Appendix A: Derivation of mass upper limit of debris disk

Following the treatment of Paper I, we define $C_{\nu}$, the contrast between the dust emission and the photospheric emission in the infrared:

$$C_{\nu} = \frac{F_{\rm\nu,dust}}{F_{\rm\nu,star}} = \frac{L_{\rm\nu,dust}}{L_{\rm\nu,star}}\cdot$$ (A.1)

Assuming a single sized spherical particle, the dust emission is given by:

$$L_{\rm\nu,dust} = 4{\pi}^2a^2Q_{\nu}B_{\nu}(T_{\rm d})N,$$ (A.2)

where a is the radius, $T_{\rm d}$ the temperature, and ${Q_{\nu}}$ the absorption coefficient of a dust particle. N is the total number of such particles in the disk. The mass of the disk is readily computed from ${M_{\rm dust} = \frac{4}{3}{\pi}a^3 N\rho_{\rm dust}}$where ${\rho_{\rm dust}}$ is the specific mass of the grain material. For dust particles smaller than the wavelength ( ${a<0.1{\lambda}}$) one can write (e.g. Draine & Lee 1984):

$$Q = \frac{a~q_{\rm o}}{\lambda^2}\cdot$$ (A.3)

Approximating ${L_{\rm star}\approx A~T_{\rm eff}^{8.2}}$ for main-sequence dwarfs, and assuming that the Rayleigh approximation is valid in the infrared, we obtain for the star:

$$L_{\rm\nu,star} = A\frac{2{\pi}k}{{\sigma_{\rm SB}}c^2}\nu^2T_{\rm eff}^{5.2},$$ (A.4)

where ${\sigma_{\rm SB}}$ is the Stefan-Bolzmann constant. The constant A can be derived from the temperature and luminosity of known stars like the Sun or Vega, we adopt ${A=1.33\times10^{-31}}~L_{\odot}$K^-8.2. Combining the above equations:

$$M_{\rm dust} = A\frac{2k}{3{\sigma}}\frac{\rho_{\rm dust}T_{\rm eff}^{5.2}} {q_{\rm o}B_{\nu}(T_{\rm d})} C_{\nu}.$$ (A.5)

To estimate ${M_{\rm dust}}$ we assume astronomical silicate (Draine & Lee 1984) with ${q_{\rm o}~=~1.3\times10^{-4}}$ m and ${\rho_{\rm dust}~=~}$3.3 g cm^-3. Note that this relationship is independent of the grain size a as long as a is significantly smaller than the infrared wavelength.