Astronomy | Astrophysics Answers

Special relativity part-l

Space cannon

Scientists are developing a new space cannon to shoot objects from the surface of the Earth di-

rectly 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.

Problem B.1: Space Cannon (6 Points)

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/s^2

.

(b) considering the change of the gravitational force with height.

Note: Neglect the air resistance for this problem. Use 6.67×10−11 m^3kg−1

s^−2

for the gravitational

constant, 6371 km for the Earth’s radius, and 5.97 × 1024 kg for the Earth’s mass.

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.

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 ellip-

tical 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 di-

rectly 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.

This problem requires you to read the following recently published scientific article: Measuring the ionisation fraction in a jet from a massive protostar. Fedriani, R., Caratti o Garatti, A., Purser, S.J.D. et al. Nat Commun 10, 3630 (2019). Link: https://www.nature.com/articles/s41467-019-11595-x.pdf Answer the following questions related to this article: (a) Why are massive stars important for the development of the universe? (b) How can the ionised part of jets be observed? (c) What kind of region is G35.2N? Describe how it is structured. (d) What is the ionisation fraction χe and how do the authors calculate its value? (e) How is the mass-loss rate being determined for knots K3 and K4? Why not for K1 and K2? (f) Why is the ionisation fraction so small for G35.2N? 

This problem requires you to read the following recently published scientific article: Onset of Cosmic Reionization: Evidence of an Ionized Bubble Merely 680 Myr aer the Big Bang. V. Tilvi et al 2020 ApJL 891 L10. Link: https://iopscience.iop.org/article/10.3847/2041-8213/ab75ec Answer the following questions related to this article: (a) What is the so called cosmic reionization process? (b) What are Lyα lines and why did the researches want to observe them? (c) What do the authors intend to point out with Figure 1 (see article)? (d) How is confirmed that the peaks seen in Figure 3 are actually from Lyα emissions? (e) How are the bubble sizes of the galaxies estimated? (f) What is special about the findings in the article and what are the scientific implications?

Space and time are interconnected according to special relativity. Because of that, coordinates have four components (three position coordinates x, y, z, one time coordinate t ) and can be expressed as a vector with four rows as such:   ct x y z   The spaceship from problem A.4 (Special Relativity - Part I) travels away from the Earth into the deep space outside of our Milky Way. The Milky Way has a very circular shape and can be expressed as all vectors of the following form (for all 0 ≤ ϕ < 2π):   ct 0 sin ϕ cos ϕ   (a) How does the shape of the Milky Way look like for the astronauts in the fast-moving spaceship? To answer this question, apply the Lorentz transformation matrix (see A.4) on the circular shape to get the vectors (ct0 , x0 , y0 , z0 ) of the shape from the perspective of the moving spaceship. (b) Draw the shape of the Milky Way for a spaceship with a velocity of 20%, 50%, and 90% of the speed of light in the figure below (