We measure the mass of a modestly irradiated giant planet , KOI-94d .
We wish to determine whether this planet , which is in a 22-day orbit and receives 2700 times as much incident flux as Jupiter , is as dense as Jupiter or rarefied like inflated hot Jupiters .
KOI-94 also hosts at least 3 smaller transiting planets , all of which were detected by the Kepler  Mission .
With 26 radial velocities of KOI-94 from the W. M. Keck Observatory and a simultaneous fit to the Kepler  light curve , we measure the mass of the giant planet and determine that it is not inflated .
Support for the planetary interpretation of the other three candidates comes from gravitational interactions through transit timing variations , the statistical robustness of multi-planet systems against false positives , and several lines of evidence that no other star resides within the photometric aperture .
We report the properties of KOI-94b ( M _ { P } = 10.5 \pm 4.6 M _ { \earth } , R _ { P } = 1.71 \pm 0.16 R _ { \earth } , P = 3.74 days ) , KOI-94c ( M _ { P } = 15.6 ^ { +5.7 } _ { -15.6 } M _ { \earth } , R _ { P } = 4.32 \pm 0.41 R _ { \earth } , P = 10.4 days ) , KOI-94d ( M _ { P } = 106 \pm 11 M _ { \earth } , R _ { P } = 11.27 \pm 1.06 R _ { \earth } P = 22.3 days ) , and KOI-94e ( M _ { P } = 35 ^ { +18 } _ { -28 } M _ { \earth } , R _ { P } = 6.56 \pm 0.62 R _ { \earth } , P = 54.3 days ) .
The radial velocity analyses of KOI-94b and KOI-94e offer marginal ( > 2 \sigma ) mass detections , whereas the observations of KOI-94c offer only an upper limit to its mass .
Using the KOI-94Â system and other planets with published values for both mass and radius ( 138 exoplanets total , including 35 with M _ { P } < 150 M _ { \earth } ) , we establish two fundamental planes for exoplanets that relate their mass , incident flux , and radius from a few Earth masses up to ten Jupiter masses : \frac { R _ { P } } { R _ { \earth } } = 1.78 ~ { } \left ( \frac { M _ { P } } { M _ { \earth } } \right ) ^ % { 0.53 ~ { } } \left ( \frac { F } { erg s ^ { -1 } cm ^ { -2 } } \right ) ^ { -0.03 ~ { } } \text { for } % M _ { P } < 150 M _ { \earth } , and \frac { R _ { P } } { R _ { \earth } } = 2.45 ~ { } \left ( \frac { M _ { P } } { M _ { \earth } } \right ) ^ % { -0.039 ~ { } } \left ( \frac { F } { erg s ^ { -1 } cm ^ { -2 } } \right ) ^ { 0.094 ~ { } } \text { for% } M _ { P } > 150 M _ { \earth } .
These equations can be used to predict the radius or mass of a planet .