Auxiliary Tidal

Force Effects [AUTHOR(S) OMITTED FOR WEB POSTING]

h

Background Tides

Gravity

Celestial bodies

Constructive Gravitational Forces Tidal Phases

Outline Problem Scenario

Assumptions

Mathematical Development and Solution

Results and Conclusion

Extensions

Problem Scenario The Sun’s size is increasing.

What are the tidal effects?

Compare tidal shift to the RMS Titanic.

Assumptions 1. Considering only 3 bodies: The Earth, Sun, and Moon

2. Neap Tide Position

3. These bodies are stationary

4. Sun’s size increases, all other sizes remain constant

5. Each body possesses uniform density

6. As the Sun grows, it will continually have uniform density

7. Traveling momentum of the Titanic is negated

Mathematical Development 1: Modeling Tides

Newton’s Law of Universal Gravitation:

A New Formula:

center of mass (G)

center of buoyancy (B) (the center of the submerged

volume of the boat).

metacenter (M)

Mathematical Development 2: Capsizing Vessels and GZ Curves

Righting Arm (GZ length)

Mathematical Development 2: Capsizing Vessels and GZ Curves

Solution Assume time variable, and assume initial Lunar

displacement

Find a path from increasing Sun mass to Solar tidal

displacement by use of formulae:

Sun’s Mass(t) ⇒

Volume ⇒

Radius ⇒

Distance from Sun to Earth ⇒

Ratio of Sun's gravitational force to the Moon's ⇒

Solar Tidal Displacement

Results Refer to Excel Tables

Table with Mass*e^(t)

t = 6

t = 17

y = 0.2419e1.0224x R² = 0.9999

0

500

1000

1500

2000

2500

3000

0 5 10

Dis

pla

cem

en

t (m

)

Time t

Tidal Displacement

Expon.…

Conclusions 1. At t = 7, Tidal displacement exceeds record wave

height

2. At t = 17, Sun engulfs Earth (negative distance)

3. With the Sun tripled in size, tidal displacement is not

very significant

The Titanic has fallen

Extensions

Measure danger of capsizing due to size waves and length

of righting arm:

Formula for relating size of waves to a vessel’s GZ curve

and righting arm

Probability of recovering from capsizing