon }_{\mathrm{rot}}=2\pi R/{P}_{\mathrm{rot}}$, the equatorial rotation velocity ${\upsilon }_{\mathrm{rot}}$ is estimated to be ${113.6}_{-4.1}^{+2.7}$ km s⁻¹. Based on the inclination angle i = 34 ± 4 deg (Breger et al. 2015), the value of ${\upsilon }_{\mathrm{rot}}\sin i$ is deduced to be ${63.5}_{-8.7}^{+7.8}$ km s⁻¹, which is in accordance with the value $\upsilon \sin i=51\pm 8\,\mathrm{km}\,{{\rm{s}}}^{-1}$ of Bush & Hintz (2008) and slight higher than the value $\upsilon \sin i$ = 40 ± 3 km s⁻¹ of Breger et al. (2007).

The δ Scuti star EE Cam is an evolved star. As seen in Table 2, the frequency spectrum of EE Cam is very dense. The effects of rotational splitting will make the frequency spectrum much more complicated. Dziembowski et al. (1993) analyzed the effects of rotation on the frequency spectra of SPB stars and found that the rotationally split multiplets that are already begin to overlap at a rotational velocity of about a few km s⁻¹. We searched for frequency differences ranging from 1 to 30 μHz in the observed frequencies of EE Cam. If oscillation modes with ℓ = 3 are considered, we also found another possible scheme of mode identifications. There are 14 sets of rotationally split multiplets, including three sets of multiplets with ℓ = 1: (i.e., (f₂₂, f₈, f₉), (f₆, f₁₅), and (f₂₇, f₃₁)); seven sets of multiplets with ℓ = 2 (i.e., (f₁₉, f₂₁, f₃), (f₁₀, f₂₈, f₃₃), (f₃₂, f₁₂), (f₁₁, f₃₄), (f₁₃, f₅), (f₁₆, f₂₀), (f₂₅, f₃₅)); and four sets of multiplets with ℓ = 3 (i.e., (f₂₃, f₂₄, f₃₀), (f₁₈, f₇), (f₁₄, f₁₇), (f₃₇, f₃₉)). Four observed frequencies (i.e., f₂, f₄, f₃₆, and f₃₈) do not show frequency splitting. In this case, we found that most of the frequency differences in those multiplets are two or more times that of the averaged rotational splittings. In addition, the dipole mode f₂ identified by Breger et al. (2015) does not show frequency splittings. The frequency f₃ is identified as a mode with l = 2 based on the regularities of rotational splitting. However, Breger et al. (2015) suggested that f₃ is a dipole mode based on the line-profile variations. As a consequence, we suggested the mode identifications in Section 2 of this paper.

In our work, we fit three frequencies (f₁, f₁₁, and f₃₂) for each stellar model by changing three independent physical parameters (i.e., the stellar mass M, the metallicity Z, and the effective temperature T_eff). To test effects of other physical parameters, like the convective core overshooting on our fitting results, much more stellar models have been computed, f_ov ranging from 0.001 to 0.010 with a step of 0.001. The parameter of f_ov describes the efficiency of the overshooting mixing, and the definition of f_ov is identical to that of Chen et al. (2017). After doing model fittings, we find 13 other stellar models fitting well to the three frequencies (f₁, f₁₁, and f₃₂). Then we compare their theoretical frequencies with the observed frequencies in Table 3. Among these 13 stellar models, the structures of two stellar models with $(M,Z,{f}_{\mathrm{ov}},{\chi }^{2})$ = (2.03, 0.026, 0.001, 0.0065) and (2.01, 0.025, 0.002, 0.0079) are alike with that of our best-fitting model. Their acoustic radius τ₀ and period separation Π₀ are (5.82 hr, 463.5 s) and (5.79 hr, 463.8 s), respectively. They can reproduce the multiplets in Table 3. Howerver, their vaules of ${\chi }^{2}$ are higher than that of our best-fitting model by one order of magnitude. The other 11 stellar models can not reproduce all of those multiplets in Table 3.