## Astronomy C

Test your knowledge of various Science Olympiad events.
astronomybuff
Member
Posts: 36
Joined: January 25th, 2020, 9:19 am
Division: C
State: NC
Has thanked: 1 time
Been thanked: 2 times

### Re: Astronomy C

0sm0sis wrote:
December 9th, 2020, 8:59 am
Yay orbits!
Because the orbit is circular, we can equate the centripetal acceleration to the gravitational force to find that v^2 = GM/R.

From there, we can use the Vis-Viva equation to find the semimajor axis of the new orbit. The equation is V^2 = GM ( 2/R - 1/a ), but we must use v/2 in place of V. This gives us

GM/4R = 2GM/R - GM/a

Which simplifies to:

4/7 R = a

From there we should use reasoning with how orbits work. Because the velocity is perfectly tangential from the beginning, the initial distance from the star is either the perigee or apogee. Because a larger velocity is required to have circular motion for this position, this distance is probably the apogee, the furthest distance from the star.

We also know that the perigee distance + apogee distance = 2a (this can be seen easily by drawing a diagram of an elliptical orbit). Therefore, R + x = 2(4/7 R), meaning that our perigee x is:

x = R/7
Absolutely correct!. I love orbits too!

I used a slightly different method with energy. The total energy is -GMm/2a. Kinetic energy can be derived using your method of equating centripretal force to gravitational force: mv^2/r=GMm/r^2, mv^2=GMm/r, but we need a 1/2 for kinetic energy, so kinetic energy= GMm/2r. But we have v/2, which makes the kinetic energy GMm/8r. The total energy is kinetc+potential, or GMm/8r-GMm/r= -7/8 GMm/r. Setting this equal to -GMm/2a, we find a= 4/7 R. Then we can use the method you gave us. The total major axis is 8/7 R, and since the satellite started a distance R, we subtract R to get R/7.
These users thanked the author astronomybuff for the post:
0sm0sis (December 9th, 2020, 11:51 am)

0sm0sis
Member
Posts: 6
Joined: July 25th, 2020, 3:30 pm
Division: C
State: OH
Has thanked: 2 times
Been thanked: 4 times

### Re: Astronomy C

(sorry to people who don't like orbits)

Imagine that the solar system consists of just the Earth circularly orbiting the Sun. If you wanted to get a spaceship to escape the solar system from the surface of Earth, what is the minimum velocity with respect to the Earth needed? Use the values of:

Mass of Earth = 5.97 * 10^24 kg
Radius of Earth = 6400 km
Mass of Sun = 1.99 * 10^30 kg
Distance from Sun to Earth = 1.5 * 10^8 km

(question credit: Morin)
2020-21 Events:

Circuit Lab, Detector, Machines, SOM

"All science is either physics or stamp collecting." - Ernest Rutherford

Ttonyxx
Member
Posts: 9
Joined: February 27th, 2019, 5:32 pm
Division: C
State: CA
Pronouns: He/Him/His
Has thanked: 1 time
Been thanked: 0
Contact:

### Re: Astronomy C

0sm0sis wrote:
December 9th, 2020, 9:53 am
(sorry to people who don't like orbits)

Imagine that the solar system consists of just the Earth circularly orbiting the Sun. If you wanted to get a spaceship to escape the solar system from the surface of Earth, what is the minimum velocity with respect to the Earth needed? Use the values of:

Mass of Earth = 5.97 * 10^24 kg
Radius of Earth = 6400 km
Mass of Sun = 1.99 * 10^30 kg
Distance from Sun to Earth = 1.5 * 10^8 km

(question credit: Morin)
I don't know if this is right, but the escape velocity is equal to square root of 2GM/R. So, if I just do that using M as the mass of the sun and R as the distance between the sun and the Earth, I get an answer of 42.1km/s

nobodynobody
Member
Posts: 36
Joined: January 8th, 2020, 5:41 pm
Division: C
State: OH
Pronouns: He/Him/His
Has thanked: 9 times
Been thanked: 7 times

### Re: Astronomy C

0sm0sis wrote:
December 9th, 2020, 9:53 am
(sorry to people who don't like orbits)

Imagine that the solar system consists of just the Earth circularly orbiting the Sun. If you wanted to get a spaceship to escape the solar system from the surface of Earth, what is the minimum velocity with respect to the Earth needed? Use the values of:

Mass of Earth = 5.97 * 10^24 kg
Radius of Earth = 6400 km
Mass of Sun = 1.99 * 10^30 kg
Distance from Sun to Earth = 1.5 * 10^8 km

(question credit: Morin)
I think its:
16.62km/s

Escape speed from the sun is
$\sqrt \frac{2GM}{r} = 42.07km/s$
Earths velocity relative to the sun is (assume circular orbit)
$\sqrt \frac{GM}{r} = 29.75km/s$
That means, rocket as to travel an additional 12.32 km/s on top of earths velocity. From the frame of the earth, the rocket needs to travel 12.32km/s after it leaves earth's gravitational field.
Escape speed at the surface of the earth is
$\sqrt \frac{2GM}{r} = 11.16km/s$
Since the spaceship's velocity will decrease after leaving earth, we must
$\sqrt {12.32^2 + 11.16^2} = 16.62km/s$

Class of '23
2021 events: Astro, Digi, SOM, WICI

"No." - Marie Curie

0sm0sis
Member
Posts: 6
Joined: July 25th, 2020, 3:30 pm
Division: C
State: OH
Has thanked: 2 times
Been thanked: 4 times

### Re: Astronomy C

nobodynobody wrote:
February 2nd, 2021, 3:06 pm
0sm0sis wrote:
December 9th, 2020, 9:53 am
(sorry to people who don't like orbits)

Imagine that the solar system consists of just the Earth circularly orbiting the Sun. If you wanted to get a spaceship to escape the solar system from the surface of Earth, what is the minimum velocity with respect to the Earth needed? Use the values of:

Mass of Earth = 5.97 * 10^24 kg
Radius of Earth = 6400 km
Mass of Sun = 1.99 * 10^30 kg
Distance from Sun to Earth = 1.5 * 10^8 km

(question credit: Morin)
I think its:
16.62km/s

Escape speed from the sun is
$\sqrt \frac{2GM}{r} = 42.07km/s$
Earths velocity relative to the sun is (assume circular orbit)
$\sqrt \frac{GM}{r} = 29.75km/s$
That means, rocket as to travel an additional 12.32 km/s on top of earths velocity. From the frame of the earth, the rocket needs to travel 12.32km/s after it leaves earth's gravitational field.
Escape speed at the surface of the earth is
$\sqrt \frac{2GM}{r} = 11.16km/s$
Since the spaceship's velocity will decrease after leaving earth, we must
$\sqrt {12.32^2 + 11.16^2} = 16.62km/s$

16.62 km/s is correct! To reiterate, the reason why the answer isn't simply the escape velocity from the Sun is because we have to escape Earth's gravitational field as well with some leftover velocity to escape the Sun's field. Energy conservation seems like the best approach to me, which is what I think nobodynobody did too. Great job!
2020-21 Events:

Circuit Lab, Detector, Machines, SOM

"All science is either physics or stamp collecting." - Ernest Rutherford

nobodynobody
Member
Posts: 36
Joined: January 8th, 2020, 5:41 pm
Division: C
State: OH
Pronouns: He/Him/His
Has thanked: 9 times
Been thanked: 7 times

### Re: Astronomy C

Cosmology question (attempt)

1. Calculate the critical density for Hubble's constant of 50 km/s/mpc in a flat universe with no dark energy.
2. Let's assume the universe consists only of 2 500nm photons per cubic meter at a=1. What is the average energy density of the universe? If the universe expands to a=3, what will be the average photon wavelength, number density, and energy density of the universe?
3. Let's assume that the density parameter for baryonic matter is .3. Using the values above, find the redshift at which the density of baryonic matter and radiation are equal. Estimate how long ago this was. Assume the universe is matter-dominated right now.
Last edited by nobodynobody on February 3rd, 2021, 7:22 am, edited 1 time in total.
Class of '23
2021 events: Astro, Digi, SOM, WICI

"No." - Marie Curie

AstroClarinet
Member
Posts: 40
Joined: August 6th, 2020, 9:56 am
Division: C
Has thanked: 50 times
Been thanked: 22 times

### Re: Astronomy C

nobodynobody wrote:
February 3rd, 2021, 7:14 am
Cosmology question (attempt)

1. Calculate the critical density for Hubble's constant of 50 km/s/mpc in a flat universe with no dark energy.
2. Let's assume the universe consists only of 2 500nm photons per cubic meter at a=1. What is the average energy density of the universe? If the universe expands to a=3, what will be the average photon wavelength, number density, and energy density of the universe?
3. Let's assume that the density parameter for baryonic matter is .3. Using the values above, find the redshift at which the density of baryonic matter and radiation are equal. Estimate how long ago this was. Assume the universe is matter-dominated right now.
1. 5*10-27 kg/m3
2. initial energy density 8*10-19 J/m3, new avg wavelength 1500 nm, new number density 0.07 photons/m3, new energy density 1*10-20 J/m3
3. I assume you mean H=50 km/s/Mpc, so 2*108 (did I mess up somewhere along the line?)

2019-2021 events: Astronomy, Designer Genes, Water Quality, Digital Structures
1st place at state in Solar System B (2018) and Ecology B (2017)
 Solar System, Boomilever, Density Lab, Ecology, Optics, Towers, Microbe Mission, Mission Possible
AstroClarinet's Userpage

nobodynobody
Member
Posts: 36
Joined: January 8th, 2020, 5:41 pm
Division: C
State: OH
Pronouns: He/Him/His
Has thanked: 9 times
Been thanked: 7 times

### Re: Astronomy C

AstroClarinet wrote:
February 4th, 2021, 12:42 pm
nobodynobody wrote:
February 3rd, 2021, 7:14 am
Cosmology question (attempt)

1. Calculate the critical density for Hubble's constant of 50 km/s/mpc in a flat universe with no dark energy.
2. Let's assume the universe consists only of 2 500nm photons per cubic meter at a=1. What is the average energy density of the universe? If the universe expands to a=3, what will be the average photon wavelength, number density, and energy density of the universe?
3. Let's assume that the density parameter for baryonic matter is .3. Using the values above, find the redshift at which the density of baryonic matter and radiation are equal. Estimate how long ago this was. Assume the universe is matter-dominated right now.
1. 5*10-27 kg/m3
2. initial energy density 8*10-19 J/m3, new avg wavelength 1500 nm, new number density 0.07 photons/m3, new energy density 1*10-20 J/m3
3. I assume you mean H=50 km/s/Mpc, so 2*108 (did I mess up somewhere along the line?)

1&2 are correct!!!

Hint: For number 3, first calculate the redshift(or scale factor) at which their scale factors are equal, then you can estimate the time it takes under a matter-dominated universe to expand from that to today. I should specify that at a = 1 (present day), hubbles constant is 50/km/s/mpc, baryonic density parameter is .3 and the density of readiation is 2 photons (500nm) per cubic meter. For anyone trying to follow along, here is how to do the first two:
Solution for 1:
first convert hubbles constant from km/s/mpc to 1/s
$50 \frac{km}{s*Mpc} * \frac {3.24e-20Mpc}{1km} = \frac{1.62e-18}{s}$
Then we can plug in the values into the critical density formula:
$\rho = \frac{3H^2}{8\pi G} = 4.69e-27 \frac{kg}{m^3}$

Solution for 2:
Energy per photon is given by it's frequency and planck's constant. Its wavelength can be calculated from its frequency.
$E = hf, E = h \frac{c}{\lambda} = 3.97e-19 joules$

This means the energy density is  7.94e-19 joules
wavelegnth scales linearly with scale factor, so 3 times the scale factor means 3 times the wavelegnth. This results in a wavelegnth that is 3 times larger, 1500nm.
Density is inversely related to volume, which is then related to scalefactor^3. That means the density is related to a^-3. Multiplying 2 photons per cubic meter by 3^-3 gives 0.074 photons/cubicmeter.
Finally, for the last step, combine the previous two answers.
$E = \frac{hc}{\lambda} = 1.32e-19J$
$1.32e-19J * .074 = 9.77e-21\frac{J}{m^3}$


Class of '23
2021 events: Astro, Digi, SOM, WICI

"No." - Marie Curie

AstroClarinet
Member
Posts: 40
Joined: August 6th, 2020, 9:56 am
Division: C
Has thanked: 50 times
Been thanked: 22 times

### Re: Astronomy C

nobodynobody wrote:
February 4th, 2021, 7:45 pm
Hint: For number 3, first calculate the redshift(or scale factor) at which their scale factors are equal, then you can estimate the time it takes under a matter-dominated universe to expand from that to today. I should specify that at a = 1 (present day), hubbles constant is 50/km/s/mpc, baryonic density parameter is .3 and the density of readiation is 2 photons (500nm) per cubic meter.
Here's my work, I'm not quite sure what's wrong with it
$\rho_E=\frac{\rho_{E,0}}{a^4}$  and  $\rho_M=\frac{\rho_{M,0}}{a^3}$  so  $\frac{\rho_{E,0}}{a^4}=\frac{\rho_{M,0}}{a^3}$  or  $a=\frac{\rho_{E,0}}{\rho_{M,0}}$
$\rho_{M,0}=0.3\cdot 4.69\times 10^{-27}\mathrm{\:kg/m^3}\cdot (3\times 10^8\mathrm{m/s})^2=1.27\times 10^{-10}\mathrm{\:J/m^3}$
$\rho_{E,0}=7.94\times 10^{-19} \mathrm{\:J/m^3}$
$a=\frac{7.94\times 10^{-19}\mathrm{\:J/m^3}}{1.27\times 10^{-10}\mathrm{\:J/m^3}}=6.25\times 10^{-9}$
$z=a^{-1}-1=(6.25\times 10^{-9})^{-1}-1=1.60\times 10^{8}$
Distance that this redshift corresponds to (couldn't figure out any other way to convert z to a time, though using the above z value it's so high it's just going to end up being the Hubble time): $d=\frac{c}{H_0}\left(\frac{(z+1)^2-1}{(z+1)^2+1)}\right)=1.85\times 10^{26} \:\mathrm{m}$
Lookback time: $t=\frac{d}{c}=\frac{1.85\times 10^{26}\:\mathrm{m}}{3\times 10^{8}\:\mathrm{m/s}}=6.17\times 10^{17}\:\mathrm{s}=1.96\times 10^{10}\mathrm{\:yrs}$


2019-2021 events: Astronomy, Designer Genes, Water Quality, Digital Structures
1st place at state in Solar System B (2018) and Ecology B (2017)
 Solar System, Boomilever, Density Lab, Ecology, Optics, Towers, Microbe Mission, Mission Possible
AstroClarinet's Userpage

AstroClarinet
Member
Posts: 40
Joined: August 6th, 2020, 9:56 am
Division: C
Has thanked: 50 times
Been thanked: 22 times

### Re: Astronomy C

I suppose I'll post questions.

1. Name and describe two structures formed by galactic interactions.
2. Fill in the blanks: A more massive white dwarf has a _______ [larger/smaller] radius. A more massive neutron star has a _______ [larger/smaller] radius. A more massive black hole (or any other object, really) has a _______ [larger/smaller] Schwarzschild radius.
3. If the core of a massive star has a radius of 0.6 Rsun, mass of 1.5 Msun, and angular speed of 1.8*10^-7 rad/s, and collapses (with no mass or angular momentum loss) into a neutron star with radius 2.3*10^-5 Rsun, what will be the new angular speed and rotational period of the neutron star?
2019-2021 events: Astronomy, Designer Genes, Water Quality, Digital Structures
1st place at state in Solar System B (2018) and Ecology B (2017)
 Solar System, Boomilever, Density Lab, Ecology, Optics, Towers, Microbe Mission, Mission Possible
AstroClarinet's Userpage

### Who is online

Users browsing this forum: No registered users and 2 guests