Problem A.1: Interstellar Mission

You are on an interstellar mission from the Earth to the 8.7 light-years distant star Sirius. Your spaceship can travel with 70% the speed of light and has a cylindrical shape with a diameter of 6 m at the front surface and a length of 25 m. You have to cross the interstellar medium with an approximated density of 1 hydrogen atom/m3.

(a) Calculate the time it takes your spaceship to reach Sirius.
(b) Determine the mass of interstellar gas that collides with your spaceship during the mission.

Note: Use 1.673 × 10−27 kg as proton mass.

Problem A.2: Time Dilation

Because you are moving with an enormous speed, your mission from the previous problem A.1 will be influenced by the effects of time dilation described by special relativity: Your spaceship launches in June 2020 and returns back to Earth directly after arriving at Sirius.

(a) How many years will have passed from your perspective?
(b) At which Earth date (year and month) will you arrive back to Earth?

Problem A.3: Magnitude of Stars

The star Sirius has an apparent magnitude of -1.46 and appears 95-times brighter compared to the more distant star Tau Ceti, which has an absolute magnitude of 5.69.

(a) Explain the terms apparent magnitude, absolute magnitude and bolometric magnitude. (b) Calculate the apparent magnitude of the star Tau Ceti.
(c) Find the distance between the Earth and Tau Ceti.

Problem A.4: Emergency Landing

Because your spaceship has an engine failure, you crash-land with an emergency capsule at the equator of a nearby planet. The planet is very small and the surface is a desert with some stones and small rocks laying around. You need water to survive. However, water is only available at the poles of the planet. You find the following items in your emergency capsule:

• Stopwatch
• Electronic scale

• 2m yardstick
• 1Litre oil
• Measuring cup

Describe an experiment to determine your distance to the poles by using the available items. Hint: As the planet is very small, you can assume the same density everywhere.

Problem B.1: Temperature of Earth

Our Sun shines bright with a luminosity of 3.828 x 1026 Watt. Her energy is responsible for many processes and the habitable temperatures on the Earth that make our life possible.

(a) Calculate the amount of energy arriving on the Earth in a single day.
(b) To how many litres of heating oil (energy density: 37.3 x 106 J/litre) is this equivalent?
(c) The Earth reflects 30% of this energy: Determine the temperature on Earth’s surface.
(d) What other factors should be considered to get an even more precise temperature estimate?

Note: The Earth’s radius is 6370 km; the Sun’s radius is 696 x 103 km; 1 AU is 1.495 x 108 km.

Problem B.2: Distance of the Planets

The table below lists the average distance R to the Sun and orbital period T of the first planets:

(a) Calculate the average distance of Mercury, Venus and Mars to the Earth. Which one of these planets is the closest to Earth on average?

(b) Calculate the average distance of Mercury, Venus and Earth to Mars. Which one of these planets is the closest to Mars on average?

(c) What do you expect for the other planets?
Hint: Assume circular orbits and use symmetries to make the distance calculation easier. You can

approximate the average distance by using four well-chosen points on the planet’s orbit.

Distance

Orbital Period

Mercury

0.39 AU

88 days

Venus

0.72 AU

225 days

Earth

1.00 AU

365 days

Mars

1.52 AU

687 days

Problem B.3: Mysterious Object

Your research team analysis the light of a mysterious object in space. By using a spectrometer, you can observe the following spectrum of the object. The Hα line peak is clearly visible:

(a) Mark the first four spectral lines of hydrogen (Hα, Hβ, Hγ, Hδ) in the spectrum. (b) Determine the radial velocity and the direction of the object’s movement.
(c) Calculate the distance to the observed object.
(d) What possible type of object is your team observing?

Question

Problem A.1: Interstellar Mission

You are on an interstellar mission from the Earth to the 8.7 light-years distant star Sirius. Your spaceship can travel with 70% the speed of light and has a cylindrical shape with a diameter of 6 m at the front surface and a length of 25 m. You have to cross the interstellar medium with an approximated density of 1 hydrogen atom/m3.

(a) Calculate the time it takes your spaceship to reach Sirius.
(b) Determine the mass of interstellar gas that collides with your spaceship during the mission.

Note: Use 1.673 × 10−27 kg as proton mass.

Problem A.2: Time Dilation

Because you are moving with an enormous speed, your mission from the previous problem A.1 will be influenced by the effects of time dilation described by special relativity: Your spaceship launches in June 2020 and returns back to Earth directly after arriving at Sirius.

(a) How many years will have passed from your perspective?
(b) At which Earth date (year and month) will you arrive back to Earth?

Problem A.3: Magnitude of Stars

The star Sirius has an apparent magnitude of -1.46 and appears 95-times brighter compared to the more distant star Tau Ceti, which has an absolute magnitude of 5.69.

(a) Explain the terms apparent magnitude, absolute magnitude and bolometric magnitude. (b) Calculate the apparent magnitude of the star Tau Ceti.
(c) Find the distance between the Earth and Tau Ceti.

Problem A.4: Emergency Landing

Because your spaceship has an engine failure, you crash-land with an emergency capsule at the equator of a nearby planet. The planet is very small and the surface is a desert with some stones and small rocks laying around. You need water to survive. However, water is only available at the poles of the planet. You find the following items in your emergency capsule:

• Stopwatch
• Electronic scale

• 2m yardstick
• 1Litre oil
• Measuring cup

Describe an experiment to determine your distance to the poles by using the available items. Hint: As the planet is very small, you can assume the same density everywhere.

Problem B.1: Temperature of Earth

Our Sun shines bright with a luminosity of 3.828 x 1026 Watt. Her energy is responsible for many processes and the habitable temperatures on the Earth that make our life possible.

(a) Calculate the amount of energy arriving on the Earth in a single day.
(b) To how many litres of heating oil (energy density: 37.3 x 106 J/litre) is this equivalent?
(c) The Earth reflects 30% of this energy: Determine the temperature on Earth’s surface.
(d) What other factors should be considered to get an even more precise temperature estimate?

Note: The Earth’s radius is 6370 km; the Sun’s radius is 696 x 103 km; 1 AU is 1.495 x 108 km.

Problem B.2: Distance of the Planets

The table below lists the average distance R to the Sun and orbital period T of the first planets:

(a) Calculate the average distance of Mercury, Venus and Mars to the Earth. Which one of these planets is the closest to Earth on average?

(b) Calculate the average distance of Mercury, Venus and Earth to Mars. Which one of these planets is the closest to Mars on average?

(c) What do you expect for the other planets?
Hint: Assume circular orbits and use symmetries to make the distance calculation easier. You can

approximate the average distance by using four well-chosen points on the planet’s orbit.

	Distance	Orbital Period
Mercury	0.39 AU	88 days
Venus	0.72 AU	225 days
Earth	1.00 AU	365 days
Mars	1.52 AU	687 days

Problem B.3: Mysterious Object

Your research team analysis the light of a mysterious object in space. By using a spectrometer, you can observe the following spectrum of the object. The Hα line peak is clearly visible:

(a) Mark the first four spectral lines of hydrogen (Hα, Hβ, Hγ, Hδ) in the spectrum. (b) Determine the radial velocity and the direction of the object’s movement.
(c) Calculate the distance to the observed object.
(d) What possible type of object is your team observing?

Accepted Answer

As per guidelines, the first question (Problem A.1) has been answered.