Astronomy | Astrophysics Answers

A space probe is about to launch with the objective to explore the planets Mars and Jupiter. To use the lowest amount of energy, the rocket starts from the Earth’s orbit (A) and flies in an elliptical orbit to Mars (B), such that the ellipse has its perihelion at Earth’s orbit and its aphelion at Mars’ orbit. The space probe explores Mars for some time until Mars has completed 1/4 of its orbit (C). Aer that, the space probe uses the same ellipse to get from Mars (C) to Jupiter (D). There the mission is completed, and the space probe will stay around Jupiter. The drawing below shows the trajectory of the space probe (not drawn to scale): Sun Earth Mars Jupiter A B D C Below you find the obrital period and the semi-major axis of the three planets: Orbital period Semi-major axis Earth 365 days 1.00 AU Mars 687 days 1.52 AU Jupiter 4333 days 5.20 AU How many years aer its launch from the Earth (A) will the space probe arrive at Jupiter (D)?

Scientists are developing a new space cannon to shoot objects from the surface of the Earth directly into a low orbit around the Earth. For testing purposes, a projectile is fired with an initial velocity of 2.8 km/s vertically into the sky. Calculate the height that the projectile reaches, ... (a) assuming a constant gravitational deceleration of 9.81 m/s2 . (b) considering the change of the gravitational force with height. Note: Neglect the air resistance for this problem. Use 6.67×10−11 m3kg−1 s −2 for the gravitational constant, 6371 km for the Earth’s radius, and 5.97 × 1024 kg for the Earth’s mass. 

The following Lorentz transformation matrix gives the transformation from a frame at rest to a moving frame with velocity v along the z-axis:   γ 0 0 γβ 0 1 0 0 0 0 1 0 γβ 0 0 γ   where β = v/c with c being the speed of light in a vacuum, and γ is the Lorentz factor: γ = 1 p 1 − β 2 (a) State and explain the two traditional postulates from which special relativity originates. (b) Draw a plot of the Lorentz factor for 0 ≤ β ≤ 0.9 to see how its value changes. One of the many exciting phenomena of special relativity is time dilation. Imagine astronauts in a spaceship that is passing by the Earth with a high velocity. (c) Are clocks ticking slower for the people on Earth or for the astronauts on the spaceship? (d) How fast must the spaceship travel such that the clocks go twice as slow? 

Telescopes are an essential tool for astronomers to study the universe. You plan to build your own telescope that can resolve the Great Red Spot on the surface of Jupiter at a wavelength of 600 nm. The farthest distance between the Earth and Jupiter is 968 × 106 km and the Great Red Spot has currently a diameter of 16,500 km. (a) Use the Rayleigh criterion to determine the diameter of the lens’ aperture of your telescope that is needed to resolve the Great Red Spot on Jupiter. Impacts have formed many craters on the Moon’s surface. You would like to study some of the craters with your new telescope. The distance between Moon and Earth is 384,400 km. (b) What is the smallest possible size of the craters that your telescope can resolve?

Special relativity has become a fundamental theory in the 20th century and is crucial for explaining many astrophysical phenomena. A central aspect of special relativity is the transformation from one reference frame to another. The following Lorentz transformation matrix gives the transformation from a frame at rest to a moving frame with velocity v along the z-axis: where β = v/c with c being the speed of light in a vacuum, and γ is the Lorentz factor: γ = 1 p 1 − β 2 (a) State and explain the two traditional postulates from which special relativity originates. (b) Draw a plot of the Lorentz factor for 0 ≤ β ≤ 0.9 to see how its value changes. One of the many exciting phenomena of special relativity is time dilation. Imagine astronauts in a spaceship that is passing by the Earth with a high velocity. (c) Are clocks ticking slower for the people on Earth or for the astronauts on the spaceship? (d) How fast must the spaceship travel such that the clocks go twice as slow?

A total solar eclipse occurs when the Moon moves between the Earth and the Sun and completely blocks out the Sun. This phenomenon is very spectacular and attracts people from all cultures. However, total solar eclipses can also take place on other planets of the Solar System. Determine for each of the following moons if they can create a total solar eclipse on their planet. Moon Radius Distance to Planet Planet Distance to the Sun Phobos 11 km 9376 km Mars 228 × 106 km Callisto 2410 km 1.883 × 106 km Jupiter 779 × 106 km Titan 2574 km 1.222 × 106 km Saturn 1433 × 106 km Oberon 761 km 0.584 × 106 km Uranus 2875 × 106 km Note: The radius of the Sun is 696 × 103 km.

Scientists are developing a new space cannon to shoot objects from the surface of the Earth directly into a low orbit around the Earth. For testing purposes, a projectile is fired with an initial velocity of 2.8 km/s vertically into the sky. Calculate the height that the projectile reaches, ... (a) assuming a constant gravitational deceleration of 9.81 m/s2 . (b) considering the change of the gravitational force with height. Note: Neglect the air resistance for this problem. Use 6.67×10−11 m3kg−1 s −2 for the gravitational constant, 6371 km for the Earth’s radius, and 5.97 × 1024 kg for the Earth’s mass.

Astronomers need to identify the position of objects in the sky with very high precision. For that, it is essential to have coordinate systems that specify the position of an object at a given time. One of them is the equatorial coordinate system that is widely used in astronomy. (a) Explain how the equatorial coordinate system works. (b) What is the meaning of J2000 that oen occurs together with equatorial coordinates? The object NGC 4440 is a galaxy located in the Virgo Cluster at the following equatorial coordinates (J2000): 12h 27m 53.6s (right ascension), 12◦ 170 3600 (declination). The Calar Alto Observatory is located in Spain at the geographical coordinates 37.23◦N and 2.55◦W. (c) Is the NGC 4440 galaxy observable from the Calar Alto Observatory?

A projectile is launched into space with a given initial velocity. How can calculate its maximum height? Acceleration is not constant (must use Newton's law of gravitation)

Mass is not given

Special relativity has become a fundamental theory in the 20th century and is crucial for explaining many astrophysical phenomena. A central aspect of special relativity is the transformation from one reference frame to another. The following Lorentz transformation matrix gives the transformation from a frame at rest to a moving frame with velocity v along the z-axis:

( γ 0 0 γβ )

0 1 0 0

0 0 1 0

γβ 0 0 γ

where β = v/c with c being the speed of light in a vacuum, and γ is the Lorentz factor:

γ = 1 / 1 − β 2

(a) State and explain the two traditional postulates from which special relativity originates.

(b) Draw a plot of the Lorentz factor for 0 ≤ β ≤ 0.9 to see how its value changes.