A planet has been discovered orbiting the sun-like star HD 209458. From the period and velocity measured by the Doppler method, its mass was calculated to be 0.69 times that of Jupiter. The plane of the planet's orbit is edge-on and it has been observed to transit across the face of the star. The light from the star was seen to drop by 1.7% when the planet's disc lay fully within that of the star which has a radius of 8 × 10³ km. Assuming that the brightness of the star is constant across the observed face, estimate the diameter of the planet and hence it density in comparison with that of Jupiter which has a radius of 7.1 x 10¹ km.

One of the brightest comets of the 20th century was Comet Hyakutake, which passed close to the sun in early 1996. The orbital period of this comet is estimated to be about 30,000 years. Find the semimajor axis of this comet’s orbit. Compare it to the average sun–Pluto distance and to the distance to Alpha Centauri, the nearest star to the sun, which is 4.3 light-years distant.

In 2004, astronomers reported the discovery of a large Jupiter-sized planet orbiting very close to the star HD179949. The orbit was just 6.4x106 km (about 9 less than the orbit of Mercury) and the planet takes 3.1 days to make one circular orbit. The mass of the star is Answerx_______1030 kg. (Give the number before the exponent.)

You send a probe to orbit Mercury at 135 km above the surface. What orbital velocity (in km/s) is needed to keep it in orbit? (The mass of Mercury is 3.30 ✕ 1023 kg, and the radius of Mercury is 2.44 ✕ 103 km.) What is the ratio of the time it takes a signal from Earth to reach Mercury (d = 57.9 ✕ 106 km) to the time it would take to reach the Moon (d = 384,400 km)? If your signal is at 8 cm, what is the wavelength shift (in cm) at this orbital velocity? (Assume the probe is at a point in its orbit in which it is moving directly away from the Earth.) The orbital velocity is just the circular velocity. vc =    GM r , where the distance is the distance above the surface plus the radius of Mercury.     vc =