Calculating Orbital Velocity and Probe Velocity in a Hohmann Transfer to Neptune

[garyman]

A Hohmann transfer orbit is used to send a spacecraft to neptune. However the positions of the other planets were not taken into consideration. The craft approaches Jupiter at an angle of 75 degrees to Jupiter's orbit. Calculate (a) the orbital velocity of Jupiter, (b) the probe's velocity upon reaching Jupiter.

Not really sure how to start this question. Does anyone else have any ideas?

 

[alphysicist]

Hi garyman,

A Hohmann transfer orbit is used to send a spacecraft to neptune. However the positions of the other planets were not taken into consideration. The craft approaches Jupiter at an angle of 75 degrees to Jupiter's orbit. Calculate (a) the orbital velocity of Jupiter, (b) the probe's velocity upon reaching Jupiter.

Not really sure how to start this question. Does anyone else have any ideas?

I would say the starting point is to calculate the parameters of the elliptical Hohmann orbit using the beginning and ending orbital radii, and using the equations for the velocity impulses needed. However, it's not clear to me what quantities are to be considered as "given". What quantities are you allowed to look up?

 

[thomate1]

I would like to first clarify that a Hohmann transfer orbit is a commonly used technique for interplanetary travel, but it is not the only method and it does not always consider the positions of other planets. However, in this scenario, we will assume that the Hohmann transfer is the chosen method for sending the spacecraft to Neptune.

To calculate the orbital velocity of Jupiter, we will use the formula V = √(GM/r), where G is the universal gravitational constant, M is the mass of Jupiter, and r is the distance between the spacecraft and Jupiter. According to NASA's planetary fact sheet, Jupiter's mass is approximately 1.898 x 10^27 kg and its average distance from the Sun is 778.5 million kilometers (5.20 astronomical units). Therefore, the orbital velocity of Jupiter at this distance would be:

V = √((6.67408 x 10^-11 Nm^2/kg^2) * (1.898 x 10^27 kg) / (7.785 x 10^11 m))

V = 13.07 km/s

To calculate the probe's velocity upon reaching Jupiter, we will use the Hohmann transfer equation, which states that the velocity of the probe upon reaching Jupiter will be equal to the square root of the sum of the departure planet's orbital velocity squared and the arrival planet's orbital velocity squared. In this case, the departure planet is Earth and the arrival planet is Jupiter.

Therefore, the probe's velocity upon reaching Jupiter would be:

Vprobe = √(Vearth^2 + Vjupiter^2)

Vprobe = √((29.78 km/s)^2 + (13.07 km/s)^2)

Vprobe = 32.78 km/s

It is important to note that this calculation assumes a perfectly executed Hohmann transfer and does not take into account any gravitational assists or other factors that may affect the spacecraft's velocity. Additionally, the angle of approach to Jupiter's orbit may also impact the probe's velocity. Further calculations and adjustments may be necessary to accurately determine the probe's velocity upon reaching Jupiter.

 

[baba89]

I would approach this question by first understanding the basic principles of orbital mechanics and Hohmann transfers. A Hohmann transfer is a type of orbital maneuver that uses the gravitational pull of one planet to transfer a spacecraft from one circular orbit to another. In this case, the spacecraft is being transferred from Earth's orbit to Neptune's orbit, with a flyby of Jupiter in between.

To calculate the orbital velocity of Jupiter, we need to use the formula for orbital velocity: V = √(GM/r), where G is the gravitational constant, M is the mass of Jupiter, and r is the distance from the center of Jupiter. According to NASA, Jupiter's mass is approximately 1.898 × 10^27 kg and its average distance from the Sun is 778 million km. However, since the spacecraft is approaching Jupiter at an angle of 75 degrees to its orbit, we need to use the cosine of this angle to calculate the actual distance from the center of Jupiter. This would give us a distance of approximately 194.5 million km. Plugging these values into the formula, we get an orbital velocity of approximately 13.6 km/s.

To calculate the probe's velocity upon reaching Jupiter, we need to use the concept of conservation of energy. During a Hohmann transfer, the spacecraft's velocity at the end of the first half of the transfer is equal to the velocity of the planet it is orbiting (in this case, Jupiter). However, since the spacecraft is approaching Jupiter at an angle of 75 degrees, we need to use the cosine of this angle to calculate the actual velocity at this point. This would give us a velocity of approximately 3.4 km/s. Therefore, the probe's velocity upon reaching Jupiter would be 3.4 km/s relative to Jupiter.

It is important to note that while these calculations provide a general estimate, the actual orbital and probe velocities may vary due to the influence of other planets and their positions at the time of the transfer. As a scientist, it would be important to consider and account for these factors in order to accurately calculate the orbital and probe velocities in a Hohmann transfer to Neptune.