Help me with the numbers of a landing

[mariohm1311]

I'm trying to do a programmed landing with Remote Tech in Minmus in preparation for a landig with delay. I use the VisViva equation and the gravitational potential energy equation. The Delta-V that has to be burnt is correct but the time to burn is always like 1 minute before it should. Can anyone help me?

• GM = Gravitational parameter = 1.7658*10^9
  
• Radius of planet (m) = 60,000
  
• Periapsis (m) = 20,700
  
• Terrain height (m) = 0
  
• r1 =20,700 + 60,000 = 80,700
  
• r2 = 60,000
  
• v² = 2 * ( -1.7658*10^9 / 80,700 + 1.7658*10^9 / 60,000 )
  
• v² = 15097,9182
  
• v = sqrt(15097,9182)
  
• v = 122,874 m/s (final velocity)
  
• average velocity = 122.874 /2 = 61,437
  
• fall distance = periapsis - terrain height = 20,700
  
• fall distance / average speed =20,700 m / 61.437 m/s = 336.931 seconds
  
• acceleration of craft = 12.61 m/s^2
  
• Time to cancel final velocity = 122.874 m/s / 12.61 m/s^2 = 9.744 seconds
  
• Start burn at 336.931 seconds - 9.744 seconds = 327.187 seconds from start, burn for 9.744 seconds
  

I know I can use LanderTron rockets, and I have them in my lander, but I don't want to use them because the lack the sense of acomplishment.

Edited January 18, 2015 by mariohm1311

[OhioBob]

The first thing that I notice is that your total fall time is incorrect. Since gravitational acceleration varies with height, velocity vs. time isn't a linear function. You can't simply call your average velocity 1/2 of your final velocity.

The second thing I'd recommend is to not base the start of your burn on time, but rather on altitude. Since you haven't quite reached the ground yet when you have to start the burn, your velocity at engine start will be a little less than your calculated velocity of 122.874 m/s. Let's call it 120 m/s (we can check this latter). Your velocity at touchdown is given by

v = (a + g) t + v_o

where v_o is your 120 m/s initial velocity, a is your applied acceleration, and g is the acceleration of gravity. It's not clear from your post whether your 12.61 m/s² acceleration is that resulting from thrust only or whether it has been reduced to account for gravity. I'm going to assume the former. Near the surface of Minmas, g = 0.49 m/s². Let say you want your touchdown velocity to be 1 m/s. We therefore have,

1 = (-12.61 + 0.49) t + 120

t = 9.8185 s

The distance travelled during the burn is give by

d = (a + g) t² / 2 + v_o t

Plugging in your numbers we get

d = (-12.61 + 0.49) * 9.8185² / 2 + 120 * 9.8185

d = 594 meters

Therefore you should start your burn 594 meters above the ground (allowing a little bit for the time it takes to throttle up).

We can now go back to the VisViva equation to check the velocity at an altitude of 594 m. I get 120.5 m/s, which is close enough to the assumed velocity of 120 m/s.

Also note that the acceleration of your spacecraft will increase slightly as you burn propellant, though with less than a 10 second burn this is unlikely to be much. If your decent velocity starts to go to zero before reaching the ground, just back off on the throttle a little bit to increase your fall rate.

Edited January 19, 2015 by OhioBob