The Flight Effectiveness Index

[Leganeski]

Often, you might want to take advantage of an extraterrestrial atmosphere by flying a plane or using parachutes to slow down a craft.

This can be more or less effective depending on the conditions there: a parachute that lowers a capsule down to Earth safely will crash at high speed into Mars, while a propeller plane that fails during testing on Kerbin might actually work just fine in Eve's thicker atmosphere. In this sense, Eve is better for atmospheric flight, while Mars is worse.

But by how much?

I've seen this concept quantified before in specific instances, but not in the general case, and as far as I know, it has not been given a name. I propose a new metric to measure it: the flight effectiveness index.

(If this name is already taken, or if the concept actually does already have a name I don't know about, I'd love to hear about it so that I can correct this post.)

 

Unfortunately, air is complicated, and many simplifying assumptions are often made. One unfortunately common one is the lack of distinction between atmospheric density (how much mass the air in a given volume has) and atmospheric pressure (roughly proportional to how much energy the air in a given volume has). While clearly related, these concepts can end up being very different because of the fact that air molecules move at different speeds.

KSP does maintain the distinction between pressure and density internally, but fails to convey it in situations where it is relevant, such as when parachutes open. They open at a minimum pressure, when perhaps a minimum density would be more consistent with the actual performance of the parachute.

It's not just a problem in KSP: pressure and density are both measures for "how much air there is", so the distinction between them is not intuitive and can easily lead to confusion. Even xkcd, a webcomic now known for its very good scientific accuracy, at one point got pressure and density mixed up.

(source: https://xkcd.com/620/)

The 9% figure would be correct except for the fact that Titan's atmosphere is not 50% denser than Earth's; it has 50% more pressure (and consequently over four times the density because the air is so cold).

The relationship between density and pressure can be derived from the ideal gas law. (The ideal gas law isn't perfect, but it's close enough under normal conditions.)

PV = nRT,

where P is the pressure, V is the volume, n is the number of moles of air, R is a constant, and T is the temperature.

Wait, this equation doesn't have density in it! We need to relate the density to other things that are in the equation.

The density ρ, by definition, is the mass m divided by the volume V. The total mass m is the molar mass μ times the number of moles n.

Putting this all together, we find:

PV = nRT
P = (n / V) RT
P = (m / (μV)) RT
P = (ρ / μ) RT
Pμ / RT = ρ

R is a constant, so the density is proportional to the pressure times the molar mass divided by the temperature.

 

 

The rest of the comic is basically accurate: the amount of mass held up by a given wing or parachute (at a given velocity) is proportional to the density divided by the surface gravity.

(Explanation in the spoiler box for why this works.)

Spoiler

The lift and drag equations have the same form:

F = ρv²Ac/2,

where F is the aerodynamic force on the surface, v is the airspeed velocity, A is the area exposed to airflow, and c is a dimensionless constant dependent mainly on the shape of the surface.

We're also letting A and v be constant. Velocity does change, but a craft has an approximately constant maximum landing velocity: above a certain speed, it will hit the ground with too much energy and break. (Typical values might be 5-15 m/s for a vertical descent or 40-100 m/s for a horizontal descent, depending on the terrain and how sturdy the craft is.) The performance at this speed effectively determines whether the wings / parachutes will work or not, so we'll stick to just one speed.

(It's not too hard to extend these concepts to changing speeds: a wing that is four times as effective will work at half the velocity due to the v² term.)

Under these assumptions, the force is proportional to the atmospheric density.

However, the required force to hold up a given mass increases with the local gravity. At twice the gravity, the same amount of lift will hold up only half the mass.

This gives us our final definition for the flight effectiveness index:

FEI = (Pμ) / (Tg),

where P is the pressure (in kPa),
μ is the mean molar mass (in amu = g/mol),
T is the temperature (in K),
and g is the local surface gravity (in m/s²).

Importantly, this is dependent only on the local conditions, not on any characteristics of the vessel.

(For maximum accuracy, the local surface gravity should include the effect of the body's rotation; that is, it should be reduced for rapidly spinning bodies. However, this does not make much of a difference unless the body is rotating quite quickly.)

In KSP, the pressure, temperature, and local gravity can easily be obtained in the above units from the barometer, thermometer, and gravioli detector respectively. The molar mass is not obtainable in-game, but it is globally constant across each body. For stock atmospheres, it can be found on the body's KSP wiki page. For most modded atmospheres, it can be found in the body's .cfg file as atmosphereMolarMass (in kg/mol, which must be multiplied by 1000 to obtain it in g/mol).

While this definition is based on SI units, it would be nice if there was an easy-to-remember "Earth/Kerbin standard" value. (Earth and Kerbin have basically identical conditions.) Fortunately, there is! Plugging in standard thermodynamic conditions, we find:

P = 101.325 kPa (1 atm)
T = 298.15 K (SATP standard)
μ = 28.9644 amu (US standard for Earth; Kerbin global constant)
g = 9.80665 m/s² (1 g)

FEI = (101.325 * 28.9644) / (298.15 * 9.80665) ≈ 1.00375,

which is basically 1 (to within the actual variation on Earth of any of pressure, temperature, or even gravity).

Of course, actual conditions on Earth and Kerbin vary considerably. In the thinner atmosphere at high altitude (0.8 atm, for example), the FEI falls to 0.8, meaning that a plane can only hold 0.8 times as much mass. In the cold polar weather (temperature: roughly 240 K), the FEI rises to 1.25.

 

The flight effectiveness index can now be calculated on other celestial bodies. Let's take Eve as an example of a popular destination for planes. On the equator at sea level at noon, the conditions are:

P = 506.625 kPa
T = 423.7 K
μ = 43 amu
g = 16.67 m/s²

FEI = (506.625 * 43) / (423.7 / 16.67) ≈ 3.08. This means that a parachute can hold up 3.08 times as much mass as would be safe on Kerbin, and a propeller plane can carry 3.08 times the mass (including the mass of the plane, so you get a lot more than 3.08 times as much payload).

 

While this definition is designed to be as easy as possible to calculate accurately, it still requires some work. Alternatively, I've included tables of typical FEI values on atmospheric planets and moons from a variety of systems.

The FEI varies across the surface of any body, mainly due to variations in altitude (i.e. mountains, which you probably already knew to look out for) and temperature (which doesn't change all that much unless the body is a tidally locked planet). The given values are at the datum level at a latitude of 17 degrees, with a temperature averaged across all daily and seasonal variation.

Stock system:

Spoiler

Eve		506.625	43	420.1	16.67	3.111
Kerbin		101.325	28.9644	300.8	9.758	1.000
Duna		6.755	43*	261.1	2.939	0.379
Jool		5066.25	2.2	200	7.671	7.265
Laythe		60.795	28.9644	284.6	7.839	0.789

* The KSP wiki says 42 g/mol, while Kittopia Dumps says 43 g/mol. Both values seem reasonable, so I went with the one from Kittopia Dumps (which was updated more recently).

Real Solar System:

Spoiler

Body		Pressure (kPa)	Molar mass (amu)	Temperature (K)	Gravity (m/s²)	FEI
Venus		10905.2	43.45	754.3	8.878	70.76
Earth		101.325	28.9644	299.5	9.788	1.001
Mars		1.14497	43.48	239.1	3.742	0.0556
Titan		159.018	27.55	96.10	1.356	33.62
Triton		0.00165	28.01	39.36	0.7792	0.00151
Pluto		0.00100	27.97	42.41	0.6171	0.00107

In particular, Titan's flight effectiveness index is 33.62, so you could fly there if you can lift 1 / 33.62 ≈ 2.97% of your weight on Earth.

That's really low, and easily achievable! The only problem is the temperature: at 96 K (-177 °C), you would quickly become unable to continue lifting yourself in the air because you would freeze to death in seconds.

Outer Planets Mod:

Spoiler

Body		Pressure (kPa)	Molar mass (amu)	Temperature (K)	Gravity (m/s²)	FEI
Tekto		124.63	28.9644	94.83	2.455	15.51
Thatmo		1.01325	28.9644	74.53	2.275	0.173

Galileo's Planet Pack:

Spoiler

Body		Pressure (kPa)	Molar mass (amu)	Temperature (K)	Gravity (m/s²)	FEI
Niven		12.159	43.17	361.9	4.871	0.298
Gael		101.325	28.9644	297.4	9.758	1.011
Tellumo		1013.25	29.21	275.4	18.57	5.787
Gratian		50.6625	29.73	182.4	7.354	1.123
Augustus		10.1325	28.01	126.1	3.428	0.657
Catullus		506.625	5.13	101.6	8.826	2.898
Tarsiss		141.855	27.53	94.36	1.649	25.10
Hadrian		40.53	28.01	67.36	1.763	9.560
Hox		1.01325	28.01	51.96	1.353	0.404
Leto		0.506625	28.01	45.31	1.152	0.272

Grannus Expansion Pack:

Spoiler

Body		Pressure (kPa)	Molar mass (amu)	Temperature (K)	Gravity (m/s²)	FEI
Toutatis	light side	4.053	43.33	463.3	4.413	0.0859
	terminator			339.5		0.117
	dark side			215.8		0.184
Nodens		202.65	28.69	300.7	10.79	1.792
Brovo		15.199	27.89	137.1	3.432	0.901
Epona		101.325	27.95	98.00	5.869	4.924

(Toutatis is a tidally locked planet.)

JNSQ:

Spoiler

Body		Pressure (kPa)	Molar mass (amu)	Temperature (K)	Gravity (m/s²)	FEI
Eve		1013.25	40.8	416.1	13.719	7.242
Kerbin		101.325	28.9644	297.9	9.755	1.010
Duna		4.053	42.4	242.8	3.324	0.213
Laythe		60.795	28.4	283.2	5.685	1.072
Tylo		20.265	28	125.1	3.138	1.445
Huygen		151.9875	27.5	92.13	1.471	30.84
Riga		6.0795	28	84.06	1.765	1.147
Eeloo		2.0265	28	68.90	1.446	0.570
Nara		4053	2.42	40.72	9.741	24.73

Whirligig World:

Spoiler

Body		Pressure (kPa)	Molar mass (amu)	Temperature (K)	Gravity (m/s²)	FEI
Imterril		1519.88	18.58	531.1	7.561	7.032
Tannor		0.577998	18.31	237.2	2.187	0.0204
Mesbin (latitude 89°)		101.325	42.1	248.6	129.6	0.132
Kerbmun		101.325	28.33	292.9	10.39	0.943
Derbin		790.335	18.29	382.1	21.25	1.780
Valyr		648.48	19.52	286.4	20.97	2.108
Oshan		3.09041	29.37	250.9	3.528	0.103
Egad		79.2362	28.56	200.9	5.024	2.242
Lito		24.318	26.59	191.8	7.126	0.473
Totooa		77.311	27.87	189.7	1.864	6.093
Lowel		14.1855	32.13	316.6	5.819	0.247
Ollym		0.52689	43.63	278.6	2.848	0.0290
Gannovar		176.306	28.54	293.2	4.707	3.646

(Mesbin's atmosphere is found only near the north pole.)

Strange New Worlds:

Spoiler

Body		Pressure (kPa)	Molar mass (amu)	Temperature (K)	Gravity (m/s²)	FEI
Pandemonium		10132.5	2.2	2100	10.97	0.968
Aeneas		70.9275	43	452.8	15.32	0.440
Denali		20.265	44	277.0	2.157	1.492
Oceanus		1519.88	28	293.4	11.96	12.13
Vetinghier		405.3	44	284.5	5.832	10.75
Arkhalo		303.975	29.5	305.0	5.735	5.127
Hreveldor		101.325	29.5	286.3	8.335	1.253
Kisa		50.6625	29.5	261.5	4.737	1.207
Boreal		2.0265	28	120.0	1.967	0.240

Edge of Eternity:

Spoiler

Body		Pressure (kPa)	Molar mass (amu)	Temperature (K)	Gravity (m/s²)	FEI
Revenant	Subterminal Region	6.241	41.45	820.0	7.929	0.0398
	Lightside Midlands	4.657	41.45	772.1	7.83	0.0319
	light side Highland Plateaus	3.895	41.45	699.3	7.772	0.0297
	Dry Oceans	6.89	41.45	642.2	7.963	0.0558
	Darkside Midlands	5.152	41.45	515.9	7.864	0.0526
	dark side Highland Plateaus	3.895	41.45	415.3	7.772	0.0500
	Antiterminal Region	4.279	41.45	368.3	7.802	0.0617
Achlys		741.699	47.58	847.8	12.64	3.293
Cerberus		2999.22	15.01	490.0	13.48	6.816
Warden		4.8636	51.61	351.6	2.02	0.353
Haven		161.107	28.58	282.0	8.711	1.874
Prometheus		120.577	28.25	296.0	10.68	1.078
Arenace	light side	11.1458	28.42	224.1	9.454	0.150
	terminator			161.0		0.208
	dark side			97.94		0.342
Gelis		63.8348	28.74	131.3	7.095	1.969
Heima		246.22	10.57	28.31	2.952	31.14

(Revenant and Arenace are tidally locked planets. Prometheus is also locked, but it spins rapidly enough that atmospheric mixing causes its atmosphere to behave as if it were unlocked.)

Revenant's unique topography means that the ground altitude and temperature both vary significantly across biomes, but are relatively constant within each biome. Because of this, the major biomes are listed separately in order of distance from the substellar point.

 

Edited January 8, 2023 by Leganeski
Fix grammar

[magnemoe]

Surprised me that Laythe in KSP 1 was so bad, I always found it very forgiving. Orbital velocity so low you never overheat dropping from orbit and you has to climb slower than on Kerbin with spaceplanes so you don't raise Ap to high flying with atmospheric engines. 

[Leganeski]

2 hours ago, magnemoe said:

Surprised me that Laythe in KSP 1 was so bad, I always found it very forgiving. Orbital velocity so low you never overheat dropping from orbit and you has to climb slower than on Kerbin with spaceplanes so you don't raise Ap to high flying with atmospheric engines. 

Yeah, that surprised me too, especially given how much easier it is to climb to orbit.

However, Laythe is only worse than Kerbin close to sea level. Laythe's main heat source is tidal heating from Jool, so the temperature falls very quickly as altitude increases. This raises the density, meaning that once you get a few kilometers off the ground, Laythe really is better. Also, Laythe's higher scale height means that the pressure doesn't fall as quickly, so higher altitudes are even better on Laythe compared to Kerbin all the way up to 38 km.

Altitude (m)	FEI (Kerbin)	FEI (Laythe)
0	1.000	0.789
2500	0.740	0.709
5000	0.538	0.621
7500	0.382	0.460
10000	0.251	0.332
15000	0.097	0.171
20000	0.0364	0.0837
25000	0.0140	0.0473
30000	0.0054	0.0298
35000	0.0022	0.0156
40000	0.00098	0.00745
45000	0.00048	0.00193

[JoeSchmuckatelli]

Neanderthal question: given that the atmosphere of Venus is denser than Earth's, with a slightly less dense planet, if you could overcome the temperature and acidity problems - would we be able to do a SSTO spaceplane there, easier than here?

Average wind speeds are higher, too - so lift generation should be better - but I know that denser air = greater drag.

...so maybe a slow flight up to altitude then punch it?  (would that work?)

[farmerben]

32 minutes ago, JoeSchmuckatelli said:

Neanderthal question: given that the atmosphere of Venus is denser than Earth's, with a slightly less dense planet, if you could overcome the temperature and acidity problems - would we be able to do a SSTO spaceplane there, easier than here?

Average wind speeds are higher, too - so lift generation should be better - but I know that denser air = greater drag.

...so maybe a slow flight up to altitude then punch it?  (would that work?)

I think Eve gives a fair idea.  If you have to carry oxidizer it is probably not possible.  

[Leganeski]

5 minutes ago, JoeSchmuckatelli said:

Neanderthal question: given that the atmosphere of Venus is denser than Earth's, with a slightly less dense planet, if you could overcome the temperature and acidity problems - would we be able to do a SSTO spaceplane there, easier than here?

Average wind speeds are higher, too - so lift generation should be better - but I know that denser air = greater drag.

...so maybe a slow flight up to altitude then punch it?  (would that work?)

Getting off the ground would indeed be easier. However, there are many other problems involved.

Most importantly, Venus doesn't have oxygen in its atmosphere. This means you would need propellers for the initial ascent rather than a jet engine, so the rocket stage would need start from an airspeed of at most 250 m/s rather than 1500 m/s. Also, keeping the propellers attached throughout the ascent means that they would cause extra drag in the upper atmosphere.

Slogging through the dense lower atmosphere would require a lot of energy. Keeping a propeller powered is easier than a jet, but storing hours worth of electricity in one plane would be quite challenging. (Solar panels might be possible, but good luck operating that many solar panels without causing a ton of drag.)

While the 85 m/s wind would help, it's not much compared to Earth's 465 m/s rotational velocity that Venus doesn't have.

In summary: while low orbit on Venus is 600 m/s slower than on Earth due to its smaller size, you would be need to start the rocket engines at an orbital velocity of more than 1600 m/s slower, so the actual fuel requirements would be significantly higher. Add that to all the other problems, and Venus is in fact a much worse place to fly an SSTO.

 

[magnemoe]

5 hours ago, Leganeski said:

Yeah, that surprised me too, especially given how much easier it is to climb to orbit.

However, Laythe is only worse than Kerbin close to sea level. Laythe's main heat source is tidal heating from Jool, so the temperature falls very quickly as altitude increases. This raises the density, meaning that once you get a few kilometers off the ground, Laythe really is better. Also, Laythe's higher scale height means that the pressure doesn't fall as quickly, so higher altitudes are even better on Laythe compared to Kerbin all the way up to 38 km.

Altitude (m)	FEI (Kerbin)	FEI (Laythe)
0	1.000	0.789
2500	0.740	0.709
5000	0.538	0.621
7500	0.382	0.460
10000	0.251	0.332
15000	0.097	0.171
20000	0.0364	0.0837
25000	0.0140	0.0473
30000	0.0054	0.0298
35000	0.0022	0.0156
40000	0.00098	0.00745
45000	0.00048	0.00193

Now this explains a lot like how an rapier powered spaceplane works as well, combine this with lower gravity. 
The only downside is higher landing and takeoff speeds. 

[magnemoe]

5 hours ago, farmerben said:

I think Eve gives a fair idea.  If you have to carry oxidizer it is probably not possible.  

I wonder how an nuclear jet engine would work, it don't require fuel, just reaction mass as in atmosphere. Now Eve is there to give Venus an good reputation, the only good thing about the surface of Venus is that its less radioactive than surface of Io. 
Negating this with an nuclear engine.

[tomf]

On Eve my KSP-I fusion powered ssto works well but not as well as it does on Jool

[Leganeski]

3 hours ago, tomf said:

On Eve my KSP-I fusion powered ssto works well but not as well as it does on Jool

 

On 1/7/2023 at 7:45 PM, Leganeski said:

Body		Pressure (kPa)	Molar mass (amu)	Temperature (K)	Gravity (m/s²)	FEI
Eve		506.625	43	420.1	16.67	3.111
Jool		5066.25	2.2	200	7.671	7.265

 

Jool is larger than Eve, so the ascent should always take more Δv. However, with fusion power, Δv is likely not a concern. Instead, there are other factors that make Jool easier:

• Jool has a denser atmosphere at the datum level, making it easier to start flying.
• Jool has less surface gravity than Eve, making it easier to ascend.
• Jool is less dense than Eve, making the ascent slower overall and giving you more time to react to anything that comes up.
• Jool rotates more quickly than Eve, giving you an extra boost and making the decrease in apparent surface gravity start sooner.
• The speed of sound is higher on Jool, so Mach-limited engines can go faster.