= Solution
A stable sinusoidal <radial-velocity semi-amplitude> can be stellar reflex motion caused by a companion on a nearly <circular orbit>. The amplitude is compatible with a giant <planet>; a low-<mass> <star> or <brown dwarf> viewed nearly face-on can mimic the same projected motion, but requires an increasingly unlikely orientation. <Starspots> and magnetically suppressed <convection> can produce a rotational <stellar activity> signal on a ten-day timescale; contemporaneous photometry, activity indicators and <stellar spectral-line bisector> and other profile changes distinguish these from a clean common Doppler shift. Solar-like <stellar oscillations> ordinarily have much shorter periods, so a coherent ten-day oscillation of an otherwise ordinary solar-type dwarf is less plausible. Instrumental drift, sampling aliases and blended spectra also need checking. A sinusoid alone does not establish a planetary origin, and the data do not assign numerical probabilities to these alternatives.
For the <minimum companion mass from radial velocity>, take a circular two-body orbit and initially $M_\star=M_\odot$. The exact relation and its low-<mass> limit are
$$
K=\left(\frac{2\pi G}{P}\right)^{1/3}\frac{M_p\sin i}{(M_\star+M_p)^{2/3}},\qquad M_p\sin i\simeq K\left(\frac P{2\pi G}\right)^{1/3}M_\star^{2/3}.
$$
With $\sin i=1$,
$$
\boxed{M_{p,\min}\simeq0.636\,M_J\simeq202\,M_\oplus,\qquad a\simeq0.0908\,\mathrm{au}.}
$$
Here $a=[GM_\star P^2/(4\pi^2)]^{1/3}$ is the relative orbital <semi-major axis>. Nonzero <orbital eccentricity> multiplies the inferred $M_p\sin i$ by $\sqrt{1-e^2}$ at fixed measured $K,P$, but the sinusoidal model motivates $e=0$ here.
For the <radial-velocity sensitivity limit at fixed planet mass>, use $K=M_p\sin i\sqrt{G/(M_\star a)}$ and interpret the stated detection scale as a semi-amplitude threshold $K_{\min}$. The largest distance occurs for $\sin i=1$:
$$
\boxed{a_{\max}=\frac{G M_p^2}{M_\star K_{\min}^2}\simeq0.0180\,\mathrm{au}\quad(M_\star=M_\odot).}
$$
For an inclination $i$, this boundary is smaller by $\sin^2i$. The result assumes adequate time coverage and neglects the stipulated <stellar noise>. If the detection scale instead denotes a full peak-to-peak excursion, set $K_{\min}$ to half that scale; $a_{\max}$ then increases fourfold.
The <stellar-mass scaling of radial-velocity planet estimates> changes these inferences because <Kepler's third law> and the stellar reflex response depend on the stellar <mass>. At fixed $K,P$, $M_p\sin i\propto M_\star^{2/3}$ and $a\propto M_\star^{1/3}$, whereas the fixed-<mass> detectability boundary scales as $M_\star^{-1}$. Therefore
$$
\boxed{M_{p,\min}\simeq0.401\,M_J\simeq127\,M_\oplus,\quad a\simeq0.0721\,\mathrm{au},\quad a_{\max}(3M_\oplus)\simeq0.0360\,\mathrm{au}.}
$$
The observed period and amplitude are unchanged. The <stellar radius> does not enter these leading dynamical estimates, but is crucial for the <exoplanet transit> inference.
For the transiting companion, assume an opaque spherical <planet>, negligible <transit dilution> and negligible <limb darkening>. Its <radius> ratio is $k=R_p/R_\star=\sqrt\delta=0.03873$, hence
$$
\boxed{R_p=0.01936\,R_\odot\simeq1.35\times10^7\,\mathrm m\simeq2.11\,R_\oplus.}
$$
Using the revised stellar <mass> and a <circular orbit> gives $a=0.01850\,\mathrm{au}$ and $A=a/R_\star=7.957$. Interpret the duration as the total first-to-fourth-contact duration $T_{14}$. The <impact parameter from total transit duration> follows from the <total duration of a circular planetary transit>:
$$
s=\sin\left(\frac{\pi T_{14}}P\right),\qquad b^2=\frac{(1+k)^2-A^2s^2}{1-s^2},\qquad\boxed{b\simeq0.893,\quad i=\arccos(b/A)\simeq83.56^\circ.}
$$
This is noncentral but not grazing: $b<1-k$. The small-<planet> straight-chord approximation gives $b\simeq0.845$; retaining finite $R_p$ and orbital curvature gives the stated value. A duration measured between other contacts needs the corresponding contact formula. Unknown <orbital eccentricity> and argument of periapsis change the transverse speed and prevent a unique unconstrained $b$; the numerical estimate is conditional on the stated circular model.<On the Unique Solution of Planet and Star Parameters from an Extrasolar Planet Transit Light Curve by Sara Seager and Gabriela Mallén-Ornelas>[Section3.1, equations1–4: depth, contact geometry and circular-orbit scale]{ref}
\Image[/past-exam-of-the-exoplanets-and-planetary-systems-course-of-the-university-of-cambridge/2015/paper-1-transit-geometry.png]
{title=The measured transit duration selects a short off-centre chord when finite planet size is retained}
{source=paper-1-transit-geometry.py}
Because the transit constrains $\sin i\simeq0.994$, the RV nondetection now bounds the actual planetary <mass>, rather than only its projected <mass>. The <bulk-density upper limit from a transit and an RV nondetection> is
$$
\boxed{M_p\lesssim2.16\,M_\oplus,\qquad\bar\rho_p\lesssim1.26\,\mathrm{g\,cm^{-3}}.}
$$
These bounds assume the instrumental threshold can be applied to a fitted signal at the transit period; no statistical confidence level is specified. Such a large <radius> at this low <mass> is inconsistent with an ordinary compact rock-and-iron body. A low-<density> volatile-rich <planet>, possibly with an extended H/He envelope, is indicated under the planetary interpretation. The observations do not uniquely determine interior composition, atmospheric abundance or formation history; blended eclipses and activity still require validation. There is no luminosity or effective <temperature> supplied for the host, so a numerical planetary equilibrium <temperature> cannot be inferred from these data alone.
The numerical conversions above use externally documented $(GM)_\odot=1.3271244\times10^{20}\,\mathrm{m^3\,s^{-2}}$, $R_\odot=6.957\times10^8\,\mathrm m$, $M_\oplus=5.9722\times10^{24}\,\mathrm{kg}$, $R_\oplus=6.371\times10^6\,\mathrm m$, $M_J=1.89813\times10^{27}\,\mathrm{kg}$, $G=6.67430\times10^{-11}\,\mathrm{m^3\,kg^{-1}\,s^{-2}}$ and $1\,\mathrm{au}=149597870700\,\mathrm m$. They are conversion constants, not additional measured system parameters.<Nominal solar and planetary conversion constants by the International Astronomical Union>[2015 ResolutionB3: solar <radius> and solar <mass> parameter]{ref}<Planetary Physical Parameters by NASA JPL>[Masses, mean radii and CODATA2018 gravitational constant]{ref}
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