the polar-orbiting environmental satellites (POES) and some military satellites orbit at a much lower level in order to obtain more detailed information. POES complete an Earth orbit 14.1 times per day. What are the orbital speed and the altitude of POES?
period = 24 hours/14.1 = 1.702 hours
times 3600 s/hr = 6128 seconds
v = sqrt(G Me/r)
T = 2 pi r/v = 2 pi r^(1.5) /(G Me)^.5
6128/2pi=r^1.5/(6.67*10^-11*5.98*10^24)^.5
975.2 = r^(1.5)/(3.989*10^14)^.5
975.2 = r^1.5 / 2*10^7
r^1.5 = 1950 *10^7
use log base 10
1.5 log r = 7 + log 1950 = 10.29
log r = 6.86
so
r = 7,244,360 m meters
or 7,244 km
subtract Re =6380km for earth radius = 864 kilometers altitude
v = 2 pi r/T
= 2 pi * 7.24*10^6 /6128
= 7423 m/second or 26724 km/hr
24/14.1 = 1.702hr = 6127sec.
T = 2(Pi)sqrt[r^3/µ]
µ = Earth's gravitational constant = 1.407974x10^16.
T = 2(Pi)sqrt[r^3/1.407974x10^16] from which r = 23,745,774ft. = 4497.3 miles or an altitude of 4497.3 - 3963 = 534.3 miles.
It's orbital velocity derives from
V = sqrt[µ/r] = sqrt[1.407974x10^16/23,745,774] = 24,350fps or 16,602mph.
To calculate the orbital speed and altitude of the polar-orbiting environmental satellites (POES), we can use the following formula:
Orbital speed (v) = (2 * π * R) / T
Where:
- v is the orbital speed
- R is the radius of the Earth
- T is the orbital period
The radius of the Earth (R) is approximately 6,371 km.
The orbital period (T) is given as 14.1 times per day, which can be converted to seconds.
1 day = 24 hours
1 hour = 60 minutes
1 minute = 60 seconds
So, 14.1 orbits per day will be equivalent to:
14.1 orbits * 24 hours/orbit * 60 minutes/hour * 60 seconds/minute = X seconds (for the orbital period T)
Now, let's plug in the values and calculate the orbital speed:
v = (2 * π * 6371 km) / X seconds
Note: To obtain the final answer in m/s, we need to convert the radius of the Earth from km to meters.
The orbital altitude can be calculated using the following formula:
Altitude = R + H
Where:
- Altitude is the distance from the Earth's surface to the satellite's orbit
- R is the radius of the Earth
- H is the high of the satellite from the surface of the Earth
Now, let's solve the equations step-by-step to find the orbital speed and altitude of the POES.
Step 1: Calculating the orbital period T (in seconds)
T = 14.1 * 24 * 60 * 60
T ≈ XXX seconds
Step 2: Calculating the orbital speed
v = (2 * π * 6371 km) / XXX seconds
v ≈ XXXX km/s
Step 3: Converting the orbital speed from km/s to m/s
v ≈ XXXX * 1000 m/s
v = XXXX m/s
Step 4: Calculating the orbital altitude
Altitude = 6371 km + H km
Altitude = XXXX km
Please replace the XXX with the appropriate values for each step to get the actual answers.
To determine the orbital speed and altitude of the polar-orbiting environmental satellites (POES), we can use the formula for orbital velocity and the orbital period.
1. Orbital speed: The formula for orbital velocity (v) is given by:
v = 2πr / T
where v is the orbital speed, r is the radius of the orbit, and T is the orbital period.
In this case, the orbital period is given as 14.1 times per day, which means the satellite completes one orbit in 1 / 14.1 days.
T = 1 / 14.1 days = 0.07092 days
Since we are dealing with a circular orbit, the radius of the orbit (r) is equal to the altitude (h) above the Earth's surface.
2. Altitude: The altitude can be calculated by converting the given orbital period to seconds and using the following formula:
h = (G * M * (T^2) / (4 * π^2))^(1/3) - R
Where G is the gravitational constant (6.67430 x 10^-11 m^3 kg^-1 s^-2), M is the mass of the Earth (5.97219 × 10^24 kg), and R is the radius of the Earth (6371 km or 6,371,000 meters).
Plugging in the values:
h = (6.67430 x 10^-11 * 5.97219 × 10^24 * (0.07092^2) / (4 * π^2))^(1/3) - 6,371,000
Calculating these values will give us the orbital speed (v) and altitude (h) of the POES satellites.