Difference between revisions of "Tutorial: Basic Orbiting (Technical)"

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(*G in KSP is exactly 6.674e-11. Because reasons.)
(clean up formatting)
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<math>v = 1,878,968 \cdot \sqrt{\frac{2}{r_i} - \frac{2}{(r_l+r_h)}}</math>
 
<math>v = 1,878,968 \cdot \sqrt{\frac{2}{r_i} - \frac{2}{(r_l+r_h)}}</math>
  
In this formula, ''r<sub>l</sub>'' and ''r<sub>h</sub>'' are the radii of the lower and higher orbits, respectively, and ''r<sub>i</sub>'' is the radius of the initial orbit.  If you are transferring to a higher orbit, ''r<sub>i</sub>'' will be equal to ''r<sub>l</sub>'', and ''v'' will be faster than your current speed, so burn in the direction of travel to reach ''v''.  If you are transferring to a lower orbit, ''r<sub>i</sub>'' will be equal to ''r<sub>h</sub>'', and ''v'' will be slower than your current speed, so burn in the opposite direction to reach ''v''.  Remember, ''v'' is the target speed for your initial burn that puts you into the elliptical transfer orbit.  Once you reach your new orbital altitude, you need to make a second burn to round out your orbit, using the same technique described in the [[Tutorial: Basic Orbiting#Stabilizing_your_orbit | stabilizing your orbit]] section.
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In this formula, ''r<sub>l</sub>'' and ''r<sub>h</sub>'' are the radii of the lower and higher orbits, respectively, and ''r<sub>i</sub>'' is the radius of the initial orbit.  If you are transferring to a higher orbit, ''r<sub>i</sub>'' will be equal to ''r<sub>l</sub>'', and ''v'' will be faster than your current speed, so burn in the direction of travel to reach ''v''.  If you are transferring to a lower orbit, ''r<sub>i</sub>'' will be equal to ''r<sub>h</sub>'', and ''v'' will be slower than your current speed, so burn in the opposite direction to reach ''v''.  Remember, ''v'' is the target speed for your initial burn that puts you into the elliptical transfer orbit.  Once you reach your new orbital altitude, you need to make a second burn to round out your orbit, using the same technique described in the [[Tutorial: Basic Orbiting#Stabilizing your orbit|stabilizing your orbit]] section.
  
 
Details of where this formula comes from are in the technical section below.  When using this formula, take care to remember that the radius of an orbit is equal to the orbital altitude plus Kerbin's radius (600 000 m).
 
Details of where this formula comes from are in the technical section below.  When using this formula, take care to remember that the radius of an orbit is equal to the orbital altitude plus Kerbin's radius (600 000 m).
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plot(100*1:40,1000*hohmann(100*1:40,34)$b1$to,main="Landing speeds",xlab="altitude (km)",ylab="speed (m/s)")
 
plot(100*1:40,1000*hohmann(100*1:40,34)$b1$to,main="Landing speeds",xlab="altitude (km)",ylab="speed (m/s)")
  
[[file:landingspeeds.png]]
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[[File:landingspeeds.png]]
  
R project Link[http://www.r-project.org/]
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R project Link [http://www.r-project.org/]
  
  
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=== Orbital Table ===
 
=== Orbital Table ===
'''Note:''' The atmosphere previously had a sharp cutoff at 34.5 km, but now extends to approximately 68 km.  Below this altitude, your orbit will gradually decay.  The decay becomes quite rapid below about 45 km.  The orbital parameters below 68 km are provided for reference, but understand that you will not be able to maintain these orbits without regular corrections to counteract the atmospheric drag.
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'''Note:''' The atmosphere previously had a sharp cutoff at 34.5&nbsp;km, but now extends to approximately 68&nbsp;km.  Below this altitude, your orbit will gradually decay.  The decay becomes quite rapid below about 45&nbsp;km.  The orbital parameters below 68&nbsp;km are provided for reference, but understand that you will not be able to maintain these orbits without regular corrections to counteract the atmospheric drag.
 
{| class="wikitable"
 
{| class="wikitable"
 
! Altitude (m)
 
! Altitude (m)

Revision as of 05:57, 7 April 2014

This tutorial is a stub. You can help KSP Wiki by expanding or discussing it.

Needs some math tags and general cleanup. -- N3X15 (talk) 08:46, 1 October 2012 (UTC)

Getting into space is relatively easy, but staying there without drifting endlessly into space or falling back down to Kerbin can be challenging. This tutorial will teach you how to get into and remain in orbit, how to adjust your orbit to be circular or elliptical, and how to adjust to a higher or lower orbit.

Stabilizing your orbit

During each orbit, your craft will reach maximum altitude, called apoapsis, and on the opposite side of the planet, it will reach minimum altitude, called periapsis. At both apoapsis and periapsis, your vertical speed will be zero. These points are the easiest points to make orbital corrections, because you can easily determine how fast to go when your vertical speed is zero. Note: The relative difference between your orbit's apoapsis and periapsis is called its eccentricity. Orbits that are exactly circular have zero eccentricity, and highly "flattened-out" orbits have eccentricity close to 1.

There are a number of third-party calculators available which can crunch the numbers and tell you your eccentricity, as well as provide the speeds required to circularize your orbit at your current (or future) altitude. Whether you calculate your orbits by hand, or use a third party app, the general procedures are still the same and are given below:

First, in order to get into a nice, round orbit, you need to determine how fast to go. The mathematical basis for orbital speed is determined from your current distance from your central body (), your semi-major axis from your central body (), and the mass of the central body itself (). These may be use to find the speed at an orbit around any body using the relation

where is the gravitational constant . Keep in mind that distances to the central body must account not only for altitude but also for the radius () of whatever body you are orbiting. The exact values of and may be found on their respective pages.

Returning to our case, the higher your orbit, the less gravity you'll feel from Kerbin, so the slower you'll need to go to be in a circular orbit. Determine the proper speed for your altitude at apoapsis or periapsis either by hand, by calculator, or by table. You'll probably want to watch your altimeter as you near one of the critical points, remember the altitude, determine your desired speed, and make the correction on your next pass. If you want to "round out" your orbit from apoapsis, you need to speed up to avoid falling back down to periapsis. Point your craft in the exact direction of travel (use the green circular indicator on the gimbal to line up), and apply thrust until you've gained enough speed. To round out an orbit from periapsis, you need to slow down to avoid climbing back up to apoapsis. Point your craft in the opposite direction of travel (indicated on the gimbal by a green circle with an "X" through it), and apply thrust until you have slowed to the speed indicated by the table. You should then be in an orbit that is very close to circular! Depending on how eccentric your initial orbit was, you may need to make a large correction on your first pass followed by a small correction on a subsequent pass to get very stable.

If you have version 0.11 or better, adding a set of RCS thrusters to your craft can help make minute adjustments to an orbit easier. Version 0.11 also allows you to see the current trajectory (and read periapsis and apoapsis altitudes) by switching to the Map view (M key)

Transfer Orbits

The most efficient way to transfer from a lower circular orbit to a higher circular orbit (or vice-versa) is to use an elliptical transfer orbit, also known as a Hohmann transfer orbit. To transfer, we make the periapsis of the elliptical orbit the same as the radius of the lower orbit, and the apoapsis of the elliptical orbit the same as the radius of the higher orbit. If you are going from low to high, you make a burn in the direction of travel to elongate your orbit. You will climb in altitude as you travel around the planet to the apoapsis of your transfer orbit. Then, make a second burn to round out the new, higher orbit (as described above). To go from high to low, do the opposite: Burn in the opposite direction of travel, then fall down to the periapsis of your transfer orbit, and make a second burn to round out the lower orbit (again in the opposite direction of travel).

Target Speed

The key to transfer orbits is figuring out how much speed to add or subtract to reach a desired new orbital altitude. To do this, use the formula below to determine the target velocity for your initial burn:

In this formula, rl and rh are the radii of the lower and higher orbits, respectively, and ri is the radius of the initial orbit. If you are transferring to a higher orbit, ri will be equal to rl, and v will be faster than your current speed, so burn in the direction of travel to reach v. If you are transferring to a lower orbit, ri will be equal to rh, and v will be slower than your current speed, so burn in the opposite direction to reach v. Remember, v is the target speed for your initial burn that puts you into the elliptical transfer orbit. Once you reach your new orbital altitude, you need to make a second burn to round out your orbit, using the same technique described in the stabilizing your orbit section.

Details of where this formula comes from are in the technical section below. When using this formula, take care to remember that the radius of an orbit is equal to the orbital altitude plus Kerbin's radius (600 000 m).

De-orbiting

The most efficient way to de-orbit from any altitude is to initiate a transfer orbit with a periapsis below 70000 m, the edge of Kerbin's atmosphere. Note that the upper atmosphere is very thin so if you do not want to wait for several orbits of aerobraking, aim for under 35000 m and thicker air. As you approach periapsis, the atmospheric drag will start to slow your craft and eventually it can no longer maintain orbit.

R code snippet for planning Hohmann transfer

   hohmann <- function(from_alt,to_alt){
     # provides information needed to perform
     # a hohmann transfer from a circular ortbit
     # at from_alt (km) to a circular orbit at to_alt (km)
     mu  <- 3530.394     # Gravitational parameter (km^3/s^2)
     R   <- 600          # Kerbin radius (km)
     r1  <- from_alt+R   # radius 1 (km)
     r2  <- to_alt+R     # radius 2 (km)
     vc1 <- sqrt(mu/r1)  # circular orbit velocity 1 (km/s)
     vc2 <- sqrt(mu/r2)  # circular orbit velocity 2 (km/s)
     a   <- (r1+r2)/2    # semi-major axis of transfer orbit (km)
     T   <- 2*pi*sqrt((a^3)/mu)  # period of transfer orbit (s)
     dv1 <- (sqrt(r2/a)-1)*vc1   # delta v1 (km/s)
     dv2 <- (1-sqrt(r1/a))*vc2   # delta v2 (km/s)
     b1  <- list(from=vc1,to=vc1+dv1) # burn one from-to velocities (km/s)
     t   <- T/2          # time between burns (s)
     b2  <- list(from=vc2+dv2,to=vc2) # burn two from-to velocities (km/s)
     out <- list(from_alt=from_alt,b1=b1,t=t,b2=b2,to_alt=to_alt)
   return(out)}

Example usage

Produce a graph showing the speeds need to transfer from a range of circular orbit altitudes into a landing orbit.

plot(100*1:40,1000*hohmann(100*1:40,34)$b1$to,main="Landing speeds",xlab="altitude (km)",ylab="speed (m/s)")

Landingspeeds.png

R project Link [1]


Transfer Orbits

Coming soon!

Orbital Table

Note: The atmosphere previously had a sharp cutoff at 34.5 km, but now extends to approximately 68 km. Below this altitude, your orbit will gradually decay. The decay becomes quite rapid below about 45 km. The orbital parameters below 68 km are provided for reference, but understand that you will not be able to maintain these orbits without regular corrections to counteract the atmospheric drag.

Altitude (m) Horizontal Speed (m/s) Orbital Period (min)
40000 2348.7 28.54
50000 2330.6 29.21
60000 2312.8 29.88
70000 2295.5 30.56
80000 2278.6 31.25
85000 2270.3 31.60
90000 2262.0 31.94
100000 2245.8 32.64
110000 2229.9 33.34
120000 2214.4 34.05
130000 2199.2 34.76
140000 2184.3 35.48
150000 2169.6 36.20
160000 2155.3 36.93
170000 2141.3 37.66
180000 2127.5 38.39
190000 2114.0 39.13
200000 2100.7 39.88
210000 2087.7 40.63
220000 2075.0 41.38
230000 2062.4 42.14
240000 2050.1 42.91
250000 2038.0 43.68
260000 2026.1 44.45
270000 2014.5 45.23
280000 2003.0 46.01
290000 1991.7 46.79
300000 1980.6 47.59
310000 1969.7 48.38
320000 1959.0 49.18
330000 1948.4 49.98
340000 1938.0 50.79
350000 1927.8 51.61
360000 1917.7 52.42
370000 1907.8 53.24
380000 1898.0 54.07
390000 1888.4 54.90
400000 1879.0 55.73
410000 1869.6 56.57
420000 1860.5 57.41
430000 1851.4 58.26
440000 1842.5 59.11
450000 1833.7 59.96
460000 1825.0 60.82
470000 1816.5 61.69
480000 1808.0 62.55
490000 1799.7 63.42
500000 1791.5 64.30
510000 1783.4 65.18
520000 1775.5 66.06
530000 1767.6 66.95
540000 1759.8 67.84
550000 1752.1 68.73
560000 1744.6 69.63
570000 1737.1 70.53
580000 1729.7 71.44
590000 1722.4 72.35
600000 1715.3 73.26
610000 1708.2 74.18
620000 1701.1 75.10
630000 1694.2 76.03
640000 1687.4 76.96
650000 1680.6 77.89
660000 1673.9 78.83
670000 1667.3 79.77
680000 1660.8 80.71
690000 1654.3 81.66
700000 1648.0 82.61
710000 1641.7 83.56
720000 1635.4 84.52
730000 1629.3 85.48
740000 1623.2 86.45
750000 1617.2 87.42
760000 1611.2 88.39
770000 1605.3 89.37
780000 1599.5 90.35
790000 1593.7 91.33
800000 1588.0 92.32
810000 1582.4 93.31
820000 1576.8 94.31
830000 1571.3 95.30
840000 1565.8 96.31
850000 1560.4 97.31
860000 1555.0 98.32
870000 1549.7 99.33
880000 1544.5 100.35
885000 1541.9 100.86
890000 1539.3 101.37
900000 1534.2 102.39
910000 1529.1 103.41
920000 1524.0 104.44
930000 1519.1 105.47
940000 1514.1 106.51
950000 1509.2 107.55
960000 1504.4 108.59
970000 1499.6 109.64
980000 1494.8 110.69
990000 1490.1 111.74
1000000 1485.5 112.79
2 868 378 1008.910 6 hours
8 140 000 635.4 24 hours