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

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(R code snippet for planning Hohmann transfer: fixed error in code)
(R code snippet for planning Hohmann transfer)
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       out <- list(from_alt=from_alt,b1=b1,t=t,b2=b2,to_alt=to_alt)
 
       out <- list(from_alt=from_alt,b1=b1,t=t,b2=b2,to_alt=to_alt)
 
     return(out)}
 
     return(out)}
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 +
== 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)")
 +
 +
[[file:landingspeeds.png]]
 +
 +
R project Link[http://www.r-project.org/]
  
 
== Technical Details ==
 
== Technical Details ==

Revision as of 18:16, 2 September 2011

Getting into space is relatively easy, but staying there without drifting endlessly into space or falling back down to Kearth 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.

Your first orbit

A good procedure for getting into orbit was posted by HarvesteR on the KSP forums (edited with a more accurate horizontal speed based on the orbital tables below):

  1. Launch straight up, and continue to climb up to about 10000 meters.
  2. Then, gradually start leveling off. You will start gaining horizontal speed.
  3. Now you're basically trading vertical speed for horizontal speed. The idea is to get to 0 vertical speed at about 40000 meters up, and be moving horizontally at about 2350 m/s.

This will put you in a stable orbit. The altitude at which you start leveling off and the altitude at which you reach orbital velocity will depend a lot on how your rocket is designed. The idea, however, is to get out of the thickest part of the atmosphere before you start adding horizontal speed, so that you aren't wasting energy adding horizontal speed which will just bleed off due to air resistance.

It is likely that you will overshoot or undershoot these figures by a wide margin on your first couple attempts. Don't worry! Manual orbit insertion is difficult, that's why NASA uses computer guidance! When you finally do get into a stable orbit, you'll probably be on an elliptical trajectory; that is, your ship will coast away from the planet, gradually losing speed. When it reaches its maximum altitude, it will start to fall back toward Kearth, picking up speed again. If you're going fast enough, you'll fall "past" Kearth instead of into it, and that's orbiting.

Video Tutorial

Video Example Of Building A Rocket And Taking It To 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.

First, in order to get into a nice, round orbit, you need to determine how fast to go. The higher your orbit, the less gravity you'll feel from Kearth, so the slower you'll need to go to be in a circular orbit. Consult the table below to determine the proper speed for your altitude at apoapsis or periapsis. You'll probably want to watch your altimiter as you near one of the critical points, remember the altitude, look up the speed in the table, 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.

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 perikee of the elliptical orbit the same as the radius of the lower orbit, and the apokee 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 apokee 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 perikee 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:

v = 1 878 968 * sqrt(2/ri - 2/(rl + rh))

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 Kearth'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 perikee just below 34 500 m, the edge of Kearth's atmosphere. Just as you reach perikee, the atmospheric drag will start to slow your craft until 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          # Kearth 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]

Technical Details

Below are the formulae used to determine orbital speeds, and a table of orbital speeds up to 1000 km.

Orbital Speed

The relation between orbital speed and acceleration is given by the formula:

a = v2 / r,

where a is the acceleration due to gravity, v is the horizontal speed, and r is the radius of orbit.

Of course, gravity varies depending on your distance from the planet, so we also need the following formula to determine a based on your altitude:

a = g * (R / (R + h))2,

where g is the acceleration due to gravity at sea level (9.807 m/s2), R is the radius of Kearth (600 km), and h is the altitude of your orbit.

Note: From here, we will substitute R + h for r, since the radius of your orbit is equal to the radius of Kearth plus your altitude.

Substituting for a and simplifying, we get:

g * (R / (R + h))2 = v2 / (R + h)

g * R2 / (R + h) = v2

v = R * sqrt(g / (R + h))

Finally, substituting known values for g and R,

v = 600 000 m * sqrt(9.807 m/s2 / (600 000 m + h))

Orbital Period

When the orbital speed is known, the orbital period is simple to determine. From the basic mechanics formula:

d = v * t

We know v from the above, and d is simply the circumference of a circle with a radius equal to your orbital altitude plus the radius of Kearth:

t = 2π * (600 000 m + h) / v

Transfer Orbits

Coming soon!

Orbital Table

Note: At altitudes below 34.5 km, the atmosphere will quickly drag you out of orbit.

Altitude (m) Horizontal Speed (m/s) Orbital Period (min)
35000 2357.9 28.20
36000 2356.1 28.27
37000 2354.2 28.33
38000 2352.4 28.40
39000 2350.5 28.47
40000 2348.7 28.54
41000 2346.9 28.60
42000 2345.0 28.67
43000 2343.2 28.74
44000 2341.4 28.80
45000 2339.6 28.87
46000 2337.8 28.94
47000 2336.0 29.00
48000 2334.2 29.07
49000 2332.4 29.14
50000 2330.6 29.21
51000 2328.8 29.27
52000 2327.0 29.34
53000 2325.2 29.41
54000 2323.4 29.48
55000 2321.7 29.54
56000 2319.9 29.61
57000 2318.1 29.68
58000 2316.4 29.75
59000 2314.6 29.82
60000 2312.8 29.88
61000 2311.1 29.95
62000 2309.4 30.02
63000 2307.6 30.09
64000 2305.9 30.16
65000 2304.1 30.22
66000 2302.4 30.29
67000 2300.7 30.36
68000 2299.0 30.43
69000 2297.2 30.50
70000 2295.5 30.56
71000 2293.8 30.63
72000 2292.1 30.70
73000 2290.4 30.77
74000 2288.7 30.84
75000 2287.0 30.91
76000 2285.3 30.98
77000 2283.6 31.05
78000 2281.9 31.11
79000 2280.3 31.18
80000 2278.6 31.25
81000 2276.9 31.32
82000 2275.2 31.39
83000 2273.6 31.46
84000 2271.9 31.53
85000 2270.3 31.60
86000 2268.6 31.67
87000 2266.9 31.74
88000 2265.3 31.80
89000 2263.7 31.87
90000 2262.0 31.94
91000 2260.4 32.01
92000 2258.7 32.08
93000 2257.1 32.15
94000 2255.5 32.22
95000 2253.9 32.29
96000 2252.2 32.36
97000 2250.6 32.43
98000 2249.0 32.50
99000 2247.4 32.57
100000 2245.8 32.64
105000 2237.8 32.99
110000 2229.9 33.34
115000 2222.1 33.70
120000 2214.4 34.05
125000 2206.7 34.40
130000 2199.2 34.76
135000 2191.7 35.12
140000 2184.3 35.48
145000 2176.9 35.84
150000 2169.6 36.20
155000 2162.4 36.56
160000 2155.3 36.93
165000 2148.3 37.29
170000 2141.3 37.66
175000 2134.4 38.02
180000 2127.5 38.39
185000 2120.7 38.76
190000 2114.0 39.13
195000 2107.3 39.51
200000 2100.7 39.88
205000 2094.2 40.25
210000 2087.7 40.63
215000 2081.3 41.01
220000 2075.0 41.38
225000 2068.7 41.76
230000 2062.4 42.14
235000 2056.3 42.52
240000 2050.1 42.91
245000 2044.0 43.29
250000 2038.0 43.68
255000 2032.1 44.06
260000 2026.1 44.45
265000 2020.3 44.84
270000 2014.5 45.23
275000 2008.7 45.62
280000 2003.0 46.01
285000 1997.3 46.40
290000 1991.7 46.79
295000 1986.1 47.19
300000 1980.6 47.59
305000 1975.1 47.98
310000 1969.7 48.38
315000 1964.3 48.78
320000 1959.0 49.18
325000 1953.7 49.58
330000 1948.4 49.98
335000 1943.2 50.39
340000 1938.0 50.79
345000 1932.9 51.20
350000 1927.8 51.61
355000 1922.7 52.01
360000 1917.7 52.42
365000 1912.7 52.83
370000 1907.8 53.24
375000 1902.9 53.66
380000 1898.0 54.07
385000 1893.2 54.48
390000 1888.4 54.90
395000 1883.7 55.32
400000 1879.0 55.73
405000 1874.3 56.15
410000 1869.6 56.57
415000 1865.0 56.99
420000 1860.5 57.41
425000 1855.9 57.84
430000 1851.4 58.26
435000 1846.9 58.68
440000 1842.5 59.11
445000 1838.1 59.54
450000 1833.7 59.96
455000 1829.3 60.39
460000 1825.0 60.82
465000 1820.7 61.25
470000 1816.5 61.69
475000 1812.2 62.12
480000 1808.0 62.55
485000 1803.9 62.99
490000 1799.7 63.42
495000 1795.6 63.86
500000 1791.5 64.30
505000 1787.5 64.74
510000 1783.4 65.18
515000 1779.4 65.62
520000 1775.5 66.06
525000 1771.5 66.50
530000 1767.6 66.95
535000 1763.7 67.39
540000 1759.8 67.84
545000 1756.0 68.28
550000 1752.1 68.73
555000 1748.3 69.18
560000 1744.6 69.63
565000 1740.8 70.08
570000 1737.1 70.53
575000 1733.4 70.98
580000 1729.7 71.44
585000 1726.1 71.89
590000 1722.4 72.35
595000 1718.8 72.80
600000 1715.3 73.26
605000 1711.7 73.72
610000 1708.2 74.18
615000 1704.6 74.64
620000 1701.1 75.10
625000 1697.7 75.56
630000 1694.2 76.03
635000 1690.8 76.49
640000 1687.4 76.96
645000 1684.0 77.42
650000 1680.6 77.89
655000 1677.2 78.36
660000 1673.9 78.83
665000 1670.6 79.29
670000 1667.3 79.77
675000 1664.0 80.24
680000 1660.8 80.71
685000 1657.6 81.18
690000 1654.3 81.66
695000 1651.1 82.13
700000 1648.0 82.61
705000 1644.8 83.09
710000 1641.7 83.56
715000 1638.5 84.04
720000 1635.4 84.52
725000 1632.3 85.00
730000 1629.3 85.48
735000 1626.2 85.97
740000 1623.2 86.45
745000 1620.2 86.93
750000 1617.2 87.42
755000 1614.2 87.91
760000 1611.2 88.39
765000 1608.2 88.88
770000 1605.3 89.37
775000 1602.4 89.86
780000 1599.5 90.35
785000 1596.6 90.84
790000 1593.7 91.33
795000 1590.9 91.83
800000 1588.0 92.32
805000 1585.2 92.82
810000 1582.4 93.31
815000 1579.6 93.81
820000 1576.8 94.31
825000 1574.0 94.81
830000 1571.3 95.30
835000 1568.5 95.80
840000 1565.8 96.31
845000 1563.1 96.81
850000 1560.4 97.31
855000 1557.7 97.81
860000 1555.0 98.32
865000 1552.4 98.82
870000 1549.7 99.33
875000 1547.1 99.84
880000 1544.5 100.35
885000 1541.9 100.86
890000 1539.3 101.37
895000 1536.7 101.88
900000 1534.2 102.39
905000 1531.6 102.90
910000 1529.1 103.41
915000 1526.6 103.93
920000 1524.0 104.44
925000 1521.5 104.96
930000 1519.1 105.47
935000 1516.6 105.99
940000 1514.1 106.51
945000 1511.7 107.03
950000 1509.2 107.55
955000 1506.8 108.07
960000 1504.4 108.59
965000 1502.0 109.11
970000 1499.6 109.64
975000 1497.2 110.16
980000 1494.8 110.69
985000 1492.5 111.21
990000 1490.1 111.74
995000 1487.8 112.27
1000000 1485.5 112.79
8 140 000 635.4 24 hours