You've made two obvious mistakes and a poor choice of algorithm.
First, you've modeled the position of the moon as a single number rather than a vector in 3-space, so what you're modeling here is simple harmonic motion, like a spring constrained to a single dimension, and not an orbit. Start by modeling positions and velocities as 3-space vectors, not doubles.
Second, you've made the sign of the gravitational force positive, so that it is a repelling force between two bodies, rather than attractive. The sign of the force has to be in the direction of the earth.
Third, your implementation of Euler's algorithm seems to be correct, but Euler is a poor choice for numerically solving orbital problems because it is not conservative; you can easily get into situations where the moon gains or loses a little bit of momentum, and that adds up, and wrecks your nice elliptical orbit.
Since the Moon's orbit is Hamiltonian, use a symplectic algorithm instead; it's designed to simulate conservative systems.
https://en.wikipedia.org/wiki/Symplectic_integrator
This approach, and your Eulerian approach, are fundamentally about finding the state vectors:
https://en.wikipedia.org/wiki/Orbital_state_vectors
However, for your simple 2-body system there are easier methods. If what you want to do is make a simulation like Kerbal Space Program, where only the body you are orbiting affects your orbit, and planets with multiple moons are "on rails", you don't need to simulate the system on every time unit to work out the sequence of state vectors; you can compute them directly given the Keplerian elements:
https://en.wikipedia.org/wiki/Orbital_elements
We know the six elements for the moon to a high degree of precision; from those you can compute the position in 3-space of the moon at any time, again, assuming that nothing perturbs its orbit. (In reality, the moon's orbit is changed by the sun, by the fact that the earth is not a sphere, by the tides, and so on.)
UPDATE:
First off, since this is for coursework you are required to cite all your sources, and that includes getting help from the internet. Your instructors must know what work is yours and what work you had someone else do for you.
You asked how to do this work in two dimensions; that seems wrong, but whatever, do what the course work says.
The rule that I wish more beginners were taught is: make a type, a method or a variable which solves your problem. In this case, we wish to represent the behavior of a complex value, so it should be a type, and that type should be a value type. Value types are struct in C#. So let's do that.
struct Vector2
{
double X { get; }
double Y { get; }
public Vector2(double x, double y)
{
this.X = x;
this.Y = y;
}
Notice that vectors are immutable, just like numbers. You never mutate a vector. When you need a new vector, you make a new one.
What operations do we need to perform on vectors? Vector addition, scalar multiplication, and scalar division is just fancy multiplication. Let's implement those:
public static Vector2 operator +(Vector2 a, Vector2 b) =>
new Vector2(a.X + b.X, a.Y + b.Y);
public static Vector2 operator -(Vector2 a, Vector2 b) =>
new Vector2(a.X - b.X, a.Y - b.Y);
public static Vector2 operator *(Vector2 a, double b) =>
new Vector2(a.X * b, a.Y * b);
public static Vector2 operator /(Vector2 a, double b) =>
a * (1.0 / b);
We can do multiplication in the other order too, so let's implement that:
public static Vector2 operator *(double b, Vector2 a) =>
a * b;
Making a vector negative is the same as multiplying it by -1:
public static Vector2 operator -(Vector2 a) => a * -1.0;
And to help us debug:
public override string ToString() => $"({this.X},{this.Y})";
}
And we're done with vectors.
We are trying to simulate the evolution of orbital state parameters, so again make a type. What are the state parameters? Position and velocity:
struct State
{
Vector2 Position { get; }
Vector2 Velocity { get; }
public State(Vector2 position, Vector2 velocity)
{
this.Position = position;
this.Velocity = velocity;
}
Now we get to the core algorithm. What do we have? a state and an acceleration. What do we want? A new state. Make a method:
public State Euler(Vector2 acceleration, double step)
{
Vector2 newVelocity = this.Velocity + acceleration * step;
Vector2 newPosition = this.Position + this.Velocity * step;
return new State(newPosition, newVelocity);
}
}
Super. What's left? We need to work out the acceleration. Make a method:
static Vector2 AccelerationOfTheMoon(Vector2 position)
{
// You do this. Acceleration is force divided by mass,
// force is a vector, mass is a scalar. What is the force
// given the position? DO NOT MAKE A SIGN MISTAKE AGAIN.
}
And now you have all the parts you need. Starting from an initial state you can repeatedly call AccelerationOfTheMoon to get the new acceleration, and then call Euler to get the new state, and repeat.
As a beginner, study these techniques carefully. Notice what I did: I made types to represent concepts, and methods to represent operations on those concepts. Once you do that, the program becomes extremely clear to read.
Think about how you would extend this program using these techniques; we made a hard-coded AccelerationOfTheMoon method, but what if we wanted to compute accelerations of several bodies? Again, make a type to represent Body; a body is characterized by State and mass. What if we wanted to solve the n-body problem? Our acceleration computation would have to take the other bodies as a parameter. And so on.
